A-Z Listing

abs

The operation abs produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean corresponding integer
integer absolute value
real absolute value
character fault ?A
phrase fault ?A
fault argument A

If the argument is the largest negative integer for the computer, the result is the corresponding positive real number because the corresponding positive integer is too large for representation as an integer.

     abs  l  -2  3.5  `a  "abc  ??error
1 2 3.5 ?A ?A ?error

The operation abs can be used to convert a boolean value to an integer. Its primary use is in the numeric domains, where it is used to give a positive number measuring the size of a number.

Equations

   abs abs A = abs A
   abs opposite A = abs A

accumulate

The transformer accumulate is similar to the concept of scan in APL. It computes the partial reductions of the initial lists of the items of A, using reduce f (a right-to-left reduction) to compute each partial result. (A reductive operation is one which when applied to an array having a number of items returns a single result.)

In the example below, the initial lists of Z are:

     [13, 13 39, 13 39 92, 13 39 92 45];

and the right-to-left reduction of f on Z is equivalent to:

     [f 13, 13 f 39, 13 f (39 f 92), 13 f (39 f ( 92 f 45))]

     Z := 13 39 92 45 ;
     ACCUMULATE pass Z
+--+-----+----------+---------------+
|13|13 39|+--+-----+|+--+----------+|
|  |     ||13|39 92|||13|+--+-----+||
|  |     |+--+-----+||  ||39|92 45|||
|  |     |          ||  |+--+-----+||
|  |     |          |+--+----------+|
+--+-----+----------+---------------+

The first example shows that accumulate builds the result values with a right-to-left reduction.

     ACCUMULATE max Z
13 39 92 92

     ACCUMULATE min Z
13 13 13 13

     ACCUMULATE sum (count 10)
1 3 6 10 15 21 28 36 45 55

     ACCUMULATE minus (count 10)
1 -1 2 -2 3 -3 4 -4 5 -5

Useful applications of accumulate include the accumulate sum transform used to compute a running sum, and the accumulate minus transform which gives an alternating sum.

     ACCUMULATE or oollooll
oollllll

     S := '  black and white'
  black and white

     B := not (S match ` )
oolllllolllolllll

     ACCUMULATE or B sublist S
black and white

The accumulate or transform produces a bitstring that can be used with sublist to drop leading items that fail to meet a predicate.

Definition

     ACCUMULATE IS TRANSFORMER f OPERATION A {­
        heads IS OPERATION A {­
           count tally A EACHLEFT take list A};
        EACH REDUCE f heads A }

Equations

   shape ACCUMULATE f A = shape A
   ACCUMULATE f solitary A = solitary A
   ACCUMULATE f single A = single A
   ACCUMULATE f Null = Null
   ACCUMULATE f[A,B,C]=[A,A f B,A f(B f C)]

across

ACROSS is a general recursion transformer for traversing the length of an array. It has three operation arguments: endf is applied to the end argument before starting to build the result, parta computes the left value from the argument, which is stacked, and joinf combines the left and right values as the recursion unwinds.

     ACROSS [ 0 first, pass, plus ] 3 4 5 6
18
     reshape [ shape, ACROSS  [pass, pass, CONVERSE append ] ] 2 15 12 4
4 12 15 2

Definition

ACROSS IS TRANSFORMER endf parta joinf OPERATION A {­
        A := list A;
           Res := endf void last A;
           FOR I WITH reverse tell tally A DO
                     Res := parta A@I joinf Res
           ENDFOR }

Equations

   ACROSS [endf, parta, joinf] A = RECUR [empty, endf, parta first, joinf, rest ] list A
   sum A = ACROSS [ 0 first, pass, plus ] A
   reverse A = reshape [ shape, ACROSS  [pass, pass, CONVERSE append ] ] A
   EACH f A = reshape [ shape, ACROSS [ Null first, f, hitch ] ] A

action

An action is the construct that is entered in the interactive loop of the Q’Nial interpreter or accepted as an input unit within the operation loaddefs:

     action ::= definition-sequence
        | expression-sequence
        | external-declaration
        | remark

     definition-sequence ::= definition
        {­ ; definition } [ ; ]

     remark ::= # < any text >

If an action is a definition-sequence, its definitions are installed in the global environment.

If an action is an expression-sequence, it is executed and a value is returned. The value returned by an expression-sequence is displayed on the screen unless it is the fault ?noexpr.

An external-declaration assigns a role to a name in the global environment so that the name can be used in other definitions before it is completely specified.

A remark is an input to the Q’Nial interpreter that is not processed. It begins with a line that has the symbol # as the first non-blank character in the line. In direct input at the top level loop, a remark ends at the end of the line unless a backslash symbol ( \ ) is used to extend the line. In a definition file, a remark ends at the first blank line. A remark cannot appear within a definition or expression-sequence.

address

An address is an integer or a list of integers describing the location of an item in an array. An address uses 0-origin counting, i.e. the first position in a list is at address 0. All the addresses of an array can be stored in an array of the same rectangular structure as the array itself. Such an array is called the grid of an array. Consider the following example:

             GRID OF THE ARRAY                 ARRAY
+-----+-----+-----+-----+-----+-----+ +---+---+---+---+---+---+
|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+| |499|434|122|770|733|890|
||0|0|||0|1|||0|2|||0|3|||0|4|||0|5|| +---+---+---+---+---+---+
|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+| |660|160| 32|808|240|584|
+-----+-----+-----+-----+-----+-----+ +---+---+---+---+---+---+
|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+| |473|235|164|496|808|966|
||1|0|||1|1|||1|2|||1|3|||1|4|||1|5|| +---+---+---+---+---+---+
|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+| |156|205| 34|576|236|454|
+-----+-----+-----+-----+-----+-----+ +---+---+---+---+---+---+
|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|
||2|0|||2|1|||2|2|||2|3|||2|4|||2|5||
|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|
+-----+-----+-----+-----+-----+-----+
|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|
||3|0|||3|1|||3|2|||3|3|||3|4|||3|5||
|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|+-+-+|
+-----+-----+-----+-----+-----+-----+

The array on the right is a 4 by 6 table of numbers. It has 24 items arranged along two directions of length 4 and 6 respectively. The address of 496 is [2,3].

The addresses of a list are integers: 0, 1, etc. For convenience, in situations where an address of a list is expected, a solitary integer is treated as an integer.

The address of a single is the empty list Null.

allbools

The operation allbools tests whether or not A is a nonempty array of boolean items. It returns true if A is not empty and all the items of A are boolean atoms, false otherwise.

     allbools lool
l
     allbools 1 0 1
o
     allbools Null
o

Definition

allbools IS OPERATION A {­ and EACH isboolean A and not empty A }

allchars

The operation allchars tests whether or not A is a nonempty array of character items. It returns true if A is not empty and all the items of A are character atoms, false otherwise.

     allchars 'apple pie'
l
     allchars "today
o
     allchars Null
o

Definition

allchars IS OPERATION A {­ and EACH ischar A and not empty A }

allin

The operation allin tests whether or not all the items of array A are also items of array B. The result is true if the test holds and false if it does not.

     3 5 7 allin count 10
l
     'ae' allin 'where are you?'
l
     'where' allin 'hear'
o

The items of A and B are compared for exact equality, including type. The items being compared may be atoms or they may be any other array.

Definition

     allin IS OPERATION A B {­ and ( A EACHLEFT in B ) }

Equation

   A allin B = (list A) allin (list B)

allints

The operation allints tests whether or not A is a nonempty array of integer items. It returns true if A is not empty and all the items of A are integer atoms, false otherwise.

     allints lool
o
     allints 1 0 25
l
     allints Null
o

Definition

allints IS OPERATION A {­ and EACH isinteger A and not empty A }

allnumeric

The operation allnumeric tests whether or not A is a nonempty array with all atoms of the same numeric type. It returns true if A is not empty and all items are boolean atoms, or all are integer atoms or all are real atoms, false otherwise.

     allnumeric 3.5 -2.97
l
     allnumeric l 45 3.78
o
     allnumeric Null
o

Definition

allnumeric IS OPERATION A {­ allbools A or allints A or allreals A }

allreals

The operation allreals tests whether or not A is a nonempty array of real items. It returns true if A is not empty and all the items of A are real atoms, false otherwise.

     allreals 2 2.5 47.9
o
     allreals 2.5 -7.8 27.3
l
     allreals Null
o

Definition

allreals IS OPERATION A {­ and EACH isreal A and not empty A }

and

  • Class:
    logic operation
  • Properties:
    multi pervasive, reductive
  • Usage:
    and A A and B
  • See Also:
    or, not

The operation and applied to a boolean array A does the boolean product of its items. If all items of A are true, the result is true; otherwise it is false. In binary form and implements the and-connective of logic.

If A is a simple array and has a non-boolean item, the result is the logical fault ?L. The operation extends to non-simple arrays by the multi pervasive mechanism.

     l and l
l

     and llllloll
o

     lloo and lolo
looo

And is a reductive operation in that it reduces an array of booleans to a single boolean. If its argument is a pair of bitstrings or a list of bitstrings, and is applied to bits in corresponding positions of each item of the argument producing a bitstring of the same length as one of the items of its argument.

Equations

   A and B = B and A
   not and A =f= or EACH not A
   and Null = True

append

The operation append attaches B to the end of the list of items of A. It returns a list of length one greater than the tally of A.

     (2 3 4) append (5 6 7)
+-+-+-+-----+
|2|3|4|5 6 7|
+-+-+-+-----+

     append '' 'Wow'
+---+
|Wow|
+---+

The first example shows list (5 6 7) appended to list (2 3 4). The second example shows that if an array is appended to an empty list, the result is the solitary of the second argument. If append is applied to an array that is not a pair, a fault is returned.

Definition

     append IS OPERATION A B {­ A link solitary B }

Equations

   A append B = ( list A ) append B
   Null append A = solitary A
   A append B = list (A append B)

appendfile

The operation appendfile writes a list of character arrays A to the end of the file named Filename, using one line of the file for each row in the items of A. If the file does not exist, it is created by the operation. The file must not be open.

Appendfile is similar to operation putfile except that putfile overwrites an existing file. Getfile reads a file containing text records.

     putfile "Mailing ['1','2','3'];
     appendfile "Mailing ['Mr. Jones','123 Main St','New York NY 12345'];
     getfile "Mailing
+-+-+-+---------+-----------+-----------------+
|1|2|3|Mr. Jones|123 Main St|New York NY 12345|
+-+-+-+---------+-----------+-----------------+

Definition

     appendfile IS OPERATION Filename A {­
        Fnum := open Filename "a ;
        ITERATE (Fnum writefile) (link EACH rows A) ;
        close Fnum ; }

apply

The operation apply carries out the application of the operation represented by the array Op to array A. Op may be a phrase or string that names a predefined or user-defined operation or it may be the cast of an operation expression.

If the operation is represented by a phrase or string giving its name, the name is sought in the environment where apply is being used.

     apply "second (3 4 5)
4

The operation to be applied may also be represented by the parse tree or cast for the operation expression. If the cast is a name, it is sought in the environment where the cast appears. In the following example, !first is the cast of the operation first.

     (!first apply 3 4 5)
3

     apply !(first rest) (3 4 5)
4

As seen in the second example, an operation expression does not have to be named in order to be used with apply. The entire first example is enclosed in parentheses to avoid conflict with the notation for executing a host command.

arccos

The operation arccos is the inverse of cos in the numeric domain. It produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean arccosine of the corresponding real
integer arccosine of the corresponding real
real the angle B in radians such that cos B = A, if A is in the range -1 to 1; otherwise the fault ?arccos
character fault ?arccos
phrase fault ?arccos
fault argument A
     arccos l -2 0.35
0. ?arccos 1.21323

     arccos `a "abc ??error
?arccos ?arccos ?error

Definition

     arccos IS OPERATION A ( Pi / 2.0 - arcsin A )

Equation

   cos arccos cos A = cos A (within roundoff error)

arcsin

The operation arcsin is the inverse of sin in the numeric domain. It produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean arcsine of the corresponding real
integer arcsine of the corresponding real
real the angle B in radians such that sin B = A, if A is in the range -1 to 1; otherwise, the fault ?arcsin
character fault ?arcsin
phrase fault ?arcsin
fault argument A
     arcsin l -2 0.35
1.5708 ?arcsin 0.357571

     arcsin `a "abc ??error
?arcsin ?arcsin ?error

     arcsin l 1 .5
1.5708 1.5708 0.523599

Equation

   sin arcsin sin A = sin A   (within roundoff error)

arctan

The operation arctan is the inverse of tan in the numeric domain. It produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean arctangent of the corresponding real
integer arctangent of the corresponding real
real angle B in radians where tan B = A
character fault ?arctan
phrase fault ?arctan
fault argument A
     arctan l -2 0.35
0.785398 -1.10715 0.336675

     arctan `a "abc ??error
?arctan ?arctan ?error

     arctan 2. l 1 .5
1.10715 0.785398 0.785398 0.463648

Equation

   tan arctan tan A = tan A   (within roundoff error)

argument

An argument is an array value supplied to an operation. All operations in Nial take a single array as the argument, but binary operations expect that argument to be a pair. When an operation is used in infix syntax, then the two array values on the left and right are combined into a pair to be supplied as the argument to the operation.

The argument to an operation-form is treated specially. If the operation-form has only one formal parameter then the argument is assigned to the parameter. If it has two or more formal parameters, then the argument is checked to see if it has the same number of items as there are parameters. If so, the items are assigned to the parameters. If not, the result is the fault ?op_parameter.

array

The data objects of Nial are nested rectangular arrays. Atomic data objects such as numbers and characters are included within this description by virtue of an atom being considered as a self-containing array object with no dimensions.

Atomic Arrays

There are six types of atoms in Nial. They are boolean, integer, real, character, phrase and fault. The first three are numeric types and are used for arithmetic operations. The last three are literal types and are used for text and symbol manipulation. All six types of atoms are used in comparisons.

Rectangularity Structure

An array is a collection of data objects having its items held at locations in a rectangular structure. The items are viewed as being at locations that are positioned relative to a set of directions at right angles to each other. The items may be arranged along zero, one, two or more directions. For example, the following array is a 4 by 6 table of numbers. It has 24 items arranged along two directions of length 4 and 6 respectively.

                   +---+---+---+---+---+---+
                   |499|434|122|770|733|890|
                   +---+---+---+---+---+---+
                   |660| 32|808| 24|  5|473|
                   +---+---+---+---+---+---+
                   |499|434|122|770|733|890|
                   +---+---+---+---+---+---+
                   |660| 32|808| 24|  5|473|
                   +---+---+---+---+---+---+

The items of an array are themselves arrays. Thus, an array can have an arbitrarily deep nesting structure.

assign

The operation assign assigns the value of A to the variable named by Nm. Nm may be a phrase or string (e.g. "X or 'X'), in which case the named variable is sought in the current environment; or it may be the cast of a name (e.g. !X), in which case it is sought in the environment in which it was cast. If the variable is found, its value is replaced by the array A. Otherwise, a variable with that name is created in the global environment and given A as its value.

Any phrase may be used as a variable name with the operation assign. Thus, it is possible to assign a value to names which are invalid identifiers. The associated value can be retrieved using the operation value. The result of the operation is A.

     assign "X (2 3 4)
2 3 4

     (!Var assign 2 3 4)
2 3 4

In the first example, the variable X is assigned the list 2 3 4. If this is done in an environment where X is a local variable, the local variable is updated. If X does not exist, it is created in the global environment.

In the second example, a variable Var must exist in the environment where the cast is done. The cast can be formed in a more global environment and passed into an operation as a parameter (This technique is called by-variable parameter passing).

assign expression

     assign-expression ::=
        {­ variable }+ := expression
        | indexed-variable := expression

An assign-expression assigns an array value to one or more variables at the time of evaluation of the assign expression. The semantics of an assign expression is interpreted in two stages: when the expression is analyzed (parsed) and when it is executed.

During the parse of the assign-expression appearing in a block, each name on the variable list is sought in the local environment. If the name exists in the local environment, the assignment affects the local association. If a name does not exist in the local environment and no reference has been made to a nonlocal variable with the same name, a local variable is created in the block. An assign-expression parsed in the global environment creates a global variable if a variable with that name does not already exist.

When an assign expression is executed, the expression on the right of the assignment symbol ( := ) is evaluated. If the variable list on the left has only one name, the value of the expression is assigned to that variable. That is, the value is associated with that name.

If the variable list has several names, the items of the value are assigned to the variables in the order in which they appear. If the number of items does not match the number of variables, the fault ?assignment is returned as the value of the assign-expression. Otherwise, the value of the assign-expression is the value of the expression on the right.

When an indexed-variable is used on the left in an assign-expression, the parts of the array associated with the variable at the locations specified by the index are replaced by the values of the expression on the right.

If the index expression for an indexed-variable assignment specifies a number of locations (at-all or slice indexing), there are two cases: if the value on the right is a single, the item of the single is placed in each location; otherwise, the value on the right must have the same number of items as the index expression indicates and the corresponding locations are updated with the items of the array value.

atlas

     atlas ::= [ operation-expression {­ , operation-expression } ]

An atlas is an operation made up of a list of component operations. The result of applying an atlas is a list of the same length as the atlas. Each operation in the atlas is applied in turn to the argument resulting in an array value that becomes the item of the result list in the corresponding position. An atlas is used by the transformers FORK, INNER and TEAM.

atomic

The operation atomic tests whether or not its argument is an atom. It returns true if it is and false if it is not.

     atomic 3.5
l
     atomic "hello
l
     atomic 'hello'
o

The examples illustrate that a number and a phrase are atoms and that a string is not an atom.

An atom is a primitive concept in array theory and Nial. Atoms are distinguished from other arrays by the property that they are self nesting. The definition of atomic is based on this property.

Definition

     atomic IS OPERATION A {­ first A = A }

Equations

   atomic A <==> single A equal A
   and EACH atomic A = simple A

axes

The operation axes generates a list of axis numbers for the array A counting from zero.

     axes tell 3 4 5
0 1 2

Definition

     axes IS OPERATION A {­ tell valence A }

Equation

   tally axes A = tally shape A

binary

An operation is said to binary if it must have exactly two items in its argument. Many of the built-in operations of Nial are binary. They can be used in both an infix and prefix manner. If a binary operation f is used in infix syntax then the arguments on each side of f are treated as the two items of its argument.

     3 reshape 5

In prefix usage, f can precede a single array with two items, or an explicit pair formed with strand notation or bracket-comma notation.

     X := 3 5;
     reshape X
5 5 5

     reshape 3 5
5 5 5

     reshape [3,5]
5 5 5

binary pervasive

Each operation f in this class maps a pair of atoms to an atom.

A binary pervasive operation maps two arrays having identical structure to one with the same structure, mapping each pair of corresponding atoms by the function’s behaviour on pairs of atoms.

All of the binary operations of arithmetic and logic are binary pervasive.

If a binary pervasive operation is applied to a pair of arrays that do not have the same shape, the effect is to build a conformable pair by replicating an atom or solitary item of the pair to the shape of the other item. If both items are of unequal shape and if both items are made up of more than one item, the fault ?conform is returned. The replication of an argument with one item provides binary pervasive operations with a scalar extension capability. For example,

     3 4 5 6 - 5 = (3 4 5 6 - 5 5 5 5)
l

If a binary pervasive operation is applied to an array that is not a pair, a fault is returned.

The following table lists the binary pervasive operations.

Operation Function
divide division of numbers
gt greater than comparison
gte greater than or equal comparison
lt less than comparison
lte less than or equal comparison
match equality of atoms without type coercion
mate equality of atoms with type coercion
minus subtraction of numbers
mod remainder on division of integers
plus addition of numbers
quotient quotient on division of integers
times multiplication of numbers divide division of numbers

Equations

   A f B = A EACHBOTH f B
   A f B = EACH f (A pack B)
   shape (A f B) = shape pack A B

blend

The operation blend combines the top two levels of an array B into a single level blending the axes of the second level of B into the combined level according to the axis numbers given in A. The items of B must be of the same shape.

If the array B is equivalent to A split C for some array C, the result of A blend B is C.

The tally of A is the valence of an item of B. The items of A indicate where the axes of the items of B are placed in the shape of the result.

     C := (2 4 3 reshape count 24) ;
     B := 2 0 split C
+----+----+----+-----+
|1 13|4 16|7 19|10 22|
|2 14|5 17|8 20|11 23|
|3 15|6 18|9 21|12 24|
+----+----+----+-----+

     C := 2 0 blend B
 1  2  3   13 14 15
 4  5  6   16 17 18
 7  8  9   19 20 21
10 11 12   22 23 24

In the example, B is a list of four tables of shape 3 2 created by a 2 0 split. The result of 2 0 blend B is the array of valence 3 that is the second argument to split.

Equations

   SORT <= list I = axes A and not empty A ==> I blend (I split A) = A
   equal EACH shape A and not empty A ==> (valence A+axes first A) blend A = mix A
   A blend B =f= [Null,A] PARTITION first B

block

     block ::=
        {­ [ LOCAL {­ identifier-sequence }+ ; ]
     [ NONLOCAL {­ identifier-sequence }+ ; ]
     [ definition-sequence ; ]
       expression-sequence }

A block is a scope-creating mechanism that permits an expression-sequence to be created so that it has local definitions and variables which are visible only inside the block. A block may appear as a primary-expression or as the body of an operation-form.

A local environment is a collection of associations that are known within a limited section of program text. These limited sections are formed by blocks, operation-forms and transformer-forms. A name that has a local association in one of these forms is said to have local scope.

If the definition appears within a block, the association is made in the local environment. Otherwise, the association is made in the global environment and assigns a role to the name as representing that kind of expression.

If a block is used as a primary-expression, the local environment created by a block is determined by the block itself. If it is the body of an operation-form, the local environment includes the formal parameter names of the operation-form as variables.

Local and Nonlocal Declaration

The identifiers included in the local and nonlocal declarations are declared to be variables. Both forms of declarations are optional, but if both are given, local declarations must be made first. If the block is the body of a globally defined operation-form or expression, a nonlocal declaration effectively declares its variables as global ones.

A block delimits a local environment. It allows new uses of names which do not interfere with uses of those names outside the block. For example, within a block, a predefined operation name can be redefined and used for a different purpose. Only the reserved words of Q’Nial cannot be reused in this fashion.

Definitions that appear within the block have local scope. That is, the definitions can be referenced only in the body of the block. Variables assigned within the block may or may not have local scope, depending on the appearance of a local and/or a nonlocal declaration. If there is no declaration, all assigned variables have local scope. Declaring some variables as local does not change the effect on undeclared variables that are used on the left of assignment. They are automatically localized.

If a nonlocal declaration is used, an assigned name that is on the nonlocal list is sought in surrounding scopes. If the name is not found, a variable is created in the global environment.

bracket-comma notation

A list may be constructed directly by using bracket-comma notation. In this notation, the items of a list are separated by commas and the list is bounded by square brackets. If an item is omitted before or after a comma, then the fault value ?noexpr is used for the value of the missing item. The notation denotes the Null if their are no items, and a solitary if there is only one item.

     [2,3 4,5]
+-+---+-+
|2|3 4|5|
+-+---+-+

     [,4 5]
+-------+---+
|?noexpr|4 5|
+-------+---+

     [] = Null
l

     ['hello world']
+-----------+
|hello world|
+-----------+

     [3,[4,5,6],7]
+-+-----+-+
|3|4 5 6|7|
+-+-----+-+

break

The execution of Break causes the interpreter to interrupt normal execution in an expression sequence and to display the current callstack. It then prompts for input with the prompt --> followed by the default command in brackets. The visible environment is that of the expression in which the break occurs. Thus, it is possible to examine the values of local variables in break mode.

At a break you can type any expression to inspect the value of a variable or to see a portion of its value. The operation see can also be used to view any of the definitions in the environment.

The debugging capability allows one to step forward in expression sequences using commands step, stepin, next or toend to control whether you step into or over other definitions or to the end of a loop or a definition. The command resume ends the break and normal execution is restarted.

     foo is op A {­ Break; A }
     foo 3
-------------------------------------------------------------
    Break debug loop: enter debug commands, expressions or
      type: resume    to exit debug loop
      <Return> executes the indicated debug command
    current call stack :
foo
-------------------------------------------------------------
?.. A
-->[stepv]

In the example, the operation foo has a Break which is executed when foo is applied to 3. If Return is pressed at the prompt the expression A is evaluated and 3 is displayed followed by another prompt.

breakin

The operation breakin installs a break point prior to the first executable expression in a definition named by the string or phrase Defname. The named definition must be an expression or an operation form. The effect is that when the definition is executed a break interrupt occurs prior to the execution of the first expression in the expression sequence in the body of the definition.

The optional argument Mode is provided to set the internal flag explicitly. If it is omitted, the internal flag value is toggled. All breakin flags are initially false. The value of the breakin flag is retained if the definition is replaced by a loaddefs so that editing a definition and reloading it does not change its breakin status. However, if definitions are reloaded using a restart then the breakin status is set to false.

The operation returns the previous setting. If the result of breakin is assigned to a variable, the previous setting can be restored later. The names of definitions that have breakin set can be viewed by the expression Breaklist.

     library"labeltab

     see "labeltable

labeltable IS OPERATION Corner Rowlabel
   Columnlabel Table {­
   % Combine the corner label with the column
      labels for first line;
   Firstrow := Corner hitch Columnlabel;
   % Hitch the row labels to the rows of
      the table;
   Labeledrows := Rowlabel EACHBOTH hitch
      rows Table;
   % Hitch the first row of labels to the labeled
      rows and mix them;
   mix (Firstrow hitch Labeledrows) }

     X gets count 3
1 2 3

# execute labeltable without break;

     labeltable "TIMES X X (X OUTER times X)
TIMES 1 2 3
    1 1 2 3
    2 2 4 6
    3 3 6 9

# set breakin and execute again

     breakin "labeltable
o

     labeltable "TIMES X X (X OUTER times X)

-------------------------------------------------------------
    Break debug loop: enter debug commands, expressions or
      type: resume    to exit debug loop
      <Return> executes the indicated debug command
    current call stack :
labeltable
-------------------------------------------------------------
?.. Firstrow := Corner hitch Columnlabel
-->[stepv] TIMES 1 2 3
?..   % Hitch the row labels to the rows of the table
-->[stepv] resume
TIMES 1 2 3
    1 1 2 3
    2 2 4 6
    3 3 6 9

To clear all breaks use:

     EACH breakin Breaklist

breaklist

The execution of Breaklist prints out a list of the definition names for which the breakin flag has been set.

Its main use is to assist in clearing the breaks as debugging proceeds.

     Breaklist
labeltable

To clear all breaks use:

     EACH breakin Breaklist

bycols

The transformer BYCOLS applies an operation f to the columns of a table A, where f is an operation that maps lists to lists.

     setformat '%5.3f';
     BYCOLS (SORT <=) (5 6 reshape random 30)
0.726 0.009 0.210 0.385 0.123 0.119
0.851 0.448 0.384 0.476 0.193 0.384
0.880 0.449 0.536 0.603 0.455 0.498
0.882 0.729 0.631 0.819 0.639 0.521
0.911 0.893 0.810 0.961 0.940 0.672

The example generates a random table and then sorts each column.

Definition

     BYCOLS IS TRANSFORMER f OPERATION A {­ transpose (1 RANK f transpose A) }

Equation

   shape f A = shape A ==> shape BYCOLS f A = shape A

bye

The expression Bye is used to terminate a session of Q’Nial and to return to the host environment. The current workspace is not saved.

bykey

The transformer BYKEY applies an operation f to lists gathered from A, where each list is made up of items of A that have the same value in the corresponding position in K. The result is a list with as many items as there are unique items in K.

     Str := 'a stitch in time saves nine';

     cull Str
a stichnmev

     Str BYKEY tally Str
2 5 3 2 4 1 1 3 1 3 1

     Keys := 3 5 3 7 2 5 4 2 1 3 4;
     Data := 23.1 14.2 13.5 18.9 22. 98. 3.5 28.7 19.3 16.5 43.2;

     Keys BYKEY sum Data
53.1 112.2 18.9 50.7 46.7 19.3

     average IS OP A {­ sum A / tally A }

     Keys BYKEY average Data
17.7 56.1 18.9 25.35 23.35 19.3

In the first example BYKEY tally is used to count the frequency of the letters in the string Str, giving the counts in the order of cull Str. In the remaining examples, the values in Data corresponding to equal items in Keys are added and averaged in the two uses of BYKEY.

Definition

     BYKEY IS TRANSFORMER f OPERATION K A {­
        Keys gets cull A;
        IF simple Keys THEN
           Patterns := Keys EACHLEFT match A;
        ELSE
           Patterns := Keys EACHLEFT EACHRIGHT = A;
        ENDIF;
        EACH f (Patterns EACHLEFT sublist B) }

Equations

   tally BYKEY f K A = tally cull K
   valence BYKEY f K A = 1

byrows

The transformer BYROWS applies an operation f to the rows of a table A, where f is an operation that maps lists to lists.

     BYROWS reverse (5 6 reshape count 30)
 6  5  4  3  2  1
12 11 10  9  8  7
18 17 16 15 14 13
24 23 22 21 20 19
30 29 28 27 26 25

The example reverses the rows of the generated table.

Definition

     BYROWS IS TRANSFORMER f OPERATION A {­ 1 RANK f A }

Equation

   shape f A = shape A ==> shape BYROWS f A = shape A

by-variable

One use of the operation assign is to mimic a by-variable form of parameter passing in place of Nial’s by-value form. The result depends on what kind of name is provided, a phrase or a cast. If the name is provided as a phrase, the variable that is selected is determined by assign when it does the assignment by looking first in the local environment and then in the surrounding ones. If the name is provided as a cast, the variable selected is the one that exists at the point where the cast is formed. Thus, by-variable parameter passing is achieved by using the cast of the variable as an argument in the call. In the body of the operation the formal parameter is assigned using assign and evaluated using value.

     foo is op A Nm {­
       B := A + value Nm;
       Nm assign (B + sum count 5); }

     X := 100;
     foo 1000 "X;
     X
1115

callstack

The expression Callstack displays the sequence of active definition calls at the point it is invoked. It is usually used in conjunction with Break. Callstack can be used to see the execution path by which the computation reached the current state while computation is suspended during a break.

canonical

There is a canonical way of displaying program text in Nial. This is done automatically by the routines descan and deparse used by see. The canonical form sets the case of all identifiers used in Nial program text according to their role in order to assist visual parsing of Nial text. The following table summarizes the rules:

Role Case rule
Variable first letter upper case, the rest lower case
Expression first letter upper case, the rest lower case
Operation all lower case
Transformer all upper case
Reserved Word all upper case

Definition

     canonical IS link descan deparse parse scan

cart

The operation cart corresponds to the cartesian product of set theory. Its purpose is to form all possible combinations of the items of its argument and return them in a structured result having as many axes as the sum of the number of axes of the items of argument.

     2 3 4 cart 5 6
+---+---+
|2 5|2 6|
+---+---+
|3 5|3 6|
+---+---+
|4 5|4 6|
+---+---+

In the example above, the cart of a triple with a pair yields a table of shape 3 2 of pairs.

     cart ['abc', 4 2 reshape count 8]
+---+---+  +---+---+  +---+---+
|a 1|a 2|  |b 1|b 2|  |c 1|c 2|
+---+---+  +---+---+  +---+---+
|a 3|a 4|  |b 3|b 4|  |c 3|c 4|
+---+---+  +---+---+  +---+---+
|a 5|a 6|  |b 5|b 6|  |c 5|c 6|
+---+---+  +---+---+  +---+---+
|a 7|a 8|  |b 7|b 8|  |c 7|c 8|
+---+---+  +---+---+  +---+---+

In the above example, the shape of the result is 3 4 2, getting an axis of length 3 from the first item and axes of length 4 and 2 from the second.

Equations

   shape cart A = link EACH shape A
   valence cart A = sum EACH valence A
   tally cart A = product EACH tally A
   equal EACH shape cart A = True
   A OUTER pair B = A cart B
   cart link A = EACH link cart EACH cart A
   cart EACH EACH f A = EACH EACH f cart A
   cart single A = EACH single A
   simple A ==> cart A = single A
   cart Null = single Null
   empty A ==> cart A = single A
   or EACH empty A ==> empty cart A
   not empty A and not empty cart A ==> shape first cart A = shape A
   not empty A and not empty cart A ==> first cart A = EACH first A
   isshape A ==> tell A = cart EACH tell A

case-expr

The expression following case is evaluated. If the result matches one of the constants, C1 … Cn, the corresponding expression sequence is executed. If the result does not match any constant, the expression sequence following else is executed. Alternates may be matched on with the label A1 | A2, that is, placing the vertical bar between both constants you wish to match on.

Example:

     Month := 4 ;
     CASE Month FROM
        1:     Month_name := "January;       END
        2:     Month_name := "February;      END
        3 | 4: Month_name := "March/April;   END
     ELSE  Month_name := phrase 'May through December';
     ENDCASE ;
     Mname
March/April

cast

A cast is an array expression that denotes an internal representation of a valid fragment of Q’Nial program text:

     cast ::= ! identifier
        | ! ( expression-sequence)
        | ! ( operation-expression )
        | ! ( transformer-expression )

The use of the exclamation symbol ! before an identifier causes Q’Nial to select the internal representation for the identifier rather than the value of the array associated with the identifier. Its use before a parenthesized program fragment selects the internal representation of the program fragment.

The major use of casts is in conjunction with the operations assign and apply. These operations mimic the Q’Nial constructs for assignment to a variable and application of an operation to an array. Casts permit passing an argument to an operation by variable name rather than by value. They also permit evaluation of a program fragment that has been stored in its internal form using the operation eval rather than requiring the use of the operation execute on the corresponding program text stored as a string.

     Salary := 90000.
     A gets 'Salary > 100000.' ;
     Rule1 := execute A
o
     Rule1 := eval !A
o

The details of the internal representation is not specified as part of the Nial language.

The cast notation !Name is used to denote the parse tree that represents the name. At the top level loop, parentheses must be included around the use of the cast notation, e.g. (!Name), to avoid ambiguity with the use of ! to indicate a host command.

The cast, because it is analyzed in the context in which it appears, refers to a variable or definition in a static way.

Q’Nial contains operations that mimic the underlying meaning of variables, expressions and operations in Q’Nial. The operations use strings, phrases or casts to represent the name of the object under consideration (except that see and getdef do not take casts).

Operation Action
value A Return the value of a variable named by string, phrase or cast A .
A assign B Assign the array B to the variable named by string, phrase or cast A ; return B .
A apply B Apply the operation named by string, phrase or cast A to array B ; return the result of the operation.
getdef A Return the parse tree associated with the definition named by string or phrase A .
see A Display the definition named by the string or phrase A .
update P A B Put array B at address A in the variable named by the phrase, string or cast P ; return the new value of P .
updateall P A B Put items of B at addresses A in the variable named by the phrase, string or cast P ; return the new value of the variable named by P .
deepupdate P A B Put array B at path A in variable named by the string, phrase or cast P ; return new value of the variable named by P .

catch

The transformer CATCH is used to regain control when the called argument function f applied to the argument A terminates its computation early by the application of the control flow function throw, either directly by f, or by a function that is called during the execution of f.

If f executes without applying a throw the result of CATCH f A is the value computed at the completion of f. Otherwise it is the value of the argument to throw in the function that executes the throw.

The CATCH-throw mechanism was added in Version 7 of Q’Nial to provide a dynamic mechanism for changing the flow of control when an exceptional situation is encountered.

catenate

The operation catenate joins the items of A along axis I. The items of A must conform in all other axes.

     1 catenate (tell 2 3) (count 2 5)
+---+---+---+---+---+---+---+---+
|0 0|0 1|0 2|1 1|1 2|1 3|1 4|1 5|
+---+---+---+---+---+---+---+---+
|1 0|1 1|1 2|2 1|2 2|2 3|2 4|2 5|
+---+---+---+---+---+---+---+---+

The example joins two tables along the rows. Each table has two rows.

Definition

     catenate IS OPERATION I A {­
        % "push down" I axis of items of A;
        B := EACH ( I split ) A ;
        IF equal EACH shape B THEN
           I blend EACH link pack B
        ELSE
           fault '?conform error in catenate'
        ENDIF }

ceiling

The operation ceiling produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean corresponding integer
integer argument
real next higher integer or the fault ?A, if the result is outside the range of integers
character fault ?A
phrase fault ?A
fault argument A

Examples

     ceiling  l  -2  3.5
1 -2 4
     ceiling  `a  "abc  ??error
 ?A ?A ?error
     ceiling  3.5  -4.6  7.0  25.3e20
4 -4 7 ?A

Definition

     ceiling IS OPERATION A {­
        opposite floor opposite A }

char

The operation char is used to convert an integer in the range 0 to 255 to the character that has the integer as its internal representation. The result is system dependent.

     char 66
B
     char 66 67 68 69
BCDE

The major purpose of char is to create special characters for a specific system. For example, the characters that control cursor motion differ from one terminal to another. Programs using char may not be portable.

Equations

   N in tell 256 ==> charrep char N = N
   ischar C ==> char charrep C = C

charrep

The operation charrep is used to convert a character to its internal representation as an integer. This operation is system dependent.

     charrep `A
65
     charrep 'hello'
104 101 108 108 111

The major purpose of charrep is to permit determination of special characters for a specific system. For example, the characters that control cursor motion differ from one terminal to another.

Programs using charrep may not be portable.

Equations

   N in tell 256 ==> charrep char N = N
   ischar C ==> char charrep C = C

choose

The operation choose is used to select a subarray from array A specified by the array of addresses I. The result is an array of the same shape as I with items chosen from A. If an item of I is not an address of A, the corresponding position in the result is the fault ?address.

The operation choose is related to the at all notation V#I, which selects items from the array associated with the variable V. The differences are that choose may select from an array that has not been assigned to a variable and that it handles out-of-range in a different manner.

     3 1 0 1 3 4 choose 'range'
garage

The following example shows that the valence of the array of addresses can be higher than that of the array from which the selection is made.

     I gets (2 4 reshape tell 8)
0 1 2 3
4 5 6 7
     I choose 'some words as a string'
some
 wor

Definition

     choose IS OPERATION I A {­ I EACHLEFT pick A }

Equations

   shape (I choose A) = shape I
   J choose (I choose A) = (J choose I) choose A
   tell shape A choose A = A
   (list I) choose A = list (I choose A)
   (EACH list I) choose A = I choose A
   I allin grid A ==> EACH f (I choose A) = I choose EACH f A

clear workspace

Q’Nial organizes the data and code objects available for use into a logical structure called a workspace . It consists of a symbol table to hold associations between names and the predefined and user defined objects, a heap to store data and parse trees, a stack used to hold values temporarily during execution, and an atom table for uniquely storing phrases and faults.

When Q’Nial is invoked for interactive use it starts the session with a clear workspace unless a specific saved workspace is requested. The clear workspace is created by an initialization process.

clearprofile

The expression Clearprofile is used to clear the internal data structures that are used in the gathering of profiling statistics. It should be called when one profiling session has been completed and profile has been called, before starting another one.

A detailed explanation of the profiling mechanism is given in the help entry on profiling.

Example

     Clearprofile

close

The operation close is used to close a file previously opened with the open operation. The argument is the file designator, an integer returned previously by open for that file.

The result of close is the fault ?noexpr or a fault message indicating an error.

     Fd := open "foo "w
3
     close Fd
     close Fd
?file is not open

In the examples above, the first use of close was successful. The second attempt to close the file resulted in the fault message.

cols

The operation cols rearranges the axes of a table of shape M by N to form a list of length N of the columns of length M of the table. Cols is generalized to arrays of all valences as follows: if A is a list or a single, the result is single A; if A has valence three or higher, the result is an array of valence one less than the valence of A with the second last axis pushed down.

     A := 3 4 reshape count 12
1  2  3  4
5  6  7  8
9 10 11 12

     cols A
+-----+------+------+------+
|1 5 9|2 6 10|3 7 11|4 8 12|
+-----+------+------+------+

If only column I of a table is needed, it can be selected by I pick cols A or by A|[,I] using the at slice indexing notation. The second way is more efficient for large arrays because it avoids restructuring the array. If a table A has no columns, cols A results in the empty list Null.

Definition

     cols IS OPERATION A {­
        IF valence A = 0 THEN single A
        ELSE
           valence A - 2 max 0 split A
        ENDIF }

Equations

   valence A = 2 and not empty A ==> cols A = pack rows A
   valence A = 2 and not empty A ==> cols A = rows transpose A
   valence A = 2 and not empty A ==> mix cols A = transpose A

comment

     comment ::=
        % <any text excluding a semicolon> ;

     remark ::= # < any text >

A comment is a brief section of text included in a program fragment to assist readability. Comments may be placed anywhere in a block before or after declarations, definitions or expressions. Their purpose is to provide an explanation of the program fragment for the programmer who may be required to modify the program at a later date. The value of a comment as an expression is the ?noexpr fault. Comments are retained when a definition is translated into internal form and they appear in its creation in the canonical form used by the operations see and defedit.

A remark is an input to the Q’Nial interpreter that is not processed. It begins with a line that has the symbol # as the first non-blank character in the line. In direct input at the top level loop, a remark ends at the end of the line unless a backslash symbol ( \ ) is used to extend the line. In a definition file, a remark ends at the first blank line. A remark cannot appear within a definition or expression-sequence.

conform

The operation pack is used in evaluating binary pervasive and multi pervasive operations. Its task is to interchange the top two levels of the argument to such operations if the items of the argument conform . For the binary case there are two items; they conform if the items have the same shape, or if one or both items have only one item. In the latter case, the item with only one item is replicated to the shape of the other item.

For the multi pervasive case, all the items that do not have only one item must be of the same shape and all the items with one item are replicated to that shape.

     pack [2 3 4,10 11 12]
+----+----+----+
|2 10|3 11|4 12|
+----+----+----+

     pack [2 3,10 11 12]
?conform

     pack [3 4 5,3,10 11 12,[4]]
+--------+--------+--------+
|3 3 10 4|4 3 11 4|5 3 12 4|
+--------+--------+--------+

     pack [tell 4 5,count 4 5]
+---------+---------+---------+---------+---------+
|+---+---+|+---+---+|+---+---+|+---+---+|+---+---+|
||0 0|1 1|||0 1|1 2|||0 2|1 3|||0 3|1 4|||0 4|1 5||
|+---+---+|+---+---+|+---+---+|+---+---+|+---+---+|
+---------+---------+---------+---------+---------+
|+---+---+|+---+---+|+---+---+|+---+---+|+---+---+|
||1 0|2 1|||1 1|2 2|||1 2|2 3|||1 3|2 4|||1 4|2 5||
|+---+---+|+---+---+|+---+---+|+---+---+|+---+---+|
+---------+---------+---------+---------+---------+
|+---+---+|+---+---+|+---+---+|+---+---+|+---+---+|
||2 0|3 1|||2 1|3 2|||2 2|3 3|||2 3|3 4|||2 4|3 5||
|+---+---+|+---+---+|+---+---+|+---+---+|+---+---+|
+---------+---------+---------+---------+---------+
|+---+---+|+---+---+|+---+---+|+---+---+|+---+---+|
||3 0|4 1|||3 1|4 2|||3 2|4 3|||3 3|4 4|||3 4|4 5||
|+---+---+|+---+---+|+---+---+|+---+---+|+---+---+|
+---------+---------+---------+---------+---------+

Definition

     conform IS OP A {­
         equal EACH shape
           (not (EACH tally A match 1) sublist A) }

content

The operation content returns the list of atoms of A. The effect of content is to remove all structure from an array, returning a list of the atoms in a depth first, row-major order. The effect on an atom is to produce the solitary of the atom.

     A := (2 3) (4 5 (6 7))
+---+---------+
|2 3|+-+-+---+|
|   ||4|5|6 7||
|   |+-+-+---+|
+---+---------+

     content A
2 3 4 5 6 7

Definition

     content IS OPERATION A {­
        IF atomic A THEN
           list A
        ELSE
           link EACH content A
        ENDIF }

Equations

   content A = list content A
   content A = content list A
   content A = link EACH content A

converse

The transformer CONVERSE is used with a binary operation f and applies f to the pair formed by reversing the arguments of CONVERSE f.

     at IS CONVERSE pick
     'abcde' at 2
c

     holds IS CONVERSE in
     count 20 holds 5
l

In the above examples, at and holds are defined in terms of pick and in, respectively. The first example shows that 'abcde' at 2 gives the result expected for 2 pick 'abcde'. The second example shows that count 20 holds 5 gives the same result as 5 in (count 20). Thus, at and holds do the same work as pick and in; the former operations simply use their arguments in the reverse order.

Definition

     CONVERSE IS TRANSFORMER f OPERATION A B {­
        B f A }

Equations

   A CONVERSE CONVERSE f B = A f B
   A EACHLEFT f B = B EACHRIGHT CONVERSE f A

cos

The operation cos implements the cosine function of mathematics. It produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean cosine of the corresponding real
integer cosine of the corresponding real
real cosine of angle A given in radians
character fault ?cos
phrase fault ?cos
fault argument A
     cos  l  -1  0.5  `a  "abc  ??error
0.54030 0.54030 0.87758 ?cos ?cos ?error

Equation

   cos opposite A = cos A

cosh

The operation cosh implements the hyperbolic cosine function of mathematics. It produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean hyperbolic cosine of the corresponding real
integer hyperbolic cosine of the corresponding real
real hyperbolic cosine of angle A given in radians
character fault ?cosh
phrase fault ?cosh
fault argument A
     cosh  l  -1  0.5
1.54308 1.54308 1.12763

     cosh `a  "abc  ??error
?cosh ?cosh ?error

Equation

   cosh opposite A = cosh A

count

The operation count generates a list of integers starting at 1 and going up to and including N. It differs from tell by counting from 1 instead of from 0.

     count 5
1 2 3 4 5

     .1 times count 5
0.1 0.2 0.3 0.4 0.5

The first example shows count generating the sequence of integers from 1 to 5. The second example shows how count can be used to generate a sequence of five real numbers with an interval of .1 between each number.

     count 2 3
+---+---+---+
|1 1|1 2|1 3|
+---+---+---+
|2 1|2 2|2 3|
+---+---+---+

The third example shows how count generalizes to other arguments in the same manner as the operation tell.

Definition

     count IS OPERATION A {­ 1 + tell A }

Equations

   isshape A ==> shape count A = A
   isshape A ==> count A = cart EACH count A

cull

The operation cull returns a list whose items are those of A with duplicates removed. The order of the items is maintained.

     cull 3 5 4 3 5 2 4
3 5 4 2

     cull 'a few letters with duplicates'
a fewltrsihdupc

Definition

     cull IS OPERATION A {­
        grid A EACHLEFT in (A EACHLEFT find A) sublist A }

Equations

   cull A = list cull A
   diverse A <==> cull A = list A
   sortup cull A = cull sortup A

Pragmatics

The operation cull executes faster when a large array has been sorted, hence if the ordering of the result is unimportant it is better to use sortup to order the array prior to applying cull.

curried operation

A curried-operation is an operation in which the left item of a two-item argument is combined with a given operation to form an operation expression. Examples are: 1+ and 3 reshape.

A curried operation can be named or grouped in parentheses as an argument to a transformer.

     incr IS 1+

     EACH (5 take) Lines

In general, the syntax of a curried operation is:

       curried-operation ::= simple-expression simple-operation

The result of applying a curried-operation is determined by applying the simple-operation to the pair formed from the simple-expression and the argument to the curried-operation. Thus,

     (1+) 5
6

is interpreted as

     + (1 5)
6

cut

The operation cut converts an array A into a list of items formed from the items of A according to the bitstring B. The list of items is divided where true values occur in the corresponding positions in the bitstring B. The items of A where the divisions occur are not included in the items of the result and any empty segments are not included.

     A := 'The boy stood   on the burning deck' ;
     `  match A cut A
+---+---+-----+--+---+-------+----+
|The|boy|stood|on|the|burning|deck|
+---+---+-----+--+---+-------+----+

     B := 'Formula 1: 3,5,7,,,9';

     `, match B cut B
+------------+-+-+-+
|Formula 1: 3|5|7|9|
+------------+-+-+-+

     EACH equal tell 4 4 cut tell 4 4
+-----------------+-----------------+-----------------+
|+---+---+---+---+|+---+---+---+---+|+---+---+---+---+|
||0 1|0 2|0 3|1 0|||1 2|1 3|2 0|2 1|||2 3|3 0|3 1|3 2||
|+---+---+---+---+|+---+---+---+---+|+---+---+---+---+|
+-----------------+-----------------+-----------------+

Cut can be used to cut a string into a list of substrings. In the first example, the string is cut at blank characters. Several adjacent blanks are treated as one blank character. In the second example, the cut is done where commas occur. The third example shows that the items of A do not have to be atoms.

Definition

     cut IS OPERATION B A {­
        C := EACH rest (B cutall A);
        not EACH empty C sublist C }

Equations

   B cut A = (shape A reshape B) cut A
   list (B cut A) = B cut A
   B cut list A = B cut A

cutall

The operation cutall converts an array A into a list of items formed from the items of A according to the bitstring B. The list of items is divided where true values occur in the corresponding positions in the bitstring B. The items where the divisions occur are kept as the first item of each group.

     A := 'The boy stood on the deck' ;
     `  match A cutall A
+---+----+------+---+----+-----+
|The| boy| stood| on| the| deck|
+---+----+------+---+----+-----+

     B := 'Formula 1: 3,5,7,,,9';
     `, match B cutall B
+------------+--+--+-+-+--+
|Formula 1: 3|,5|,7|,|,|,9|
+------------+--+--+-+-+--+

     EACH equal tell 4 4 cutall tell 4 4
+---------------------+---------------------+---------------------+-----+
|+---+---+---+---+---+|+---+---+---+---+---+|+---+---+---+---+---+|+---+|
||0 0|0 1|0 2|0 3|1 0|||1 1|1 2|1 3|2 0|2 1|||2 2|2 3|3 0|3 1|3 2|||3 3||
|+---+---+---+---+---+|+---+---+---+---+---+|+---+---+---+---+---+|+---+|
+---------------------+---------------------+---------------------+-----+

Equations

   tally (B cutall A) = sum (shape A reshape B)
   B cutall A = (shape A reshape B) cutall A
   list (B cutall A) = B cutall A
   B cutall list A = B cutall A

debugging

Debugging Definitions

The Q’Nial system provides an optional debugging facility that aids interactive debugging of definitions. It is active by default, but can be turned off for running production applications. See the detailed documentation for the various versions on how to turn off debugging.

The debugging system is based on the idea of placing breaks in the code and stepping through the program code in a number of different ways. Due to constraints in the way Q’Nial is implemented, debugging is always done in the context of an expression sequence. A break point occurs either before the execution of the expression sequence in a definition, or at an explicit break expression within an expression sequence. There is also a watch mechanism that executes a defined action whenever the value of a variable changes, and an ability monitor all use of user defined objects and of the predefined operations.

Defining a Break Point

There are two ways to cause a break in a Nial definition: by using the expression Break in an expression sequence, or by using the operation breakin to set a break on entry to the operation. The following table summarizes the break related primitives:

Expression Action
Break Suspend evaluation of the expression and pass control to an evaluation loop in the environment at the point of the break. Variables accessible at that point can be displayed. This loop recognizes a number of commands described below.
breakin Nm [M] Set or rest an internal break flag for the definition of Nm . If the boolean value M is omitted, the flag is toggled. If set, a break occurs before the execution of the expression sequence of the definition. The Nm must be the name of a defined expression or a defined operation using the operation form style of operation expression.
Breaklist Display the list of names of definitions with break flag set.

deepplace

The operation deepplace returns an array the same as A except that the array at path P is replaced by B. It is the insertion operation corresponding to the selection operation reach and generalizes the operation place from addresses to paths.

     A1 := 1 2 (3 4 (5 6 (7 (8 9))))
+-+-+-------------------+
|1|2|+-+-+-------------+|
| | ||3|4|+-+-+-------+||
| | || | ||5|6|+-+---+|||
| | || | || | ||7|8 9||||
| | || | || | |+-+---+|||
| | || | |+-+-+-------+||
| | |+-+-+-------------+|
+-+-+-------------------+
     A1 := "M (2 2 2 0) deepplace A1
+-+-+-------------------+
|1|2|+-+-+-------------+|
| | ||3|4|+-+-+-------+||
| | || | ||5|6|+-+---+|||
| | || | || | ||M|8 9||||
| | || | || | |+-+---+|||
| | || | |+-+-+-------+||
| | |+-+-+-------------+|
+-+-+-------------------+

The array 7 is replaced with M. Since the result of the insertion is assigned to variable A1, the insertion of M is retained.

Definition

     deepplace IS OPERATION C A {­
        B Path := C ;
        IF empty Path THEN
           B
        ELSE
           I := first Path ;
           ((B (rest Path)) deepplace (I pick A)) I place A
        ENDIF }

Equations

   Path a valid path into A ==> ( Path reach A ) Path deepplace A = A
   B Null deepplace A = B, I in grid A ==> B \[I] deepplace A = B I place A

deepupdate

The operation deepupdate provides the semantics of the Nm@@P := A form of assignment expression. Nm must be an existing variable represented by a string, phrase or a cast; P is the path of addresses to the location to be updated; and A is the array to be placed in the variable.

     Array1 := (1 2(3 4 (5 6 (7 (8 9)))))
+-+-+-------------------+
|1|2|+-+-+-------------+|
| | ||3|4|+-+-+-------+||
| | || | ||5|6|+-+---+|||
| | || | || | ||7|8 9||||
| | || | || | |+-+---+|||
| | || | |+-+-+-------+||
| | |+-+-+-------------+|
+-+-+-------------------+

     deepupdate "Array1 (2 2 2 1 0) "Tom
+-+-+---------------------+
|1|2|+-+-+---------------+|
| | ||3|4|+-+-+---------+||
| | || | ||5|6|+-+-----+|||
| | || | || | ||7|Tom 9||||
| | || | || | |+-+-----+|||
| | || | |+-+-+---------+||
| | |+-+-+---------------+|
+-+-+---------------------+

The major purpose of deepupdate is to allow a selective update with a path on a global variable without forcing a copy. By passing the name of the variable to the operation that is doing the update, rather than its value, no sharing of the internal data is made and hence the update can be made “in place”.

definition

A definition in Nial is a syntactic construct that names a program fragment. The syntax is one of the three forms:

     <name>  IS  <array expression>
     <name>  IS  <operation expression>
     <name>  IS  <transformer expression>

A definition is used to associate a name (identifier) with a program fragment that is an array expression, an operation expression or a transformer expression. If the definition appears within a block, the association is made in the local environment. Otherwise, the association is made in the global environment and assigns a role to the name as representing that kind of expression.

If the program fragment is syntactically correct, the name is associated with the program fragment in the environment and no result is given. If a syntax error is detected in the analysis of the program fragment, an explanatory fault message is returned and the name association is not made.

If the name being associated in a definition is already in use, the new definition must be for a construct of the same role and the earlier definition is replaced. The use of a defined name always refers to its most recent definition.

deparse

The operation deparse is used to convert a parse tree representation of a Nial definition or program fragment into a token stream which can be converted into text corresponding to the Nial definition. The argument to deparse must be either a cast or the result of parse or getdef.

     deparse !(Pi * cos 0.5)
99 1 ( 2 Pi 2 * 2 cos 18 0.5 1 )

The principal use of deparse is in displaying Nial definitions. It is used in the definition of defedit and see.

The token stream returned by deparse also includes indicators to identify where new lines are to begin and to control indentation. The tokens corresponding to identifiers are given in the canonical form, indicating what role each token plays.

The result of deparse is implementation dependent and should be viewed as an internal representation provided as an interface to editing. It is subject to change as Q’Nial evolves.

Equations

   Pt a parse tree ==> parse deparse Pt = Pt
   Ts a token stream ==> parse deparse parse Ts = parse Ts

depth

The operation depth returns an integer indicating the number of levels of nesting of the array. This is called the depth of the array. The depth of an atom is 0. The depth of a simple nonempty is 1. In general, the depth of an array is 1 plus the maximum of the depths of the items.

     depth "abc
0
     depth 3 4 5
1
     depth 1 (2 3) (4 5 6)
2
     depth [2, [3, 4, [5],8],24]
3

Definition

     depth IS OPERATION A
     {­ IF atomic A THEN
          0
       ELSE
          1 + (max EACH depth A)
       ENDIF
     }

Equations

   depth Null = max Null
   tally depth A = 1
   depth A = FORK [atomic, 0 first, 1 plus max EACH depth] A

descan

The operation descan converts a token stream to a list of strings that represents the program fragment given by the token stream. The input to descan must be a token stream produced by scan or deparse or an equivalent list of tokens.

     descan scan 'A + 32'
+-------+
|A + 32 |
+-------+

     descan deparse !(Pi * cos 0.5)
+-----------------+
|( Pi * cos 0.5 ) |
+-----------------+

The principal use of descan is in displaying Nial definitions. It is used to define defedit and see. The argument to descan may also include tokens indicating the beginning of new lines and controlling indentation used to display a structured definition.

Equations

   S a string of Nial text ==> scan link descan scan S = scan S
   S a string of Nial text ==> canonical canonical S = canonical S
   S a string of Nial text ==> execute canonical S = execute S

diagram

The operation diagram computes a character table that gives the fully boxed picture of A with the decoration of the atoms determined by the current setting of the decor switch. An array is displayed as a frame with cells for each item. It is arranged in two dimensions, using groupings of table frames to picture arrays of higher dimension. Each cell is large enough to hold the diagram of the corresponding item of A.

The diagram of an atom is a picture that indicates its value. The diagram of a non-atomic single has an “o” in its upper left corner.

Diagram returns the display given by the operation picture when in diagram mode of display.

The decor or nodecor mode switch controls the display of atoms. With decor set, it gives a picture that distinguishes all atoms.

     set "decor;
     A := diagram (2 3 2 reshape 3 'abc' (2 1 reshape count 3) "apple 8.5
          (3 4));
     set "nodecor; A
+---+----------+  +---+----------+
|  3|+--+--+--+|  |  3|+--+--+--+|
|   ||`a|`b|`c||  |   ||`a|`b|`c||
|   |+--+--+--+|  |   |+--+--+--+|
+---+----------+  +---+----------+
|+-+|"apple    |  |+-+|"apple    |
||1||          |  ||1||          |
|+-+|          |  |+-+|          |
||2||          |  ||2||          |
|+-+|          |  |+-+|          |
+---+----------+  +---+----------+
|8.5|+-+-+     |  |8.5|+-+-+     |
|   ||3|4|     |  |   ||3|4|     |
|   |+-+-+     |  |   |+-+-+     |
+---+----------+  +---+----------+

The display of the result of a picture operation, such as diagram, makes sense when it itself is pictured in sketch-nodecor mode. In other modes, the characters making up the table would be boxed and/or decorated.

Definition

     diagram IS OPERATION A {­
        Old_setting := set "diagram ;
        Result := picture A ;
        set Old_setting ;
        Result }

dimensions

The number of axes of an array is referred to as its dimensionality. In array theory terminology the dimensionality is called the valence of the array. The following terms describe arrays by their valence:

Valence Description
0 single
1 list, vector
2 table, matrix
2 or more multivalent

display

The operation display returns a string which, when executed, returns the value A.

     set "decor; display 23.5
'23.5'

Display inserts the operation to construct values that cannot be described directly by constants. In the example below, since the phrase containing a blank character cannot be specified simply using the phrase mark, display inserts the operation phrase and the string that will create the desired phrase An answer.

     display "Nm (phrase 'An answer') 'Nm'
'["Nm,(phrase 'An answer'),'Nm']'

     X := "Name (2 3 (4 5) 'Queen''s' (3 2 reshape count 6))
+-----+-----------------------+
|"Name|+-+-+---+---------+---+|
|     ||2|3|4 5|'Queen's'|1 2||
|     || | |   |         |3 4||
|     || | |   |         |5 6||
|     |+-+-+---+---------+---+|
+-----+-----------------------+

     display X
'["Name,[2,3,4 5,'Queen''s',3 2 reshape 1 2 3 4 5 6]]'

     execute display X = X
l

In the definition of X, strand notation was used. In the display of X, brackets notation is created to represent X. The display is seen to be correct by the last example that shows that the execute of the display of X is X.

Equation

   execute display A = A

diverse

The operation diverse tests whether or not the items of A are all different. It returns true if they are and false if they are not.

     diverse 2 3 5
l

     diverse 'hello world'
o

     diverse 2 3 (2 3)
l

In the last example, the items are all different because the last item is a pair of items (2 3) which is not the same as 2 or 3.

Definition

     diverse IS OPERATION A {­ cull A = list A }

Pragmatics

The operation diverse executes faster when a large array has been sorted, hence it is better to use sortup to order a large array prior to applying diverse.

divide

The operation divide returns the result of dividing two numeric atoms. It coerces the type of the atoms to be real and gives a real number result. If B is a numeric zero, the result is the fault ?div.

If one argument is numeric and the other is a fault or if both arguments are the same fault, the answer is the fault. In all other cases when one or more of the arguments is not numeric, the result is the arithmetic fault ?A.

     R gets 1 2 2.5 `a "abc ??error
1 2 2.5 `a "abc ??error

     R outer divide R
    1.    0.5    0.4 ?A ?A ?error
    2.     1.    0.8 ?A ?A ?error
   2.5   1.25     1. ?A ?A ?error
?A     ?A     ?A     ?A ?A ?A
?A     ?A     ?A     ?A ?A ?A
?error ?error ?error ?A ?A ?error

The above example illustrates all combinations of atom types for the two arguments to divide.

Equations

   1.0 divide A = reciprocal A
   A divide B = A times reciprocal B (within roundoff error)

down

DOWN is a general transformer that recurs over the depth of an array to some arbitrary level. DOWN has four operation arguments: test tests when the recursion has gone as deep as necessary, endf is applied to the argument that satisfies test, structf rearranges the argument before recurring on each item, and joinf combines the results of the recursion on the items.

     DOWN [ atomic, 0 minus, pass, pass]  3 -4 (5 6)
+--+-+-----+
|-3|4|-5 -6|
+--+-+-----+
     DOWN [ simple, dosum, pack, pass ] 3 4 5 6
18
     DOWN [ atomic, 0 first, pass, 1 plus max ] [2, [3, 4, [5],8],24]
3

Definition

DOWN IS TRANSFORMER test endf structf joinf OPERATION A {­
   Candidates := [A];
   Results := Null;
   WHILE not empty Candidates DO
     Candidates B := [front, last] Candidates;
     IF B = "Start THEN
        Candidates Shp := [front, last] Candidates;
        N := prod Shp;
        IF N = 0 THEN N := N + 1; ENDIF;
        Results Items := opposite N [drop, take] Results;
        Results := Results append joinf (Shp reshape Items);
     ELSEIF test B THEN
        Results := results append endf B;
     ELSE
        B := structf B;
        Candidates := Candidates link [shape B, "Start] link reverse list B;
       IF empty B THEN Candidates := Candidates append first B; ENDIF;
ENDIF;
   ENDWHILE;
   first Results }

Equations

   DOWN [test, endf, structf, joinf ] A
        = FORK [test, endf, joinf EACH DOWN [test, endf, structf, joinf] structf ] A
   opposite A = DOWN [ atomic, 0 minus, pass, pass] A
   sum A = DOWN [ simple, dosum, pack, pass ] A
   depth A = DOWN [ atomic, 0 minus, pass, pass] A

drop

The operation drop selects the items of an array after a specified number of items have been dropped. If B is a list and A is a non-negative integer, the result is the list formed from dropping A items from the front of B. If A is a negative integer, the result is formed by dropping abs A items from the right end of B.

If B is a table and A is a pair of non-negative integers, the result is lower right corner of B that remains after dropping the number of rows and columns indicated by A from the upper left corner of B. If one or both items of A are negative, the dropping occurs from the other end of the extent of the corresponding axis.

For a higher dimensional array B, tally A must equal valence B; and the result is obtained by dropping from the front or back of the extents along each axis. If B is a single and product A is zero, the result is B; otherwise the result is an empty array with valence of tally A.

     2 drop "Able "Baker "Charlie "Dog
Charlie Dog

     T1 := tell 2 4
+---+---+---+---+
|0 0|0 1|0 2|0 3|
+---+---+---+---+
|1 0|1 1|1 2|1 3|
+---+---+---+---+

     1 2 drop T1
+---+---+
|1 2|1 3|
+---+---+

     1 -2 drop T1
+---+---+
|1 0|1 1|
+---+---+

Definition

     drop IS OPERATION A B {­
        IF not and EACH isinteger A THEN
           fault '?left arg of drop must be integers'
        ELSEIF valence B = 0 THEN
           IF product A > 0 THEN
              (tally A reshape 0) reshape B
           ELSE
              B
           ENDIF
        ELSEIF tally A ~= valence B THEN
           fault '?valence error in drop'
        ELSE
           ((A < 0) + (-1 * (A >= 0) )) * (shape B - abs A max 0) take B
        ENDIF }

Equations

   0 times shape B drop B = B
   tally B drop list B = Null

dropright

The operation dropright is an obsolete operation that drops items from the ends of extents. It is provided to retain compatibility with earlier versions of Q’Nial.

Definition

     dropright IS OPERATION A B {­ opposite A drop B }

each

The transformer EACH forms an operation, called the EACH transform of f, that applies f to every item of the argument A. The shape of the result of the application of EACH f to A is the same shape as A.

     EACH reverse ('abc' 'def' 'ghi')
+---+---+---+
|cba|fed|ihg|
+---+---+---+

     EACH first tell 3 4
0 0 0 0
1 1 1 1
2 2 2 2

EACH is used to distribute an operation f across an entire array A. Its use avoids the need to explicitly allocate space for the result and to write an explicit loop to apply f to each item of A individually.

Equations

   shape EACH f A = shape A
   I in grid A ==> I pick (EACH f A) = f (I pick A)
   (EACH f) (EACH g) A = EACH (f g) A
   EACH f A = shape A reshape (f first A hitch EACH f rest A)
   EACH f single A = single f A
   EACH f solitary A = solitary f A
   EACH f list A = list EACH f A
   isshape A ==> EACH f (A reshape B) = A reshape EACH f B
   EACH f link A = link EACH EACH f A
   EACH EACH f cart A = cart EACH EACH f A

eachall

The transformer EACHALL forms an operation which applies f to arrays formed by selecting the items of the items of A in corresponding positions, assuming that the items of A are all of the same shape.

When the items of A are not all the same shape, A is examined to see if all the items with tally greater than one have the same shape. If so, all the items of tally 1 are replicated to that shape. If not, the fault ?conform is returned. EACHALL is implicitly applied in multi pervasive operations.

     A := 9 [2] (4 4 reshape (5+tell 16))
+-+-+-----------+
|9|2| 5  6  7  8|
| | | 9 10 11 12|
| | |13 14 15 16|
| | |17 18 19 20|
+-+-+-----------+
     pack A
+------+------+------+------+
|9 2 5 |9 2 6 |9 2 7 |9 2 8 |
+------+------+------+------+
|9 2 9 |9 2 10|9 2 11|9 2 12|
+------+------+------+------+
|9 2 13|9 2 14|9 2 15|9 2 16|
+------+------+------+------+
|9 2 17|9 2 18|9 2 19|9 2 20|
+------+------+------+------+

     (EACH sum) pack A
16 17 18 19
20 21 22 23
24 25 26 27
28 29 30 31

     EACHALL sum A
16 17 18 19
20 21 22 23
24 25 26 27
28 29 30 31

     sum A
16 17 18 19
20 21 22 23
24 25 26 27
28 29 30 31

The example shows that EACHALL is defined in terms of pack and is implicitly used in the multi pervasive operation sum.

Definition

     EACHALL IS TRANSFORMER f OPERATION A {­ EACH f pack A }

Equations

   shape EACHALL f A = shape pack A
   I in grid pack A ==> I pick (EACHALL f A) = f (I pick pack A)

eachboth

The transformer EACHBOTH forms an operation which applies f to pairs formed by selecting the items of A and B in corresponding positions, assuming that A and B have the same shape.

When A and B do not have the same shape, if either A or B has only one item, that item is replicated to the shape of the other argument; otherwise, the fault ?conform is returned.

EACHBOTH is used implicitly in all binary pervasive operations.

     7 2 1 EACHBOTH post 4 5 3
+-+-+-+
|7|2|1|
|4|5|3|
+-+-+-+

Definition

     EACHBOTH IS TRANSFORMER f OPERATION A B {­ EACHALL f A B }

Equations

   shape (A EACHBOTH f B) = shape (A pack B)
   I in grid pack A B ==> I pick (A EACHBOTH f B) = f (I pick (A pack B))

eachleft

The transformer EACHLEFT forms an operation which applies f to pairs formed by pairing the items of A with B. The shape of the result is the shape of A.

     2 3 4 EACHLEFT reshape 5
+---+-----+-------+
|5 5|5 5 5|5 5 5 5|
+---+-----+-------+

     (2 3 reshape 'abcdef') EACHLEFT hitch '123'
+----+----+----+
|a123|b123|c123|
+----+----+----+
|d123|e123|f123|
+----+----+----+

Definition

     EACHLEFT IS TRANSFORMER f OPERATION A B {­ EACH (B CONVERSE f) A }

Equations

   shape ( A EACHLEFT f B ) = shape A
   I in grid A ==> I pick (A EACHLEFT f B) = f (I pick A) B
   A EACHLEFT f B = A EACHBOTH f single B
   single A EACHLEFT f B = single (A f B)
   solitary A EACHLEFT f B = solitary (A f B)
   list A EACHLEFT B = list (A EACHLEFT f B)
   isshape S ==> (S reshape A) EACHLEFT f B = S reshape (A EACHLEFT f B)

eachright

The transformer EACHRIGHT forms an operation which applies f to pairs formed by pairing A with the items of B. The shape of the result is the shape of B.

     2 EACHRIGHT reshape 3 4 5
+---+---+---+
|3 3|4 4|5 5|
+---+---+---+

     `X EACHRIGHT hitch (2 3 reshape 'ab' 'cd')
+---+---+---+
|Xab|Xcd|Xab|
+---+---+---+
|Xcd|Xab|Xcd|
+---+---+---+

Definition

     EACHRIGHT IS TRANSFORMER f OPERATION A B {­ EACH ( A f ) B }

Equations

   I in grid B ==> I pick (A EACHRIGHT f B) = f A (I pick B)
   shape ( A EACHRIGHT f B ) = shape B
   A EACHRIGHT f B = single A EACHBOTH f B
   A EACHRIGHT f single B = single (A f B)
   A EACHRIGHT f solitary B = solitary (A f B)
   A EACHRIGHT f list B = list(A EACHRIGHT fB)
   isshape S ==> A EACHRIGHT f (S reshape B) = S reshape (A EACHRIGHT f B)

edit

The operation edit passes control to the standard editor, requesting it to edit the file named by the string or phrase Filename. The editor used is determined by the environment variable EDITOR or by a default chosen for each version. When the editing task is completed, the editor returns control to Q’Nial. The result is the fault ?noexpr.

     edit "test.ndf
     edit 'data_records'

If edit is used to edit a file of Nial definitions, the operation loaddefs must be executed on return from the editor to load the definitions.

empty

The operation empty tests whether or not an array A has any items. If A has no items, it returns true; otherwise, false. The predefined expression Null is empty as is any array with a zero in its shape.

Since there are empty arrays with more than one dimension, it is better to test for an empty array by using empty rather than by direct comparison with Null.

     empty Null
l

     empty (0 3 2 reshape 5)
l

     empty solitary 5
o

     empty solitary Null
o

The last two examples show that a solitary is not empty even if the item it contains is empty.

Definition

     empty IS OPERATION A {­ tally A = 0 }

Equation

   empty A = 0 in shape A

equal

The operation equal is normally used to compare two arrays A and B to see whether or not they are identical. It returns true if they are equal; false otherwise. (Two nonempty arrays are identical if they have identical shapes and hold identical items at each location. Two empty arrays are identical if they have identical shapes.) Empty is extended to arbitrary arrays by returning true if all items of an array are identical.

The symbol = is a synonym for equal and can be used in both infix and prefix application. In the Equations sections of the dictionary entries, the symbol = is used to separate two sides of an equation. To test such an equation, it may be necessary to replace the expression following = with the same expression in parentheses in order to force correct parsing of the equation as a Nial expression.

     2 3 4 = [2,3,4]
l
     Null = ''
l
     2 (3 4) = (2 3) 4
o
     equal EACH shape tell 2 3 4
l

The first example shows that a triple formed using strand notation is identical to that formed using brackets notation. The second shows that Null is equal to the empty string. The third shows that grouping items of a list in different ways creates different arrays. The last shows the convenience of the more general form of equal.

Equations

   A equal B <==> shape A = shape B
   A equal B and (I in grid A) ==> I pick A = I pick B

equations

The equations in the Nial Dictionary and in the on-line help provide an abbreviated way of stating properties of the term or object being described. They could be expanded into an explanation in English but that would lengthen the manual considerably.

The equations use variables such as A,B and C that take on array values; and variables such as f, g and h that denote operations that map arrays to arrays. An equation holds for all arrays and for all computable operations without side effects, unless a specific qualification is made. Thus, the equation:

   (EACH f) (EACH g) A = EACH (f g) A

says that for all operations f and g and all arrays A, the use of the EACH transform of f on the result of the use of the EACH transform of g on A has the same value as the use of the EACH transform of the composition of f and g on A. In mathematical terms, the EACH transformer distributes over operation composition.

The symbol = is used in its mathematical sense in equations and separates two Nial expressions. The symbol = has higher precedence than the two expressions it is separating. Where the Nial equivalent to = is needed to state the equation, the term equal is used.

To test an equation using Q’Nial, in order to force the correct parsing of the equation as a Nial expression, it may be necessary to replace

               Expr1 = Expr2

with

               Expr1 = ( Expr2 )

The equality used in equations assumes that both sides of the equations compute without triggering a fault and produce equal arrays; or both produce the same fault value if fault triggering is off. In some cases, the equality is inexact due to roundoff errors.

The symbol =f= is used to denote an equality where both sides produce the same non-fault value; but in some cases one or both of the sides may fault and the equality is no longer valid.

Some of the equations are qualified by a constraint on the variables. The constraint is written in English or as a conditional expression in Nial that must hold for the equation to be true. In a statement of the form

               Expr ==> Eqn

Expr is the qualification written as a Nial boolean expression, the symbol ==> is used for “implies”, and Eqn is the equation that holds under the qualification.

The symbol <==> denotes an if and only if implication. Thus a statement of the form

              Expr1 <==> Expr2

states that both expressions have the same truth value; either both are true or both are false.

Reading the Equations

The equations related to abs, the operation that finds the absolute value of a number, are:

   abs A = EACH abs A
   shape abs A = shape A
   abs abs A = abs A

The first two come from the property that abs is unary pervasive. The first one says that applying abs to an array A is the same as applying abs to the items of array A. It also implies that the shape of the result of abs A is the same as shape of A since EACH transforms always preserve shape. The second equation says that the shape of the result of abs A is the same as the shape of A. The third equation indicates that subsequent applications of abs after the first do not change the result.

The equations related to minus, denoted by -, are:

   A - B = EACH - pack A B
   shape (A - B) = shape pack A B
   A - B = A + opposite B

The first equation says that pair of arrays A and B are packed at each level of nesting to bring corresponding items together. Then, the items are subtracted.

The second equation states that the shape of the result of subtracting A from B is the shape formed by packing A and B. The third equation states that subtracting A from B is equivalent to adding the opposite of B to A.

The equations related to hitch are as follows:

   A hitch B = A hitch list B
   list (A hitch B) = A hitch B

The first says that hitch treats its right argument as though it were a list, and the second states that the result of hitch is a list.

The following equations illustrate the use of qualifications:

   atomic A ==> single A = A
   not empty A ==> mix rows A = A
   diverse A <==> cull A = list A

The first equation says that if A is atomic, the single of A is equal to A. That is, the single of an atom is the atom. The second equation says that if A is not empty, forming the rows of A and applying mix to recombine them, results in the original array. The third equation says that if A is diverse, the cull of A is equal to the list of A; and that if A is not diverse, the cull of A is not equal to the list of A.

There are many identities that hold for all arrays in Nial. Three general equations are the following:

A unary pervasive operation f satisfies the equation:

   f A = EACH f A

A binary pervasive operation f satisfies the equation:

   A f B = A EACHBOTH f B

A multi pervasive operation f satisfies the equations:

   f A = EACHALL f A
   f A = REDUCE f A

erase

The operation erase is used to remove unwanted global definitions and variables from the workspace. The argument Nm must be a phrase or string giving the object to be removed. Predefined names and names in local scopes cannot be removed.

When a name has been erased, it remains in the symbol table with its original role. It can be redefined in the same role or, if a variable, reassigned a value by a subsequent action. A name cannot change its role once established.

     average IS divide[sum,tally];
     calc IS EACH average;
     Var3 := calc (2 3 4) (5 6 7)
3. 6.

     EACH erase "average 'Var3' "sum
?noexpr ?noexpr ?system name

In the example above, the operation sum cannot be erased as it is a predefined operation. In the next three examples below, faults are returned because of missing variables or operations that were erased. The effect of erasing an object is not propagated to other definitions referring to the erased name.

     Var3
?no_value

     average count 10
?missing_op

     calc (2 3 4) (5 6 7)
?missing_op ?missing_op

     average := 5 6 7
?invalid assignment: AVERAGE :=  <***> 5 6

In the last example, an attempt is made to redefine the role of average and Q’Nial indicated that average is not a suitable name for such an assignment.

eraserecord

The operation eraserecord is used to erase component N of direct access file designated by the number Filenum. In a file with record components, eraserecord replaces the erased component with an empty string. In a file with array components, it replaces the erased component with the fault ?missing.

N may be a single record number or a list of record numbers.

If the last component in a file is erased, any immediately preceding components that are empty in a record component file or contain the fault ?missing in an array component file are also removed. The file tally, which records the number of records in the file, is adjusted accordingly.

The operation returns the fault ?noexpr or a fault indicating improper use of the operation.

     File_num := open "Names "d ;
     eraserecord File_num 23 ;

eval

The operation eval provides access to the underlying expression evaluator of Q’Nial. The argument to eval must be a parse tree returned by parse or getdef, a named expression or a cast which represents a Nial expression. The effect is to evaluate the expression and return the resulting value. As seen below, the parse tree can be one formed by using the cast mechanism or one built explicitly using parse.

     eval parse scan '23 + 45'
68

     eval !(23 + 45)
68

The next example shows that eval can be used to evaluate a named expression using either the cast or the phrase of its name.

     A := 100 ;
     Addone IS (A := A + 1);
     (eval !Addone)(eval "Addone)
101 102

     (eval !A)(eval "A)(value "A)
102 ?not an expression 102

The last example shows that eval can be used to evaluate the cast of a variable but not a phrase naming the variable. The operation value should be used instead.

Equations

   S a string holding a Nial expression ==> eval parse scan S = execute S
   Pt a parse tree ==> eval parse deparse Pt = eval P

except

The operation except returns the list of items of A that are not in B. It corresponds to set difference if A and B are viewed as set representations. The result is a list with items in the same order as in A.

     count 10 except 3 5 7
1 2 4 6 8 9 10

     'a list of words' except 'aeiou'
 lst f wrds

     tell 2 3 except [0 0]
+---+---+---+---+---+
|0 1|0 2|1 0|1 1|1 2|
+---+---+---+---+---+

     tell 2 3 except (0 0)
+---+---+---+---+---+---+
|0 0|0 1|0 2|1 0|1 1|1 2|
+---+---+---+---+---+---+

The last two examples show that if the left argument has nonatomic items, to remove a single non-atomic array, the right argument must be made into a solitary list. Otherwise the items of the right argument are used.

Definition

     except IS OPERATION A B {­ A EACHLEFT notin B sublist A }

Equations

   list (A except B) = A except B
   list A except list B = A except B
   (A except B) except C = A except (B link C)
   A except cull B = A except B

Pragmatics

The operation except uses an internal sort on its arguments to reduce the algorithmic complexity of the above definition. It executes considerably faster if its arguments have already been sorted.

execute

The operation execute evaluates a Nial expression given as a string S. The result of the evaluation is returned. The execution of the string takes place in the environment where execute is applied.

If the string contains a name, the local meaning of that name will be used if a local meaning exists. If execution of the string results in the creation of a new variable or defined name, the new object is placed in the global environment.

If a string being executed contains an integer representation that would be converted to an integer outside the range for the computer, the result is converted to the corresponding real number.

     execute '23 + 40045'
40068.

     A := count 5; opname := 'product';
     execute link opname ' A'
120

The last example shows that Nial program text can be constructed under program control and then executed. This feature is useful in applications where responses are generated by combining text elements based on the user input and the current state of the computation. The technique is useful for applications involving computer assisted instruction and knowledge based systems.

It is more efficient to use eval to evaluate the cast of an expression than to use execute to evaluate the corresponding string. In the former approach the scan and parse is done once, whereas in the latter it is repeated on each execution.

Definition

     execute IS OPERATION A {­ eval parse scan A }

Equations

   execute readscreen Prompt = read Prompt
   Txt, a Nial expression ==> eval !(Txt) = execute 'Txt'
   S a string holding a Nial expression ==> execute canonical S = execute S

where canonical IS link descan deparse parse scan.

exp

The operation exp implements the exponential function of mathematics. It produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean exponential of the corresponding real
integer exponential of the corresponding real
real exponential of A
character fault ?exp
phrase fault ?exp
fault argument A
     exp  o  1  0.5  `a  "abc  ??error
1. 2.71828 1.64872 ?exp ?exp ?error

Equations

   exp ln A = A   (within roundoff error)
   exp A = exp 1.0 power A   (within roundoff error)

expression

The term expression is used in its most general sense to describe a program fragment that denotes one of the three primary objects of Nial: an array, an operation, or a transformer. However, in most contexts we use the term as an abbreviation of an array expression.

An array expression denotes an array value. That is, it is a program fragment that when evaluated in the proper context will produce an array. The predefined expressions of Q’Nial either produce a constant value, or they carry out some system action and return the fault value ?noexpr.

The control constructs of Nial are array expressions made up of keywords, simple-expressions and expression-sequences.

A named expression is either a predefined expression or an expression that has been given an explicit name using the IS definition mechanism.

expression sequence

An expression-sequence is the main construct used for program text that produces a value. It consists of one or more expressions separated by semi-colons and possible followed by a semi-colon.

     expression-sequence ::= expression
        {­ ; expression } [ ; ]

The expressions in an expression-sequence are evaluated in left-to-right order. If the sequence does not terminate with a semicolon, the array returned is the result of the last expression. If the sequence does end with a semicolon, the array returned is the fault ?noexpr. At the top level loop, if the array returned is the fault ?noexpr, it is not displayed.

exprs

The expression Exprs returns a list of phrases giving the names of all user defined expressions in the workspace.

Definition

     Exprs IS {­
        Names Roles := pack symbols 0;
        "expr match Roles sublist Names }

extent

The term extent is used to describe the length of an axis of an array in a particular dimension. Thus a 4 by 6 table is said to have extent 4 in the first dimension and extent 6 in the second dimension.

external declaration

     external-declaration ::= identifier IS
        EXTERNAL ( EXPRESSION
                         | OPERATION
                         | TRANSFORMER
                         | VARIABLE )

An external declaration assigns a role to a name, allowing it to be used in a definition before its own definition is given. This mechanism is useful for creating mutually recursive definitions. An external declaration is made only in the global environment.

If the name is already defined with the same role, the declaration has no effect. If the name has another role, a fault is reported. If the name is not currently defined, a default object is associated with it.

false

The constant expression False denotes the boolean atom for false, which Nial also denotes by o. It is the result of comparing two arrays that are not identical for equality.

     False (not False)
ol

Equations

   tally False = 1
   shape False = Null
   single False = False
   not False = True

fault

The operation fault converts a string or phrase into the fault value with the string as its message. If the argument is not a string, phrase or a fault, the result is the fault ?type error.

Fault is used to construct faults that contain blanks or other characters than cannot appear in a literal fault or ones constructed from a message provided from the host system.

By default, fault triggering is on when the interpreter is initialized. This causes the creation of a fault to interrupt the flow of execution. The operation quiet_fault can be used to create a fault value without triggering an interrupt.

     settrigger o;
     fault '?missing data'
?missing data

     fault "Notastring
Notastring

     fault 3.4
?type error

The convention is that all faults generated by Q’Nial internally begin with the character ?. The second example shows that this is not a requirement.

Equations

   isfault F ==> fault string F = F
   isstring S ==> string fault S = S

fault triggering

Nial assumes that every computation that terminates results in an array value. However, there are many cases where a computation does not have a sensible answer. If division by zero occurs, for example, there is no suitable number to return. Nial uses special atomic arrays called faults to indicate such results. For division by zero it is ?div.

Q’Nial has two ways of handling a fault: either a trigger mechanism is executed that causes an interruption when a fault is created, or during execution of a defined operation, expression or transformer the fault is treated as a normal atomic array.

When Q’Nial is invoked for interactive execution with the -i option, the fault triggering mechanism is turned on by default. When it is invoked for execution only the fault triggering capbility is turned off. During interactive execution, the state of the triggering mechanism can be turned on or off using the operation settrigger. The operation quiet_fault can be used to create a fault without causing fault triggering.

If fault triggering is set and a fault is generated during execution of a defined operation, execution is interrupted. On an interruption caused by a fault, a display message appears giving the call stack of definitions currently executing and the line of text that caused the fault. For example, the definition:

     foo is op A B {­ A / B + 1 }

followed by the evaluation of the expression

     foo 3 0

results in the output:

-------------------------------------------------------------
    Fault interruption loop:  enter expressions or
      type: <Return>   to jump to top level
    current call stack :
foo
      ?div triggered in : ... A / B
-------------------------------------------------------------
>>>

where the string ‘>>>’ is a special prompt indicating that a fault has occurred and execution has been interrupted. The prompt permits you to query the value of variables in the expression and its surrounding computation or to view the operation that has triggered the fault. The above session might continue as:

>>> see "foo
foo IS OPERATION A B {­
    A / B + 1 }
>>> A
3
>>> B
0
>>>

A variable in a definition that called the current one can be referenced by preceding the variables name by the definition name and a colon, e.g. G:X denotes variable X in definition G. You can execute any expressions you want at the prompt. A useful thing to do is to see the definition that has interrupted. When you are ready to resume, reply to the prompt with a Return and control returns to the interactive loop.

fault values

The Role of Faults

A fault is a special kind of atomic value used by Q’Nial to signal special values or to indicate that an operation has been given an argument it cannot handle in a normal way. The special value faults are:

Fault Meaning
?noexpr indicates that no answer is expected
?eof end of file indication
?I Zenith which is greater than all atoms
?O Nadir which is less than all atoms

filelength

The operation filelength returns the length in bytes of the host file named by string or phrase Filename. It is used in conjunction with readfield and writefield in processing host files.

     Len := filelength "Myfile;
     Data := readfield "Myfile 0 Len;

In the example, filelength is used to determine the size of the file and readfield is used to read it in as raw byte data. If the file corresponds to a text file, the data will include end of line indications appropriate for the host system.

filestatus

The expression Filestatus gives information on the files currently open in a Q’Nial session. It returns a list of triples, one for each open file, giving the file number as an integer, the filename as a phrase and the mode as a character.

The modes are r, w, a, d, pr, pw and c, standing for read, write, append, direct, pipe_read, pipe_write and communications respectively.

The files for standard input, standard output and standard error are opened with modes of r, w and w respectively using file numbers 0, 1 and 2.

     Fnum := open "F "a;
     Filestatus
+---------+----------+----------+-----+
|0 stdin r|1 stdout w|2 stderr w|3 F a|
+---------+----------+----------+-----+

filetally

The operation filetally returns the number of records in the file designated by Filenum, an integer returned by an earlier call to open. The filetally is one higher than the highest component number of a record or array that is written but not erased.

     Fn := open "newfile `d;
     writearray Fn (9 99) ('The' 'End');
     filetally Fn
100

filter

The transformer FILTER is given a predicate operation f that is used to select items from A. The result of applying FILTER f is to produce a list of the items of A that satisfy the predicate f.

     FILTER (0<) (4 -2 3 -5)
4 3

     FILTER (not isphrase) (37 4.5 "cat)
37 4.5

Definition

     FILTER IS TRANSFORMER f OPERATION A {­ EACH f A sublist A }

Equations

   tally FILTER f A = sum EACH f A

find

The operation find returns the address of the first occurrence of A as an item of B, searching B in row major order. If A does not occur in B, the result is the gage of shape B. The result of find is an integer if B is a list; and a list of integers of tally equal to valence B otherwise.

     set "diagram ;
     3 find 56 34 3 23 3 57 3
2

The result of find is 2.

     `a find 'hello world'
11

     2 3 find count 3 4
+-+-+
|1|2|
+-+-+

In the second example, the character a is not in the string ‘hello world’ and the result is the tally of ‘hello world’. In the last example, the result is a list of two integers because the valence of count 3 4 is 2.

Definition

     find IS OPERATION A B {­ gage first ( A findall B append shape B ) }

Equations

   A seek B = A [in,find] B
   A in B ==> A find B pick B = A
   I in grid A and diverse A ==> I pick A find A = I

Pragmatics

The operation find uses a linear search on the items of B if the array has not been sorted, or uses a binary search algorithm if it has. The latter fact suggests that an array that is searched frequently should be kept in lexicographical order by applying sortup to it when it is created or changed.

findall

The operation findall returns a list of the addresses of all occurrences of A as an item of B, searching B in row major order. If A does not occur in B, the result is the empty list Null.

The result of findall is a list of integers if B is a list and a list of pairs if B is a table.

     3 findall 56 34 3 23 3 57 3
2 4 6

     X := 3 4 reshape 1 7 3 2 3 4 3 2 6 3
1 7 3 2
3 4 3 2
6 3 1 7

     3 findall X
+---+---+---+---+
|0 2|1 0|1 2|2 1|
+---+---+---+---+

     `a findall 'hello world'

     1 1 findall tell 3 4
+---+
|1 1|
+---+

In the first example findall returns the list of integers where 3 is found in the list; in the second a list of pairs is returned. The last example shows that if only one occurrence is found the result is a solitary list.

Definition

     findall IS OPERATION A B {­ A EACHRIGHT equal B sublist grid B }

Equations

   (list A) findall B = A findall B
   list (A findall B) = A findall B
   A in B ==> A find B = first (A findall B)

Pragmatics

The operation findall uses a linear search on the items of B if the array has not been sorted, or uses a binary search algorithm if it has. The latter fact suggests that an array that is searched frequently should be kept in lexicographical order by applying sortup to it when it is created or changed.

first

The operation first returns the first of the items of A. If A is empty, it returns the fault ?address.

     first 4 5 6
4

     first count 3 4 5
1 1 1

     first Null
?address

Definition

     first IS OPERATION A {­ 0 pick list A }

Equations

   first single A = A
   first solitary A = A
   first list A = first A
   first (S reshape A) = first A
   (valence A reshape 0) pick A = first A

flip

The operation flip is a synonym for pack. It has been retained for compatibility with earlier versions.

floor

The operation floor produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean corresponding integer
integer argument A
real next lower integer, or the fault ?A if the result is outside the range of integers
char fault ?A
phrase fault ?A
fault argument A
     floor  l  -2  3.5  `a  "abc  ??error
1 -2 3 ?A ?A ?error

     floor  3.5  -4.6  7.0  25.3e20
3 -5 7 ?A

Equation

   floor A = opposite ceiling opposite A

fold

The transformer FOLD modifies an operation f to one that takes a pair of arguments consisting of an integer N and an arbitrary array A. The result of applying N FOLD A is to apply f, N times, applying it first to A and subsequently to the result of the previous application.

     3 FOLD rest 4 5 6 7 8 9
7 8 9
     2 FOLD sum (2 3 4) (5 6 7)
27
     sum sum (2 3 4) (5 6 7)
27

Definition

     FOLD IS TRANSFORMER f OPERATION N A {­
        IF N > 0 THEN
           N - 1 FOLD f ( f A )
        ELSE
           A
        ENDIF }

Equations

   0 FOLD f A = A
   1 FOLD f A = f A
   2 FOLD f A = f f A

fork

The transformer FORK implements a conditional functional mechanism corresponding to the if-expression. The argument to FORK must be an atlas of length two or more. The operation in the first position is a predicate. If there are exactly three items in the atlas, the first is a predicate; the second is applied to A if the predicate returns true; and the third is applied to A if the predicate returns false. That is, if f A is true, the result is g A; otherwise it is h A.

If there are more than three operations in the atlas, they are taken in pairs from left to right. The first of each pair must be a predicate and is applied to A in turn until a result is true. The result of the transform is the result of applying the second of that pair to A. If no application results in true and the number of operations in the atlas is an odd number, the last operation is evaluated. Otherwise the result is the fault ?noexpr.

     FORK [atomic, opposite, 1 +] (3 -4 (5 6 7))
+-+--+-----+
|4|-3|6 7 8|
+-+--+-----+

     FORK [isreal,floor,simple,sum,5+] 2 4 6
12

Equation

   FORK [A first, B first, C first, D first, E first] Null = IF A THEN B ELSEIF C THEN D ELSE E ENDIF

for-loop

The FOR-loop control structure is used to execute the expression sequence ExpSeq repeatedly while variable Var takes on the values specified as items in the simple expression Exp.

     FOR X WITH 1. 2. 3. DO
        write (X) (X*X) (X power X) ;
     ENDFOR
1. 1. 1.
2. 4. 4.
3. 9. 27.

In the example, X takes on the values 1., 2. and 3. in successive loops and the values of X, X squared and X to the power X are displayed.

fromraw

The operation fromraw converts the boolean array B to a simple array of the same type as atom A.

     fromraw `a olloooololloooloolloooll
abc

     fromraw 0 ooooooooooooooooooooooololollool
345

fromraw 0.0 olooooooolooooololoooooooooooooooooooooooooooooooooooooooooooooo
34.5

     fromraw l lollo
lollo

Equation

   simple A and and (type A match type first A) => A = fromraw (toraw A) (first A)

Pragmatics

Fromraw is used to convert raw bit data into atomic data that can be manipulated by Nial. If the data was created from Nial data using toraw then it will work in a system independent way. However, if raw byte data is obtained using readfile and then converted using toraw the host system byte ordering may need to be allowed for.

front

The operation front returns a list of all items but the last of A. If A is a solitary or is empty, the result is the empty list Null.

     front 3 4 5 6
3 4 5
     front tell 2 3
+---+---+---+---+---+
|0 0|0 1|0 2|1 0|1 1|
+---+---+---+---+---+

Definition

     front IS OPERATION A {­ tally A - 1 max 0 reshape list A }

Equations

   front A = front list A
   list front A = front A
   front Null = Null
   not empty A ==> front A append last A = list A
   shape A reshape (front A append last A) = A

functions in nial

A function is a mathematical name for an object that maps an argument in a given domain to a result in a given range. In Nial, an operation is an object in the set of functions from the domain of Nial arrays to the range of Nial arrays. Thus, an operation always applies to an array and returns an array.

A transformer in Nial is also a function. It domain is Nial operations and its range is also Nial operations. Since its argument is itself a function, a transformer is said to be a second order function. A transformer always applies to an operation and results in an operation.

Definitions in which the associated object is a simple-expression are used to name program fragments that return an array value but which do not need parameters. The resulting named-expression behaves like a function having no parameters.

Nial is considered to be a functional language, but it is not purely functional in that it has assignments, loops and other non-functional concepts.

fuse

The operation fuse is used for two distinct purposes. If I is simple and contains all the axes of A without repetition, the result is an array formed by a permutation of the axes of A by I. The shape of the result is I choose shape A. If I is not simple but link I is simple and contains all of the axes of A without repetition, the result is obtained by diagonalizing along axes that are grouped together, ordering them according to the ordering in I.

If link I does not contain all the axes or if there are repetitions of the axes in link I, the fault ?invalid fuse is returned.

     A := 2 3 4 reshape count 24
1  2  3  4   13 14 15 16
5  6  7  8   17 18 19 20
9 10 11 12   21 22 23 24

     1 2 0 fuse A
1 13   5 17    9 21
2 14   6 18   10 22
3 15   7 19   11 23
4 16   8 20   12 24

     0 (1 2) fuse A
 1  6 11
13 18 23
     [0 1 2] fuse A
1 18

The first example is an axis permutation; the last two are diagonalizations.

Equations

   axes A fuse A = A
   transpose A = ( reverse axes A fuse A )
   sortup I = axes A and not empty A ==> shape ( I fuse A ) = I choose shape A

gage

The operation gage is used to convert an array of integers into an atomic integer if there is only one item or a list if there is more than one. The integers must be non-negative otherwise the fault ?gage is produced.

     gage 4 5 6
4 5 6

     gage find 5 (tell 10)
6

     gage 3 -4 5
?gage

The main purpose of gage is to express a list of non-negative integers in the form an address takes, i.e. it converts a solitary integer to the integer itself and leaves all other lists alone.

Definition

     gage is OPERATION A {­
      IF and EACH isinteger A THEN
         IF tally A = 1 THEN
            IF first A >= 0 THEN first A ELSE ??gage ENDIF
         ELSE
            IF and EACH (0<=) A THEN list A ELSE ??gage ENDIF
         ENDIF
      ELSE
        ??gage
      ENDIF }

getdef

The operation getdef retrieves the parse tree associated with the global definition named Nm. Nm may be a phrase or a string. Only the parse trees associated with global user definitions can be retrieved.

     foo IS first ; getdef "foo
+---+-----------------------------+
|100|+--+---------------+------+-+|
|   ||11|+--+----------+|3 47 3|1||
|   ||  ||22|2 160 7459||      | ||
|   ||  |+--+----------+|      | ||
|   |+--+---------------+------+-+|
+---+-----------------------------+

The example shows a parse tree for a simple definition. For definitions of any reasonable size the diagram of the parse tree becomes too large to examine easily.

The primary use of getdef is to provide an interface between internal representations and the editing facilities. It is used in see and defedit. The detailed form of the parse tree is implementation specific and may change in future releases.

getfile

The operation getfile returns the records of the text file named by the string or phrase Filename as a list of strings. The file must not be open.

     Recs := getfile "Myfile;

Definition

     getfile IS OPERATION Filename {­
        Fnum := open Filename `r;
        IF isfault Fnum THEN
           Fnum
        ELSE
           Lines := '';
           Line := readfile Fnum;
           WHILE Line ~= ??eof DO
              Lines := Lines append Line;
              Line := readfile Fnum;
           ENDWHILE;
           Lines
        ENDIF }

getname

The operation getname retrieves the variable or definition name associated with a name reference triple within a parse tree. The argument must be a triple with first item 2 that has been produced by the Q’Nial parser. The operation is useful when analyzing the structure of a parse tree representation. This is a specialized task and getname is used only with considerable knowledge of the internal workings of Q’Nial.

getsyms

The operation getsyms retrieves the parameters and local variables of a defined operation or expression named by Nm. The operation is useful when analyzing the name usage of a definition in the context of analyzing the name interaction among a set of definitions. This is a specialized task and getsyms is used only with considerable knowledge of the semantics of Q’Nial.

global environment

The global environment is the set of associations between names and objects formed in the workspace that are either predefined in Nial or have been created by actions that have taken place during a session.

grade

The transformer GRADE modifies a comparator f to produce an operation that, when applied to A, returns an array of addresses of the same shape as A that orders A according to the comparator. The addresses can be used to select the items of A (using choose) so that the items are in order according to f.

A comparator is an operation that compares two arrays and returns true if they are in the desired ordering or false otherwise. Operations lte ( <=) and gte ( >=) are the most commonly used comparators.

     A gets 3 2 reshape 65 77 4 19 22 11
65 77
 4 19
22 11

     Addrs := GRADE >= A
+---+---+
|0 1|0 0|
+---+---+
|2 0|1 1|
+---+---+
|2 1|1 0|
+---+---+

     Addrs choose A
77 65
22 19
11  4

     GRADE <= ("some "not "in "order)
3 1 0 2

     GRADE <= ['xyz','abc','mno','cat']
?invalid comparison in GRADE

     GRADE up ['xyz','abc','mno','cat']
1 3 2 0

The first three expressions illustrate that GRADE returns the addresses that re-order the items of table A in descending order. The next expression shows that the GRADE transform of <= can be applied to a list of phrases. The second last expression shows that GRADE <= cannot be applied to a list of strings. This is because <= is used itemwise on the characters of the strings and hence the comparator yields a bitstring rather than an atomic boolean result. The last expression shows that the comparator up can be used with strings.

Equations

   SORT f A = GRADE f A choose A
   f a comparator ==> shape GRADE f A = shape A

gradeup

The operation

     gradeup 3 7 5 4 9 8 2 1 6 10
7 6 0 3 2 8 1 5 4 9

     gradeup ("some "words "not "in "order)
3 2 4 0 1

Definition

     gradeup IS GRADE up

Equation

   gradeup A choose A = sortup A

grid

The operation grid returns the array of addresses of the array A. The result has the same shape as A. Each item of the result is the address of the corresponding item in A. The grid of a list is a list of integers. The grid of a table is a table of pairs of integers.

     grid 3 6 4 7 4
0 1 2 3 4

     A :=  2 3 reshape count 6; A (grid A)

+-----+-------------+
|1 2 3|+---+---+---+|
|4 5 6||0 0|0 1|0 2||
|     |+---+---+---+|
|     ||1 0|1 1|1 2||
|     |+---+---+---+|
+-----+-------------+

Definition

     grid IS OPERATION A {­ tell gage shape A }

Equations

   grid A choose A = A
   shape grid A = shape A

gt

The operation gt compares two atoms A and B with the greater than relation, returning true if A is greater than B and false otherwise. The symbol > is a synonym for gt.

The atoms in Nial are organized as a lattice using <= for the ordering. The numeric atoms are comparable across types but numeric and literal atoms are incomparable. The literal types are not comparable across types. A comparison between incomparable objects results in false.

     R := l 2 2.5 `a "abc ??error
l 2 2.5 a abc ?error

     R OUTER > R
oooooo
looooo
lloooo
oooooo
oooooo
oooooo

     'apple' > 'above'
olloo

     "apple > "above
l

The use of OUTER > shows the comparisons between various atom types. The last two examples show the difference between comparing two strings, where the operation is distributed by its pervasive property; and comparing the corresponding phrases.

Definition

     gt IS OPERATION A B {­ (A gte B) and not (A mate B) }

gte

The operation gte compares two atoms A and B with the greater than or equal relation, returning true if A is greater than or equal to B and false otherwise. The symbol >= is a synonym for gte.

The atoms in Nial are organized as a lattice using <= for the ordering. The numeric atoms are comparable across types but numeric and literal atoms are incomparable. The literal types are not comparable across types. A comparison between incomparable objects results in false.

     R := l 2 2.5 `a "abc ??error
l 2 2.5 a abc ?error

     R OUTER >= R
looooo
lloooo
lllooo
oooloo
oooolo
oooool

     'apple' >= 'above'
lllol

     "apple >= "above
l

The use of OUTER >= shows the comparisons between various atom types. The last two examples show the difference between comparing two strings, where the operation is distributed by its pervasive property; and comparing the corresponding phrases.

Definition

gte IS OPERATION A B {­ B lte A }

hitch

The operation hitch attaches A to the front of the list of items of B. It returns a list of length one greater than the tally of B.

     (2 3 4) hitch (5 6 7)
+-----+-+-+-+
|2 3 4|5|6|7|
+-----+-+-+-+
     7 hitch 3
7 3
     hitch 'Wow' ''
+---+
|Wow|
+---+

The first example shows that the list 2 3 4 becomes an item on the front of the list 5 6 7. In the next example, the right argument of hitch is treated as a list. The last example shows that if the right argument is an empty list, the result is the left argument as a solitary.

Definition

     hitch IS OPERATION A B {­ solitary A link B }

Equations

   A hitch B = A hitch (list B)
   A hitch Null = solitary A
   list (A hitch B) = A hitch B
   not empty A ==> first A hitch rest A = list A
   shape A reshape (first A hitch rest A) = A

host

The operation host executes S as a host command language instruction. The argument S is a string or a phrase. If the action carried out by the host system produces output, the output is displayed on the screen. In window mode, Q’Nial is unable to capture this output and hence it may scramble the screen output. The screen is restored by executing Refresh.

The result of host is the fault ?noexpr if the command has returned normally; or a fault generated from a system dependent error message supplied by the host operating system. At the top level loop, a line beginning with !, the exclamation mark, is interpreted as a host command.

if-expr

The IF-expr construct is a notation for executing one of a number of possible expression sequences Es1, Es2 , … Esn. The sequence selected depends on the result of the conditional expressions C1, C2 , … Cn. In the general case, whichever condition is first found to return true specifies the expression sequence to be performed. If all the conditional expressions return false, expression sequence Esx is selected. The ELSEIF and ELSE clauses are optional.

In the following example, the result is one of phrase Adult, Minor or Juvenile, depending on the value of Age.

     Age := 17;
     IF Age > 18 THEN
        "Adult
     ELSEIF Age < 16 THEN
        "Minor
     ELSE
        "Juvenile
     ENDIF
Juvenile

in

The operation in returns true if A is an item of B and returns false if it is not.

     3 in 56 34 23 3 57 3
l
     1 1 in tell 3 4
l
     `a in 'hello world'
o
     `a in "apple
o
     `a in string "apple
l

The fourth and fifth examples show that a letter is not an item of a phrase but is an item of the corresponding string.

Definition

     in IS OPERATION A B {­ or (A EACHRIGHT equal B }

Equations

   A in B = A in (list B)
   A in Null = False
   A in solitary A = True
   A in (A hitch B) = True
   A in (B append A) = True
   A in (A pair B) = True

Pragmatics

The operation in uses a linear search on the items of B if the array has not been sorted, or uses a binary search algorithm if it has. The latter fact suggests that an array that is searched frequently should be kept in lexicographical order by applying sortup to it when it is created.

indexing

The term indexing is used to describe notations that can be used to select from or insert into a variable.

There are four indexing methods in Nial: at, at all, at path and slice represented by @, #, @@ and | respectively. The different indexing methods return different subsets of the array. The following is a summary of the indexing methods:

Method Name Description  
@ at indexes one item  
# at all indexes several items  
@@ at path indexes a part at depth  
    slice indexes cross-section of items
     Cities := "London "Washington "Ottawa "Moscow "Paris ;
     Cities@0
London
     Cities#[2,3]
Ottawa Moscow

     Alpha := 5 5 reshape 'ABCDEFGHIJKLMNOPQRSTUVWXY'
ABCDE
FGHIJ
KLMNO
PQRST
UVWXY

     Alpha@[0,2]
C

     Alpha|[1,]
FGHIJ

     Alpha|[,1]
BGLQV

     Alpha|[,[1,3]]
BD
GI
LN
QS
VX

The following example shows the slightly different structure which occurs when a comma is either present or missing before the last item. The library operation findpaths is used to indicate the path to the integer 10 in each case.

     set "diagram ; Nest1 := [1, 2, [3, 4, 5, [6, 7], 8, 9], 10]
+-+-+-----------------+--+
|1|2|+-+-+-+-----+-+-+|10|
| | ||3|4|5|+-+-+|8|9||  |
| | || | | ||6|7|| | ||  |
| | || | | |+-+-+| | ||  |
| | |+-+-+-+-----+-+-+|  |
+-+-+-----------------+--+

     set "diagram ; Nest2 := [1, 2, [3, 4, 5, [6, 7], 8, 9] 10]
+-+-+----------------------+
|1|2|+-----------------+--+|
| | ||+-+-+-+-----+-+-+|10||
| | |||3|4|5|+-+-+|8|9||  ||
| | ||| | | ||6|7|| | ||  ||
| | ||| | | |+-+-+| | ||  ||
| | ||+-+-+-+-----+-+-+|  ||
| | |+-----------------+--+|
+-+-+----------------------+

     findpaths 10 Nest1
+---+
|+-+|
||3||
|+-+|
+---+

     findpaths 10 Nest2
+-----+
|+-+-+|
||2|1||
|+-+-+|
+-----+

     (Nest2@@[2,0,3,0])
6

     (Nest2@@[2,0,4])
8

Address Validity

The index used in selecting a part of an array must be an expression that evaluates to a valid address. An invalid index returns a fault as follows:

Indexing Method Fault  
A@I ?address  
A#I ?addresses  
A@@P ?path  
A I ?slice

The operation pick works the same as the at method of indexing. If the index is invalid, pick returns the fault ?address. Similarly, choose works the same as at all indexing.

     3 pick Cities
Moscow

     2 3 choose Cities
Ottawa Moscow

     10 pick Cities
?address

     4 5 6 choose Cities
Paris ?address ?address

infix notation

In Nial an operation-expression may be placed between two array-expressions.This is called an infix use of the operation-expression.

     7 + 5
12

     2 3 reshape 1 2 3 4 5 6
1 2 3
4 5 6

In using the infix notation, one must understand that if a sequence of operations are placed between two array arguments, all but the first operation are applied to the second argument.

     2 + reverse tell 3
4 3 2

     2 (+ reverse tell) 3
3 6

     2 (+ reverse) tell 3
2 3 4

inner

The transformer INNER generalizes the inner product operation of linear algebra. For a pair of lists A and B, the result is the application of the reductive operation f to the result of applying the binary pervasive operation g to A and B. It is assumed that f is one of the seven reductive operations of Nial: sum, product, or, and, max, min , or link and that g distributes pairwise. For tables, OUTER g is applied to the rows of A and the columns of B and f is applied to each item of the outer product.

Thus, INNER [+,*] is equivalent to matrix multiplication in linear algebra and INNER [or,and] is boolean matrix product. For higher dimensional arrays, the lists are formed by “pushing down” the last axis of A and the first axis of B.

     2 3 4 INNER [+,*] 4 5 6
47
     loll INNER [or,and] olol
l
     A := 2 4 reshape count 8;
     B := 4 3 reshape tell 8;
     A B
+-------+-----+
|1 2 3 4|0 1 2|
|5 6 7 8|3 4 5|
|       |6 7 0|
|       |1 2 3|
+-------+-----+
     A INNER [sum,times] B
28 38 24
68 94 64

Definition

     INNER IS TRANSFORMER f g (OPERATION A B) {­
        rows A OUTER (f g) (0 split B) }

Equation

   shape (A INNER [f,g] B) = (front shape A) link (rest shape B)

innerproduct

The operation innerproduct computes the mathematical inner product of real vectors and matrices using special code for efficiency. For real matrices, it produces the same result as A INNER [+,*] B, but computes the result more rapidly for large arguments. It coerces boolean and integer arrays to reals. The name ip is provided as an abbreviation.

     A := 2 4 reshape count 8;
     B := 4 3 reshape tell 8;
     A B
+-------+-----+
|1 2 3 4|0 1 2|
|5 6 7 8|3 4 5|
|       |6 7 0|
|       |1 2 3|
+-------+-----+

     A innerproduct B
28. 38. 24.
68. 94. 64.

The number of columns of A must match the number of rows in B.

     B ip A
?conform in ip

interrupt

An interrupt is an event that causes the operating system to suspend its operation and address a requirement of higher priority. Typically, interrupts occur to handle input/output. However, an interrupt also occurs when a fault is detected.

In the default mode of operation of Q’Nial, most fault values are not created. Rather, an interrupt is triggered. A description of the fault triggering mechanism is given under fault triggering.

The user can interrupt execution by pressing <Ctrl-c> <Return> at the keyboard in console versions, or clicking on the STOP button in the GUI version. This capability can be turned off using setinterrrupts.

inverse

The operation inverse computes the mathematical inverse of a square matrix A returning a square matrix of the same shape. If A is singular within numerical limits, the result is the fault ?singular. The name inv is provided as an abbreviation.

     seed 0.5;
     A := ceiling ( 100. * (3 3 reshape random 9))
50 44 13
78 74 90
98 67 17

     inverse A
-0.0736442  0.00189821  0.0462669
  0.115652 -0.00654341  -0.053798
-0.0312664   0.0148461 0.00413593

     A innerproduct inverse A
         1. -6.93889e-18 -7.11237e-17
 9.4369e-16           1. -1.97758e-16
3.60822e-16 -6.93889e-18           1.

The final computation shows that the result is not always an exact inverse due to roundoff errors introduced by using floating point arithmetic.

Equations

   A a square matrix ==> A innerproduct (inverse A) innerproduct A = A   (within roundoff error)
   inverse A innerproduct B = A solve B   (within roundoff error)

invocation of QNial

QNial is invoked using the following syntax:

    SYNTAX: nial  [(+|-)size Wssize] [-defs Filename] [-i] [-h]
    
    -size Wssize     Begin with a workspace size of Wssize words.
    M or K can be used to indicate millions or thousands respectively.
    The workspace expands if space is available.
    
    +size Wssize  Fix the workspace size at Wssize words with no expansion.
    
    -defs Filenm  After loading the initial workspace the file Filenm.ndf
    is loaded using loaddefs without displaying it.
    
    -lws Wsname   A previously saved workspace file is loaded on startup.
    
    -i    Execute in interactive mode with a top level loop.
    
    -h    Display command line syntax.
    
    Examples:
    
    nial -i
    
    nial -defs app.ndf
    
    nial +size 50M -defs newfns
    

isboolean

The operation isboolean tests whether or not A is a boolean atom. It returns true if A is a boolean, false otherwise.

     isboolean false
l
     isboolean llollool
o
     isboolean 7
o

Definition

isboolean IS OPERATION A {­ type A = o }

ischar

The operation ischar tests whether or not atom A is a character atom. It returns true if A is a character, false otherwise.

     EACH ischar (`a) ('a') 7
loo

In the example 'a' is not an atom; it is a solitary holding `a.

Definition

     ischar IS OPERATION A {­ type A = ` }

isfault

The operation isfault tests whether or not A is a fault. It returns true if A is a fault, false otherwise.

     EACH isfault (??error) ("?error)
lo

Definition

     isfault IS OPERATION A {­ type A = ?? }

isinteger

The operation isinteger tests whether or not A is an integer. It returns true if A is an integer, false otherwise.

     EACH isinteger (7)(2 3 4)(`3)(3.0)
looo

     isinteger (`a find 'whale')
l

The last example returns true because the result of `a find 'whale' is an integer, since the second argument of find is a list.

Definition

     isinteger IS OPERATION A {­ type A = 0 }

isphrase

The operation isphrase tests whether or not A is a phrase. It returns true if A is a phrase, false otherwise.

     isphrase "Mike
l
     isphrase 'abc'
o
     isphrase  `a
o

Definition

     isphrase IS OPERATION A {­ type A = "" }

isreal

The operation isreal tests whether or not A is a real number. It returns true if A is a real number, false otherwise.

     isreal 3.5
l

     isreal 'abc'
o

Definition

     isreal IS OPERATION A {­ type A = 0.  }

isstring

The operation isstring tests whether or not the array A is a string. It returns true if A is a string, false otherwise.

     isstring 'A string'
l
     isstring `A
o
     isstring `2 `A `?
l
     isstring ''
l

Definition

     isstring IS OP A {­ valence A = 1 and EACH ischar A }

Equations

   atomic A ==> isstring string A = True

item

An array A is said to be an item of array B if B holds A at one or more locations. The term is a relative one; we cannot speak of item except in reference to the array that holds it. The items of an array A are the objects at the locations at the top level.

The number of items in an array is called the tally of the array. Because an array is rectangular, the tally is the product of the shape.

The following names are give to common array structures:

# of items # of axes Name
0 1 empty list
1 0 single
1 1 solitary
1 2 1 by 1 table
2 1 pair
3 1 triple
4 1 quadruple

The arrays of Nial are a recursive data type. That is, the items of an array are also arrays. Since an array has arrays as items, it may contain data at lower levels than the top one. A path is a list of addresses that describes a data object at some depth within the array.

An array is said to be simple if all its items are atomic.

A part of an array is a data object that is contained at some level within the array. The atomic parts of an array are called the leaves of the array. The simple parts are called twigs . The term level is used informally to describe the relative position of a part within the nesting structure of an array. An item is at the first or top level, an item of an item is at the second level, etc.

An empty array is one that has no items.

iterate

The transformer ITERATE is used to apply the operation f sequentially to the items of A in row major order. (Row major order means across the rows moving left to right, starting at the top row and then going down the rows. An example is given under list.) The result is the result of applying f to the final item. If A has no items, the result is the fault ?noexpr.

     ITERATE write "Hello "out "there.
Hello
out
there.

     A IS EXTERNAL VARIABLE;
     accum IS OPERATION B {­
        NONLOCAL A; A := A + B };
     A := 20;
     ITERATE accum 3 4 5 ; A
32

In the second example, the operation accum increments variable A by the argument value. The example shows ITERATE being used to apply accum to the list 3 4 5, which results in A having the values 23, 27 and 32.

With ITERATE, an operation that has a side effect can be applied to an array of arguments sequentially in a specific sequence. On the other hand, the order of application is undefined for EACH.

A second major difference between EACH and ITERATE is that ITERATE returns only the result of the last application whereas EACH returns the array of all the results.

Definition

     ITERATE IS TR f OPERATION A {­ FOR X WITH A DO f A ENDFOR }

juxtaposition

The syntax rules for simple-expressions show three uses of the side-by-side or juxtapositional notation of Nial: strand formation, prefix operation application and infix operation application. There are no syntactic restrictions as to whether or not a particular operation may be applied in infix or prefix form. A fault is returned at run time if an operation is used inappropriately.

Summary of Juxtapositional Syntax

The following table illustrates the uses of juxtaposition in Nial, where A and B are array-expressions, f and g are operation-expressions, and T is a transformer:

Form Name Object
A B strand array
A f currying operation
f A prefix use array
f g composition operation
T f transform operation
A f B infix use array
T f A transform use array

laminate

The operation laminate merges the items of A adding a new axis before axis I of the items. The items of A must be of the same shape.

The following example creates a three dimensional array from two tables, placing the new axes at the front:

     0 laminate (tell 2 3) (count 2 3)
+---+---+---+  +---+---+---+
|0 0|0 1|0 2|  |1 1|1 2|1 3|
+---+---+---+  +---+---+---+
|1 0|1 1|1 2|  |2 1|2 2|2 3|
+---+---+---+  +---+---+---+

Definition

     laminate IS OPERATION I A {­
        IF equal EACH shape A THEN
           Axesofitems := axes first A;
           link (I take Axesofitems) (I drop Axesofitems + 1) blend A
        ELSE
           fault '?conform error in laminate'
        ENDIF }

last

The operation last returns the last of the items of A. If A is empty, it returns the fault ?address. The operation last is a special case of pick (because first is defined in terms of pick) and its behaviour is determined by that of pick. Every nonempty array has a last item.

     last 4 5 6
6

     last tell 3 4
2 3

Definition

     last IS OPERATION A {­ first reverse A }

Equations

   last single A = A
   last solitary A = A
   last list A = last A

latent

The expression Latent is used to name an expression to be executed without user intervention when the workspace is loaded. Latent is used in closed applications so that an application can be started when the workspace is loaded. Latent can establish any default or initial conditions desired.

     Latent IS {­
        settrigger o;
        set "log;
        StartApp;
        Bye; }

In the example, Latent is defined to turn of triggering of faults, to turn on session logging, to start the application and then to terminate the session.

leaf

The transformer LEAF modifies an operation f into an operation that applies f to every atom of the argument A. The result of applying LEAF f to A has the same shape as A. If f maps atoms to atoms, the result has the same structure as A.

     A := tell 2 3
+---+---+---+
|0 0|0 1|0 2|
+---+---+---+
|1 0|1 1|1 2|
+---+---+---+

     LEAF tally A
+---+---+---+
|1 1|1 1|1 1|
+---+---+---+
|1 1|1 1|1 1|
+---+---+---+

     B := (2 3) (1 2)
+---+---+
|2 3|1 2|
+---+---+

     LEAF tell B
+-----------+-------+
|+---+-----+|+-+---+|
||0 1|0 1 2|||0|0 1||
|+---+-----+|+-+---+|
+-----------+-------+

The first example shows that the result of applying a LEAF transform to a table is a table of the same shape. The atoms have been mapped to 1 since the tally of an atom is 1. The second example shows that applying a LEAF transform to a list gives a list of the same length. However, the structure of the result is not preserved in this example because tell maps an integer to a list of integers.

Definition

     LEAF IS TRANSFORMER f OPERATION A {­
        IF atomic A THEN
           f A
        ELSE
           EACH (LEAF f) A
        ENDIF }

Equations

   shape LEAF f A = shape A
   f unary pervasive ==> LEAF f A = f A
   (LEAF f) (LEAF g) A = LEAF (f g) A
   LEAF f list A = list LEAF f A

level

The term level is used informally to describe the relative position of a part within the nesting structure of an array. An item is at the first or top level, an item of an item is at the second level, etc.

An atom is viewed in two ways. As an indivisible data object it is viewed as having no levels and cannot be broken into subarrays. As an array data structure it is viewed as a single holding itself and therefore has an infinity of levels. This view is necessary for atomic arrays to fit the theory of nested array mathematics.

The number of levels to reach an atom along each path need not be the same. For example, in the following array, the phrase “hello is at the first level, the integer 23 is at the second level and the character `b is at the third level.

     [ 23 'abc', "hello , tell 2 2 ]
+------------+-----+-------------+
|+--+-------+|hello|+-----+-----+|
||23|+-+-+-+||     ||+-+-+|+-+-+||
||  ||a|b|c|||     |||0|0|||0|1|||
||  |+-+-+-+||     ||+-+-+|+-+-+||
|+--+-------+|     |+-----+-----+|
|            |     ||+-+-+|+-+-+||
|            |     |||1|0|||1|1|||
|            |     ||+-+-+|+-+-+||
|            |     |+-----+-----+|
+------------+-----+-------------+

Some of the operations of Nial that operate on simple arrays are extended to arbitrarily nested arrays by being applied to the atoms at the deepest level. These are called pervasive operations.

libpath

The variable Libpath is used by the operation library as a list of paths to directories that are to be checked for the definition file named in the library argument. The directories defined by Libpath are searched before the system dependent library directories.

When you call ‘library name’ each directory in sequence in your libpath will be joined to ‘name’ with the appropriate os path separator and tried. (with .ndf addded if necessary). so it depends on ‘open’. Names that include ‘.’ or ‘..’ will work but shell expansions like ‘~’ or ‘$abc’ will not. If you want to make it relative to your home directory then use the nial expression ‘os_get_parameter 2’, Different operating systems use different environment variable names for the home directory (HOME, USER etc)

     Libpath := ['mydefs','mylib\newdefs'];

library

The operation library loads a definition from the Q’Nial library of definition files. Nm is the name of a library file as a phrase or string. The name is augmented with the path information for the library and loaded into the workspace using loaddefs.

Sw, an optional argument, is either 0 (the default) or 1. If Sw is 0, there is no display of the file as it is loaded. If Sw is 1, the file is displayed as the file is read.

Some of the programs in the library are grouped in definition files by function. When such a file is specified, all the definitions in the file are loaded. If it is desired to load only one operation from a definition file which contains several operations, it is necessary to edit the library file in order to isolate the desired operation in one file.

     library "labeltable

Library searches for files using paths that are provided in its definition. Before searching for system defined directories, it searches in the directories named in the global variable Libpath, which is empty by default.

To modify Libpath, either edit the standard definitions file defs.ndf, or assign it a value dynamically.

like

The operation like compares two arrays A and B and returns true if all the items of A are items of B and vice versa. Otherwise it returns false. The operation like corresponds to set equality at the first level of nesting.

     2 3 like 3 2 2 3 2
l
     2 3 like 3 2 4
o
     'hello world' like 'whole door'
l

Definition

     like IS OPERATION A B {­ (A allin B) and (B allin A)}

The operation link returns the list of the items of the items of its argument. If it is applied to a pair of arrays A and B, the result is the items of A followed by the items of B.

If link is applied to an arbitrary array A, the items of the first item of A are followed by the items of the second item of A, etc.

If A is empty, the result is Null. If A is not empty but there are no items in all the items of A, the result is also Null.

     2 3 5 4 7 2 link 2 4 6 8
2 3 5 4 7 2 2 4 6 8

     link 'hen' 'hello' 'eh'
henhelloeh

The first example shows that the link of two lists of length 6 and 4 is a list of length 10. The second example illustrates the linking of three strings.

Link is similar to a set-union operation, although it does not remove duplicates in the representation. The composition of cull with link may be more appropriate as a union operation.

Equations

   link A = link list A
   link A = list link A
   link A = EACH list link A
   tally link A = sum EACH tally A
   link solitary A = list A
   link Null = Null
   link EACH link A = link link A
   cart link A = EACH link cart EACH cart A
   EACH f link A = link EACH EACH f A
   simple A ==> link A = list A
   and EACH simple A ==> content A = link A

list

The operation list returns the list of the items of A in row major order.

     A := 2 3 4 reshape count 24
1  2  3  4   13 14 15 16
5  6  7  8   17 18 19 20
9 10 11 12   21 22 23 24

     list A
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
24

     set "diagram; list "abc
+---+
|abc|
+---+

The first example shows row major order for a table. The second example shows that the list of an atom is the solitary holding the atom.

Definition

     list IS OPERATION A {­ tally A reshape A }

Equations

   shape list A = tally A
   list list A = list A
   list single A = solitary A
   EACH f list A = list EACH f A
   empty A ==> list A = Null

ln

The operation ln implements the natural logarithm function of mathematics. It produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean natural logarithm of the corresponding real
integer natural logarithm of the corresponding real
real real natural logarithm if A > 0.0; ?ln if A <= 0.0
character fault ?ln
phrase fault ?ln
fault argument A
     ln  l  2  3.5   `a  "abc  ??error
0. 0.693147 1.25276 ?ln ?ln ?error

Equation

   A > 0 ==> exp ln A = A   (within roundoff error)

load

The operation load is used to retrieve the saved workspace named by the phrase or string Wsname. The convention in Q’Nial is to save workspaces with a file name extension of the form .nws to make it easy to identify workspace files in the file system. When saving or loading a workspace, the extension may be omitted.

The effect of loading a workspace is to replace the current workspace with the saved one. To keep the contents of the current workspace, save should be used prior to doing a load.

The operation load may be used within a defined expression or operation. However, in such a use, it interrupts the execution of the operation or expression and does the load as though the operation were entered at top level.

If a workspace contains an expression with the name Latent, load executes Latent when the load is complete. This mechanism can be used to have the load of a workspace automatically begin an application or to chain execution of workspaces.

There is no mechanism to obtain individual objects saved in a workspace.

     load "mywork

loaddefs

The operation loaddefs is used to load Nial actions (definitions or expressions) from the file named by the phrase or string Deffilename. The convention in Q’Nial is to name Nial definition files with a file name extension of the form .ndf to make it easy to identify such files in the file system.

When loading a file with loaddefs, the file extension may be omitted. The operation may have an optional second argument to specify the mode of loading. If mode is 0 or if it is omitted, the actions are not displayed as they are processed. If it is 1, the actions are displayed.

The file of actions consists of groups of lines that are treated as a unit. These lines are separated by a blank line. If the first character in the first line of the group is

Several lines, up to a blank line, are treated as one long line. No blank spaces are inserted between the end of one line and the beginning of another. Thus, care should be taken to ensure that identifiers are not joined together.

If a syntax error is detected when a definition is being loaded, the definition is not installed. The absence of the erroneous definition may cause subsequent definitions in the file to fail also. If any errors are detected, the number of errors is displayed when loaddefs ends. Only the first syntax error found in each action is reported.

The backslash character, ` `, which is used to continue a line of a definition in immediate mode, must not be inserted at the end of lines in a definition file.

The tab characters, if they exist in a definition file, are treated as though they are space characters. Since some editor programs insert tabs automatically, the display of the file on input may be different from the display using the editor.

A definition file may contain uses of loaddefs. For a large application with many definition files, it is a good idea to have one file which can construct the application by using loaddefs to bring in the other files in an appropriate order. Loaddefs can also be used to execute a script of Nial expressions. This feature is convenient when simulating a Q’Nial session and capturing its output in a log file.

local environment

A local environment is a collection of associations that are known within a limited section of program text. These limited sections are formed by blocks, operation-forms and transformer-forms as discussed in the relevant sections below. A name that has a local association in one of these forms is said to have local scope.

Program fragments in which local variables are being assigned can be nested, so that one local scope surrounds another. A local association is not visible outside the construct in which it is defined; and a name with local scope can hide associations that the name has in surrounding scopes.

At any point in a program fragment, there is a current environment consisting of all names whose associations are visible. It includes the names having local scope in the program fragment being executed, names that are visible in the surrounding scopes and names that have global scope.

In program text, the scope of all names is determined by the static structure of the program text. The one exception is text that has the operation execute applied to it under program control.

In a local environment, a variable identifier can be chosen the same as a predefined or user-defined global definition name. Such a choice makes the global use of the name unavailable in the local context.

In any context, an identifier can name only one of: a variable, an array- expression, an operation-expression, or a transformer-expression. During one session, the role of a name, i.e. the class of syntactic object it names, cannot be changed.

If a block is used as a primary-expression, the local environment created by a block is determined by the block itself. If it is the body of an operation-form, the local environment includes the formal parameter names of the operation-form as variables.

A block delimits a local environment. It allows new uses of names which do not interfere with uses of those names outside the block. For example, within a block, a predefined operation name can be redefined and used for a different purpose. Only the reserved words of Q’Nial cannot be reused in this fashion. Definitions that appear within the block have local scope. That is, the definitions can be referenced only in the body of the block. Variables assigned within the block may or may not have local scope, depending on the appearance of a local and/or a nonlocal declaration. If there is no declaration, all assigned variables have local scope. Declaring some variables as local does not change the effect on undeclared variables that are used on the left of assignment. They are automatically localized.

If a nonlocal declaration is used, an assigned name that is on the nonlocal list is sought in surrounding scopes. If the name is not found, a variable is created in the global environment.

During the parse of the assign-expression appearing in a block, each name on the variable list is sought in the local environment. If the name exists in the local environment, the assignment affects the local association. If a name does not exist in the local environment and no reference has been made to a nonlocal variable with the same name, a local variable is created in the block. An assign-expression parsed in the global environment creates a global variable if a variable with that name does not already exist.

An operation-form defines a local environment. The formal parameter names are names of local variables. If the body of the operation form is a block, the local environment of the block is extended to include the formal parameters. When the operation is applied, the formal parameter names are assigned from the value of the actual argument. If there is only one formal parameter, the actual argument is assigned to it as a whole; otherwise, the items of the actual argument are assigned to the formal parameters in corresponding order. If there is a length mismatch between the list of formal parameter names and the values of the actual argument, the fault ?op_parameter is returned.

The value of the application of the operation is the value of the body of the operation-form, which is evaluated with the local variables in the parameter list assigned as described above. In determining the association for a name that appears in the body of an operation form, Q’Nial looks for the name in the local environment. If the name is not found locally, the name is sought in surrounding environments until it is found or until the global environment is searched. If it is not found, a fault ?unknown identifier: is given when the operation-form is analyzed (parsed).

Operation-forms are most frequently used in definitions where they are given an associated name. However, an operation-form can appear directly in an expression provided it is enclosed in parentheses. In this usage, it can be an argument to a transformer name or can be applied to an array argument.

The operation execute can be used within the execution of a block to make an assignment to variables or to invoke the definition mechanism. If execute is used to make a new definition or to create a new variable, the resulting variable or definition is placed in the global environment. However, if the block has local variables or local definitions, execute can be used to change a local version dynamically. A similar situation occurs with dynamic alteration of variables using assign.

log

The operation log implements the base 10 logarithm function of mathematics. It produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean base 10 logarithm of the corresponding real
integer base 10 logarithm of the corresponding real
real real base 10 logarithm if A > 0.0; ?log if A <= 0.0
character fault ?log
phrase fault ?log
fault argument A
     log  l  2  3.5  `a  "abc  ??error
0. 0.30103 0.544068 ?log ?log ?error

Equation

   A > 0 ==> 10. power log A = A (within roundoff error)

log file

Q’Nial provides a facility to record the actions in a session in a text file. The default name for the log file is auto.nlg. Logging is initiated by:

     set "log
nolog

Logging is ended by:

     set "nolog
log

The log file name can be changed using:

     setlogname "newname
auto.nlg

A log file is opened and closed on each usage by the internal logging routine. As a result, the log file is always available if the session is terminated unexpectedly. If a file with the name of the log file exists when set "log is executed, the logging information is appended at the end.

lower

The operation lower is used to partition an array A along its axes by indicating that the last N axes are to become axes of the items of the result. The remaining axes become the axes of the result. N must be an integer in the range from 0 to valence A. The result is an array of shape given by dropping the last N items of shape A. The items of the result have the shape given by taking the last N items of shape A. Thus, the 1 lower of an array of shape 3 4 2 is a 3 by 4 table of pairs. The 2 lower of the same array is a triple of 4 by 2 tables.

     A := 3 4 2 reshape 'ABCDEFGHIJKLMNOPQRSTUVWX'
AB  IJ  QR
CD  KL  ST
EF  MN  UV
GH  OP  WX
     1 lower A
+--+--+--+--+
|AB|CD|EF|GH|
+--+--+--+--+
|IJ|KL|MN|OP|
+--+--+--+--+
|QR|ST|UV|WX|
+--+--+--+--+
     2 lower A
+--+--+--+
|AB|IJ|QR|
|CD|KL|ST|
|EF|MN|UV|
|GH|OP|WX|
+--+--+--+

Definition

     lower IS OPERATION A {­ valence A - N raise A }

Equations

   N a nonnegative integer <= valence A and not empty A ==> shape (N lower A) = opp N drop shape A
   shape first (N drop A) = opp N take shape A ==> mix (N lower A) = A

lt

The operation lt compares two atoms A and B with the less than relation, returning true if A is less than B and false otherwise. Symbol < is a synonym for lt.

The atoms in Nial are organized as a lattice using <= for the ordering. The numeric atoms are comparable across types but literal atoms are not comparable with numeric atoms. The literal types are not comparable across types. A comparison between incomparable objects results in false.

     R := l 2 2.5 `a "abc ??error
l 2 2.5 a abc ?error
     R OUTER < R
ollooo
oolooo
oooooo
oooooo
oooooo
oooooo

     'apple' < 'above'
ooolo

     "apple < "above
o

The use of OUTER < shows the comparisons between various atom types. The last two examples show the difference between comparing two strings, where the operation is distributed by its pervasive property; and comparing the corresponding phrases.

Definition

     lt IS OPERATION A B ( ( A lte B ) and not ( A mate B ) )

lte

The operation lte compares two atoms A and B with the less than or equal relation, returning true if A is less than B and false otherwise. The symbol <= is a synonym for lte.

The atoms in Nial are organized as a lattice using <= for the ordering. The numeric atoms are comparable across types but numeric and literal atoms are incomparable. The literal types are not comparable across types. A comparison between incomparable objects results in false.

     R := l 2 2.5 `a "abc ??error
l 2 2.5 a abc ?error

     R OUTER <= R
lllooo
ollooo
oolooo
oooloo
oooolo
oooool

     'apple' <= 'above'
loolo

     "apple < "above
o

The use of OUTER <= shows the comparisons between various atom types. The last two examples show the difference between comparing two strings, where the operation is distributed by its pervasive property; and comparing the corresponding phrases.

match

The operation match compares two atoms A and B for exact equality, returning true if A is equal to B and false otherwise. Match cannot be used to determine the equality of atoms of different numeric or literal type, as when comparing the real number 3.0 with the integer 3.

     3 match 3
l

     3.0 match 3
o

     "o match `o
o

     1.0 match l 1 1.0
ool

     `  match 'a list of letters'
olooooloolooooooo

     1 match tell 2 3
+--+--+--+
|oo|ol|oo|
+--+--+--+
|lo|ll|lo|
+--+--+--+

The second last example shows that the binary pervasive extension of match allows a lists of atoms to be compared for a value, returning a bitstring of results. The last example shows where in the structure of tell 2 3 a 1 is held.

mate

The operation mate compares two atoms A and B for equality with type coercion, returning true if A is equal to or can be coerced to B as a number; and false otherwise. Mate is used to determine the equality of atoms of different numeric type, as when comparing the real number 3.0 with the integer 3. It cannot be used to determine equality of literal types.

     3 mate 3
l

     3.0 mate 3
l

     "o mate `o
o

     1.0 mate l 1 1.0
lll

The last example shows that the binary pervasive extension of mate allows a lists of atoms to be compared for a value, returning a bitstring of results.

Definition

     mate IS OPERATION A B {­ (A lte B) and (B lte A) }

max

The operation max, when applied to a simple array A, finds the least atom that is greater than or equal to all the items. If the atoms of A are all positive numbers, for example, max A returns the biggest. If the items are not comparable, the result is the fault ?I, the atom in the lattice of Nial atoms that is greater than or equal to all other atoms (the zenith). If A is empty, the result is the fault ?O, the atom that is less than or equal to all other atoms (the nadir).

Max is extended to arbitrary arrays by its multi pervasive behaviour. It is a reductive operation in that it reduces a simple array to a single atom. Applied to a pair of simple arrays, it produces a simple array with the corresponding items compared.

If the items of A are all numeric type, they are comparable. The result is the highest numeric type represented in the array, where boolean is the lowest numeric type and real number is the highest.

     max 3 45 23 18 3.5
45.
     max 3 "abc
?I
     max "abc "def "c
def
     max 'apples' 'orange'
orpngs

The first example shows that max of a list of numbers of different type is the maximum number, coerced to the highest type. The second shows that the integer 3 and the phrase “abc” are incomparable and the result is the fault ?I. The third example shows that phrases are directly comparable. The last shows that strings are compared on a character by character basis.

Equations

   A max B = B max A
   max EACH max A =f= max link A
   atomic A ==> max B lte A = and (B EACHLEFT lte A)
   max Null = ??O

min

The operation min, when applied to a simple array A, finds the greatest atom that is less than or equal to all the items. If the atoms of A are all positive numbers, for example, min A returns the smallest number. If the items are not comparable, the result is the fault ?O, the atom in the lattice of Nial atoms that is less than or equal to all other atoms (the nadir). If A is empty, the result is the fault ?I, the atom that is greater than or equal to all other atoms (the zenith).

Min is extended to arbitrary arrays by its multi pervasive behaviour. It is a reductive operation in that it reduces a simple array to a single atom. Applied to a pair of simple arrays, it produces a simple array with the corresponding items compared.

If the items of A are all numeric type, they are comparable. The result is the highest numeric type represented in the array, where boolean is the lowest numeric type and real number is the highest.

     min 3 45 23 18 3.5
3.
     min 3 "abc
?O
     min "abc "def "c
abc
     min 'apples' 'orange'
apalee

The first example shows that min of a list of numbers of different type is the minimum number, coerced to the highest type. The second shows that the integer 3 and the phrase “abc” are incomparable and the result is the fault ?I. The third example shows that phrases are directly comparable. The last shows that strings are compared on a character by character basis.

Equations

   A min B = B min A
   min EACH min A = min link A
   atomic A ==> A lte min B = and (A EACHRIGHT lte B)
   min Null = ??I

minus

The operation minus returns the result of subtracting two numeric atoms. It coerces the type of the atoms to be the higher type, if they differ; or to integer, if both are boolean. The result is of the type of the coerced arguments.

The symbol - is a synonym for minus. Care must be taken when using this symbol because it also is used to form negative constants and the latter use has higher priority. Thus,

     A := 10;
     A - 1
9

does a subtraction, but

     A -1
10 -1

forms a pair.

If one argument is numeric and the other is a fault or if both arguments are the same fault, the answer is the fault. In all other cases when one or more of the arguments is not numeric, the result is the arithmetic fault ?A.

     R gets l 2 2.5 `a "abc ??error
l 2 2.5 a abc ?error

     R OUTER minus R
     0      -1    -1.5   ?A  ?A  ?error
     1       0    -0.5   ?A  ?A  ?error
   1.5     0.5      0.   ?A  ?A  ?error
?A      ?A      ?A       ?A  ?A  ?A
?A      ?A      ?A       ?A  ?A  ?A
?error  ?error  ?error   ?A  ?A  ?error

The above example illustrates all combinations of atom types for the two arguments to minus.

Equations

   0 minus A = opposite A
   A minus B = A plus opposite B

mix

The operation mix is intended to be used on an array A with items which are all the same shape. In this case, the shape of the result is formed by the shape of A linked with the shape of the items. The list of items of the result is the link of the items of A. If A does not have equally shaped items, the fault ?conform is returned.

     mix (2 3) (4 5) (6 7)
2 3
4 5
6 7

     A := (tell 2 3)(count 2 3)
+-------------+-------------+
|+---+---+---+|+---+---+---+|
||0 0|0 1|0 2|||1 1|1 2|1 3||
|+---+---+---+|+---+---+---+|
||1 0|1 1|1 2|||2 1|2 2|2 3||
|+---+---+---+|+---+---+---+|
+-------------+-------------+

     mix A
+---+---+---+  +---+---+---+
|0 0|0 1|0 2|  |1 1|1 2|1 3|
+---+---+---+  +---+---+---+
|1 0|1 1|1 2|  |2 1|2 2|2 3|
+---+---+---+  +---+---+---+

The first example shows that the result of mixing a triple of pairs is a 3 by 2 table. The second example shows that mixing a pair of 2 by 3 tables is a 2 by 2 by 3 array, with the tables as the planes of the array.

The operation mix is the left inverse of the operations rows and raise for nonempty arrays. It is often used to transform from a list of lists view of data to a table view of data. In practice, transformations between the two representations on large arrays should be avoided in Q’Nial since considerable work is involved in doing the restructuring.

Definition

     mix IS OPERATION A {­
        IF empty A THEN
           shape A append 0 reshape A
        ELSEIF not equal EACH shape A THEN
           ??conform
        ELSE
           shape A link shape first A reshape link A
        ENDIF }

Equations

   not empty A and equal EACH shape A ==> shape mix A = link (shape A)(shape first A)
   equal EACH shape A ==> list mix A = link A
   not empty A ==> mix rows A = A
   I in (axes A + 1) and not empty A ==> mix (I raise A) = A

mod

The operation mod returns the remainder of dividing integer A by integer B. If the divisor B is zero, the result is A. If it is negative, the result is the fault ?negative divisor.

The operation mod coerces a boolean argument to an integer. It produces a fault if either argument is not an integer or a boolean.

     R gets l 2 2.5 `a "abc ??error
l 2 2.5 a abc ?error
     R OUTER mod R
     0       1  ?A      ?A  ?A  ?error
     0       0  ?A      ?A  ?A  ?error
?A      ?A      ?A      ?A  ?A  ?error
?A      ?A      ?A      ?A  ?A  ?A
?A      ?A      ?A      ?A  ?A  ?A
?error  ?error  ?error  ?A  ?A  ?error

     (5 mod 3)(-5 mod 3)(5 mod -3)(5 mod 0)
2 1 ?negative divisor 5

The first example illustrates all combinations of atom types for the two arguments to mod.

The remainder on division by a positive integer B is always a number between 0 and B - 1. If A is negative, the integer returned obeys the rule of modular arithmetic as described in the definition. The result of A mod 0 is A rather than a fault, since that is the preferred result in mathematics.

Definition

     mod IS OPERATION A B {­ floor (A - (B * (A quotient B)) ) }

Equation

   isinteger A ==> A mod 0 = A

mold

The operation mold is provided for backward compatibility with earlier versions of Nial. The current definition is a synonym for take. The previous definition was a reshaping operation that was almost always used on lists and take has the same effect as the previous definition on lists.

Definition

   mold is take

multi pervasive

Each operation f in this class reduces a simple array (an array of atoms) to an atom.

A multi pervasive operation maps an array having items of identical structure to one with the same structure, applying the operation to the simple arrays formed from all the atoms in corresponding positions in the atoms.

There are six operations in this class. They are the reductive operations of arithmetic and logic.

If a multi pervasive operation is applied to an array that does not have items of the same shape, the effect is to build a conformable pair by replicating an atom or solitary item of the array to the common shape of the other items. If two or more of the items have tally different from one and are of unequal shape, the fault ?conform is returned. On a pair, a multi pervasive operation behaves the same way as a binary pervasive operation does.

     sum [3 4 5 6 , 5 , 7 8 9 10] = 15 17 19 21
l

The following table describes the multi pervasive operations.

Operation Function
and logical “and” of a boolean array
max highest item in the array
min lowest item in the array
or logical “or” of a boolean array
product arithmetic product of an array of numbers
sum arithmetic sum of an array of numbers

Equations

   f A = EACHALL f A
   f A = EACH f pack A
   shape f A = shape pack A
   f A = REDUCE f A

nested definition

A nested definition is one that appears in a definition sequence within a block. Its name is a local definition name. If the name is also used outside the block, the external meaning is not known in the block.

Nested definitions can be used to encapsulate support definitions within a larger definition that is to be made available to other users. This avoids cluttering up the name space with names that might interfere with the user’s other work. It is often easier to develop the large definition without encapsulation and package it in encapsulated form once the design is completed.

next

The command next is used in debugging a definition that has been suspended using Break or <Ctrl B>. The effect of next is to execute the next expression in an expression sequence and to suspend execution again. If the current expression involves a call on a defined operation or expression, execution of the entire definition is completed and execution is suspended on the expression after the current one. If the current expression is the last one in the expression sequence where the break began, the effect is the same as using resume.

The related command nextv displays the result of the expression executed on a step before displaying the next expression.

no_expr

The expression No_expr returns the fault ?missing_expr. It is the default value for an expression name declared to be external or that has been erased.

no_op

The operation no_op returns the fault ?missing_op when applied to any array A. It is the default value for an operation name declared to be external or that has been erased.

no_tr

The transformer NO_TR transforms any operation f to an operation that returns the fault ?missing_tr when applied to any array A. It is the default value for a transformer name declared to be external or that has been erased.

no_value

The fault ?no_value is a special fault value used as the default value of variables that have been declared external or that have been erased.

not

The operation not reverses the value of a boolean. It returns true if A is false and false if A is true. If A is any other atom, the result is the logical fault ?L.

     not l
o
     not lollo
olool
     not 2 "abc o
?L ?L l

Equations

   not not not A = not A
   not and A =f= or EACH not A
   not or A =f= and EACH not A

notin

The operation notin returns true if A is not an item of B and returns false if it is.

     3 notin 56 34 23 3 57 3
o

     4 1 notin tell 3 4
l

     `a notin 'hello world'
l

     `a notin "apple
l

Definition

     notin IS OPERATION A B {­ not (A in B) }

Equation

   A notin B = A notin list B

null

The expression Null denotes the empty list. It is the array returned as the shape of a single and is equal to the empty string. The empty list notation ` [ ] also denotes the empty list Null`.

     Null = shape 5
l
     Null = []
l
     Null = ''
l

Since there is an empty array for each shape containing a zero, unless you are certain A is a list, the test: A = Null should not be used in place of empty A.

     set "sketch; Null

     set "diagram; Null
+
|
+

Null is not displayed when sketch display mode is set. In diagram mode it is displayed as the left border of a list diagram.

A selection operation such as pick or first applied to Null results in the fault ?address.

Definition

     Null IS shape 0

Equations

   shape single A = Null
   shape Null = 0
   atomic A ==> Null = shape A

numeric

The operation numeric tests whether or not A is a numeric atom. It returns true if A is a boolean atom, an integer atom or a real atom, false otherwise.

     numeric 3.5
l
     numeric l 45 3.78
o
     numeric Null
o

Definition

numeric IS OPERATION A {­ isboolean A or isinteger A or isreal A }

numeric type hierarchy

The three numeric types: boolean, integer and real, are organized in a hierarchy in the order mentioned. For arithmetic and comparative binary operations, if the types of arguments are both numeric but differ in type, they are coerced to the higher of the two types. For example, if you add an integer to a real number, say 3 + 4.5, the integer is coerced to 3.0 and a real number addition is done. Boolean atoms are always coerced to integers if they are used with arithmetic operations.

There are six types of atoms in Nial. They are boolean, integer, real, character, phrase and fault. The first three are numeric types and are used for arithmetic operations. The last three are literal types and are used for symbol manipulation. All six types of atoms are used in comparisons.

A boolean atom is the result of a comparison of array values; or the result of a test relating to a characteristic of an array or the content of an array. There are two boolean atoms: true and false, denoted by l and o respectively. When booleans are treated as numbers, true corresponds to one and false to zero.

An integer atom is a positive or negative whole number representing a quantity of units. A dash symbol (-) immediately preceding the integer denotes a negative integer. No space is permitted between the dash and the number, otherwise the dash is interpreted as the arithmetical operation of subtraction. Conversely, a space is required when subtraction is intended. An integer is represented by an internal form that limits its range of values in the Q’Nial implementation of Nial.

A real atom is a number which can represent any position on the real line. It may be written with a fractional part and/or with a decimal exponent. It is represented internally by a floating point number.

open

The operation open prepares the host file named by the phrase or string Filename for use in one of seven possible modes. Mode is a character, string or phrase encoding the mode as follows:

Mode Mode Name Description
r read existing sequential text file, read only
w write create sequential text file, write only
a append existing sequential text file, write only starting at the end
d direct existing or new binary file, direct access for both read and write
c communication existing or new sequential text file, for both read and write
pr pipe read output from the command Filename is available
pw pipe write input to the command Filename is provided

In read, write, append or communication modes, Filename must include the file extension, if any. In direct mode, two host files are opened, one with extension .ndx and the other with extension .rec, standing for index and record, respectively. In this mode, Filename is the root of the file name, without any extension.

The communication mode can be used with special forms of readfile and writefile usage to access a special device or a communications port. This mode is available only on the EXTDOS Version of Nial.

The pipe modes treat the Filename as a host command. The open causes the command to be executed. In pipe read, the output from the command is obtained using readfile; in pipe write, input to the command is provided using writefile.

The result of open is a file designator, an integer specifying an input/output port. This value is used in later references to the file.

The effect of open may be system dependent. If the file exists and can be opened by the user in the requested mode, the result will be the file designator. If it cannot be opened or does not exist, the result will be a fault message generated from the error message provided by the host system. For portable programs, the specific fault message should not be used in an equality test. The result of an open should be tested for a fault value using isfault to catch unexpected failures.

operation

An operation is a functional object that is given an argument array and returns a result array. The process of executing an operation by giving it an argument value is called an operation call or an operation application.

An operation can be constructed by defining one or more parameters and giving an algorithm to compute the result in terms of the parameters. An operation is usually given a name when it is defined. There are also program fragments that construct unnamed operations by composing operations, forming a list of operations or modifying an operation by use of a transformer.

A named operation is either a predefined operation or an operation that has been given an explicit name using the IS definition mechanism.

operation composition

A sequence of simple-operation denotations forms an operation-sequence in the syntax of Nial. Its meaning is the same as the composition of the operations in the sequence which is explained by its effect when applied.

The result of applying an operation-sequence to an argument is determined by applying the simple-operations in the sequence in right-to-left order. The simple-operation on the right is applied to the argument giving an intermediate result. Then the simple-operation to the immediate left is applied to the result of the first application. Subsequent simple-operations are applied to the results in turn.

An example of this concept is the expression

     (first rest) 8 7 2 5 3
7

The expression sequence first rest is evaluated by applying rest to the argument resulting in 7 2 5 3 and then first is applied giving 7.

The relationship can be described by the equations:

   (f g) A = f (g A)
   (f g h) A = f (g (h A))
   etc.

which is mathematically known as the rule of function composition .

operation form

An operation-form is the syntactic structure used to describe an operation in terms of a parameterized expression-sequence. The identifiers following the keyword operation are called the formal parameters . The body of an operation-form is normally a block but it may be an expression-sequence without automatic localization.

     operation-form ::=
        OPERATION {­ identifiers }+ block
        | OPERATION {­ identifiers }+
          ( expression-sequence )

An operation-form defines a local environment. The formal parameter names are names of local variables. If the body of the operation form is a block, the local environment of the block is extended to include the formal parameters. When the operation is applied, the formal parameter names are assigned from the value of the actual argument. If there is only one formal parameter, the actual argument is assigned to it as a whole; otherwise, the items of the actual argument are assigned to the formal parameters in corresponding order. If there is a length mismatch between the list of formal parameter names and the values of the actual argument, the fault ?op_parameter is returned.

The value of the application of the operation is the value of the body of the operation-form, which is evaluated with the local variables in the parameter list assigned as described above. In determining the association for a name that appears in the body of an operation form, Q’Nial looks for the name in the local environment. If the name is not found locally, the name is sought in surrounding environments until it is found or until the global environment is searched. If it is not found, a fault ?unknown identifier: is given when the operation-form is analyzed (parsed).

Operation-forms are most frequently used in definitions where they are given an associated name. However, an operation-form can appear directly in an expression provided it is enclosed in parentheses. In this usage, it can be an argument to a transformer name or can be applied to an array argument.

opposite

The operation opposite returns the opposite value of the integer or real number A. The opposite value of a positive number A is the corresponding negative value and vice versa. The operation produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean opposite of the corresponding integer
integer opposite integer
real opposite real number
character fault ?A
phrase fault ?A
fault argument A

The name opp is a synonym for opposite used as an abbreviation.

     opposite  l  -2  3.5  `a  "abc  ??err
-1 2 -3.5 ?A ?A ?err

In Nial, the symbol - is used as part of negative numbers and as a synonym for minus. Care must be taken to use a space after the symbol before a constant to get the expected subtraction.

Definition

     opposite IS OPERATION A (0 minus A)

Equations

   opposite opposite opposite A = opposite A
   A - B = A + opposite B

ops

The expression Ops returns a list of phrases giving the names of all user defined operations in the workspace.

Definition

     Ops IS {­
        Names Roles := pack symbols 0;
        "op match Roles sublist Names }

or

The operation or applies to a boolean array A and does the boolean sum of its items. If any item of A is true, the result is true; otherwise it is false. Used in binary form, it implements the usual or-connective of logic. If A is a simple array but has a non-boolean item, the result is the logical fault ?L. The operation extends to non-simple arrays by the multi pervasive mechanism.

     l or o
l

     or oooolooo
l

     lloo or lolo
lllo

Or is a reductive operation in that it reduces an array of booleans to a single boolean. Applied to a pair of bitstrings, it produces a bitstring resulting from applying or to the bits of its arguments in corresponding positions.

Equations

   A or B = B or A
   not or A =f= and EACH not A

outer

The transformer OUTER transforms an operation f to an operation that applies f to each item of the cartesian product of its argument. In the case where the arguments are a pair of lists A and B, the result is a table of the applications of f with every pair chosen with the first item from A and the second from B.

     1 2 3 4 5 OUTER * 1 2 3 4 5
1  2  3  4  5
2  4  6  8 10
3  6  9 12 15
4  8 12 16 20
5 10 15 20 25

     2 3 OUTER reshape 'abcd'
+---+---+---+---+
|aa |bb |cc |dd |
+---+---+---+---+
|aaa|bbb|ccc|ddd|
+---+---+---+---+

The first example shows OUTER * being used to generate a multiplication table. The second shows that it can be applied to an operation that has a non-atomic result.

Definition

     OUTER IS TRANSFORMER f OPERATION A (EACH f cart A)

Equations

   shape OUTER f A = link EACH shape A
   and EACH (not empty) A B ==> A OUTER f B = mix (A EACHLEFT EACHRIGHT f B)

overflow

The term integer overflow is used to describe the situation when an integer that is too big for the internal representation of integers is generated. Q’Nial uses 32 bit integers and checks for overflow during computations. If an arithmetic computation produces an overflow, a fault value is produced.

An overflow can also develop during computation with real numbers. The computer hardware detects such a problem and an interrupt occurs which causes Q’Nial to jump to top level.

pack

The operation pack interchanges the top two levels of an array with equally shaped items. If A is an array with items all of the same shape, the result is an array of that shape, with items having the shape that A has. If A does not have items of the same shape but all the items that have zero items or more than one item are of the same shape, it is modified by replicating the items with one item to that shape before interchanging levels. Otherwise it returns the fault ?conform.

The items of the result are the arrays of items of A from corresponding positions. Pack is used implicitly in all binary pervasive and multi pervasive operations.

     pack (2 3 4) (4 5 6)
+---+---+---+
|2 4|3 5|4 6|
+---+---+---+
     pack 'ab' 'cd' 'ef' 'gh'
+----+----+
|aceg|bdfh|
+----+----+
     pack (tell 2 3) (count 2 3)
+---------+---------+---------+
|+---+---+|+---+---+|+---+---+|
||0 0|1 1|||0 1|1 2|||0 2|1 3||
|+---+---+|+---+---+|+---+---+|
+---------+---------+---------+
|+---+---+|+---+---+|+---+---+|
||1 0|2 1|||1 1|2 2|||1 2|2 3||
|+---+---+|+---+---+|+---+---+|
+---------+---------+---------+

     pack (2 3 4) 5 [2 3] 'abc'
+-----------+-----------+-----------+
|+-+-+---+-+|+-+-+---+-+|+-+-+---+-+|
||2|5|2 3|a|||3|5|2 3|b|||4|5|2 3|c||
|+-+-+---+-+|+-+-+---+-+|+-+-+---+-+|
+-----------+-----------+-----------+

The first example shows that pack of a pair of triples is a triple of pairs. The second shows that pack of a quadruple of pairs is a pair of quadruples. The third shows that pack of a pair of 2 by 3 tables is a 2 by 3 table of pairs. The final example shows a case where replication occurs. The first and last items have shape [3]. The second and third items are an atom and a solitary and hence can be replicated.

Equations

   equal EACH shape A and not empty pack A ==> pack pack A = A
   pack pack pack A = pack A
   pack single A = EACH single A
   pack solitary A = EACH solitary A
   empty A ==> pack A = single A
   simple A ==> pack A = single A

pair

The operation pair has no effect on a pair but converts all other arrays to a pair by reshaping the array with shape 2. Used in infix notation, it has the effect of forming a pair from the two arguments. An explicit pair can also be achieved with list or strand notation. The operation is useful in functional expressions where it is an argument to a transformer.

     3 pair 4 5
+-+---+
|3|4 5|
+-+---+
     pair 3 4 5
3 4

     pair single 3 4 5
+-----+-----+
|3 4 5|3 4 5|
+-----+-----+

The major use of pair is as an operation applied using a transformer.

Definition

     pair IS OPERATION A (2 reshape A)

Equations

   pair pair A = pair A
   A OUTER pair B = A cart B

parse

The operation parse carries out the internal parsing capability of the interpreter. Its argument is a token stream produced by scan or deparse. The result is the parse tree recording the semantic intent of the input. The parse tree is a highly encoded form of nested array designed for efficient processing.

     parse scan 'sum 23 45'
+---+-----------------------------+
|100|+-+-----+-------------------+|
|   ||4|3 4 3|+-+-------+-------+||
|   || |     ||8|1 23 23|1 45 45|||
|   || |     |+-+-------+-------+||
|   |+-+-----+-------------------+|
+---+-----------------------------+

     parse scan 'foo IS first'
+---+-----------------------------+
|100|+--+---------------+------+-+|
|   ||11|+--+----------+|3 47 3|1||
|   ||  ||22|2 160 7459||      | ||
|   ||  |+--+----------+|      | ||
|   |+--+---------------+------+-+|
+---+-----------------------------+

The first example gives the parse tree corresponding to an application of an operation to a strand of constants. The second example gives the parse tree for the definition of foo as the predefined operation first. The small integers that appear as the first items of the arrays serve as tags to indicate the kind of node that is represented in the parse tree. They are used by the eval and deparse operations. The representation of parse trees is implementation dependent and is not intended for program manipulation.

Equations

   Ts a token stream ==> parse deparse parse Ts = parse Ts
   S a string ==> eval parse scan S = execute S

partition

The transformer PARTITION is used to apply operation f to partitions of the array A, placing the results of the applications as partitions of the result array.

The partitions of A are determined by the axis numbers in I, using ( I split A) to form them implicitly. The results of applying f, viewed as an intermediate array R, are combined into the final result, using ( J blend R), to place the axes of the items of R into the final result. If the left argument is a solitary, it is assumed to be used as the argument to both split and blend implicitly.

For a table A, I = 1, f being reverse, the effect is to reverse the rows of A. In general, if I = J and is a single axis number and f maps a list to one of the same length, the effect is to apply f to the lists formed by pushing axis I down, resulting in an array of the same shape as A. Two dimensional partitions of a three dimensional array can be mapped using an I with tally 2.

     [1] PARTITION reverse (3 5 reshape count 15)
 5  4  3  2  1
10  9  8  7  6
15 14 13 12 11

     [0] PARTITION (2 rotate) (6 6 reshape tell 36)
12 13 14 15 16 17
18 19 20 21 22 23
24 25 26 27 28 29
30 31 32 33 34 35
 0  1  2  3  4  5
 6  7  8  9 10 11

     setformat '%4.2f';
     seed .5 ; A:=2 3 3 reshape random 18
0.50 0.43 0.12   0.03 0.81 0.24
0.77 0.73 0.89   0.58 0.98 0.47
0.98 0.66 0.16   0.24 0.16 0.50
     B gets 1 2 split A
+--------------+--------------+
|0.50 0.43 0.12|0.03 0.81 0.24|
|0.77 0.73 0.89|0.58 0.98 0.47|
|0.98 0.66 0.16|0.24 0.16 0.50|
+--------------+--------------+
     C gets EACH inverse B
+-----------------+-----------------+
|-7.43  0.18  4.69|-2.51  2.21 -0.90|
|11.78 -0.62 -5.55| 1.09  0.25 -0.76|
|-3.27  1.48  0.51| 0.83 -1.13  2.69|
+-----------------+-----------------+
     1 2 blend C
-7.43  0.18  4.69   -2.51  2.21 -0.90
11.78 -0.62 -5.55    1.09  0.25 -0.76
-3.27  1.48  0.51    0.83 -1.13  2.69
     [1 2] PARTITION inverse A
-7.43  0.18  4.69   -2.51  2.21 -0.90
11.78 -0.62 -5.55    1.09  0.25 -0.76
-3.27  1.48  0.51    0.83 -1.13  2.69

The transformer RANK is a simpler version of a partitioning transformer that always pushes down the right end axes.

Definition

PARTITION IS TRANSFORMER f OPERATION Ij A {­
   IF empty Ij or (tally Ij > 2) THEN
      fault 'invalid left arg of PARTITION transform'
   ELSEIF empty A THEN
      A
   ELSE
      IF tally Ij = 1 THEN
         I := J := first Ij;
      ELSEIF tally Ij = 2 THEN
         I J := Ij;
      ENDIF;
      II gets axes A except I link I;
      B := tally I RANK f (II fuse A);
      IF J = Null and (shape B = Null) THEN
         first B
      ELSEIF tally J = (valence B - (valence A - tally I)) THEN
         JJ gets axes B except J link J;
         GRADE <= JJ fuse B
      ELSE
         fault 'left arg incompatible with function in PARTITION transform'
      ENDIF
   ENDIF }

Equations

   I split A = \[I,Null] PARTITION single A
   I blend A = \[Null,I] PARTITION first A
   EACH f A = \[Null,Null] PARTITION (single f first) A

pass

The operation pass is the identity operation for arrays. It returns the argument A. Pass is used when expressing computations in a functional style using an atlas (a list of operations).

     pass 2 3 4
2 3 4
     pass 'hello world'
hello world

Definition

     pass IS OPERATION A {­ A }

Equations

   pass A = first single A
   pass A = first solitary A

paste

The operation paste combines character tables in a two dimensional layout. It is used internally by picture and can be used to create modifications of the standard picture output to improve the display of results. The left argument Sw has six parameters that control the layout and the right argument A is the array of tables to be combined. The components of argument Sw are the following:

Argument  
Vertical edge spacing an integer indicating the number of blank lines to be put between the rows of items
Horizontal edge spacing an integer indicating the number of blank spaces to be put between the columns of items
Vertical line indicator If the value is 1, vertical lines are drawn between columns. If it is 0, they are not
Horizontal line indicator If the value is 1, horizontal lines are drawn between rows. If it is 0, they are not
Vertical justification indicator This is either a single integer indicating the same justification for all fields or an array of integers of the same shape as A indicating the justification for each item of A. The codes are: 0 for top justification, 1 for centered and 2 for bottom justification
Horizontal justification indicator This is either a single integer indicating the same justification for all fields, or an array of integers of the same shape as A indicating the justification for each item of A. The codes are: 0 for left justification, 1 for centered and 2 for right justification

If either line indicator is 1, the entire result is boxed.

     1 2 1 3 1 1 paste EACH sketch (2 4 reshape 'some' 'words' 'of' 'wisdom')
+--------+---------+------+----------+
|        |         |      |          |
|  some  |  words  |  of  |  wisdom  |
|        |         |      |          |
+--------+---------+------+----------+
|        |         |      |          |
|  some  |  words  |  of  |  wisdom  |
|        |         |      |          |
+--------+---------+------+----------+

pervasive

Some of the operations of Nial that operate on simple arrays are extended to arbitrarily nested arrays by being applied to the atoms at the deepest level. These are called pervasive operations.

There are three classes of pervasive operations: unary pervasive, binary pervasive and multi pervasive. The first class applies the operations to the atoms of the array. The second class are binary operations that apply to pairs of atoms from corresponding positions in the pair of arguments. The third class applies the operation to the simple arrays formed from atoms in corresponding positions in the items of the argument, reducing the simple array to a single atom.

phrase

The operation phrase converts a string into a phrase with the same content. It returns phrases and faults unchanged and converts other atoms after coercing them to their string equivalents. The resulting phrase may contain blanks. There is a one-to-one correspondence between phrases and strings.

     phrase '(613) 549-2222'
(613) 549-2222

     set "decor ; EACH phrase o 613 3.5 `A 'Now' "Wow ?error
"o "613 "3.5 "A "Now "Wow "error

Equations

   isphrase P ==> phrase string P = P

Pragmatics

The correspondence between phrases and strings is not complete in that phrase cannot hold a null character (representation 0) and attempting to build such a phrase using the operation phrase will result in a truncated phrase.

pi

The expression Pi returns the real number which is the ratio of the circumference of a circle to its diameter. Pi is useful in scientific computing.

     setformat '%15.12f' ; Pi
 3.141592653590

Equation

   Pi = arccos -1 (within roundoff error)

pick

Pick is the fundamental selection operation for arrays. The normal use is that I is an address for A. The result is the item that is located at address I in 0-origin addressing. If I is a solitary integer and A is a list, I is converted to an integer to make it an address for a list. If I is not a valid address for A, the result is the fault ?address.

     3 pick 3 6 23 46 57 25
46

     [3] pick 3 6 23 46 57 25
46

     A := 2 3 reshape "a "list "of "words "as "phrases
a     list of
words as   phrases

     0 2 pick A
of

     5 5 pick A
?address

In the first example, pick selects the fourth item because of 0-origin indexing. In the second example, [3] is not an address of the list (its addresses are integers) but it is converted to integer 3. Thus, a solitary integer can be used as an address of a list for pick. The third example shows that an address for a table is a pair of integers. The last example shows that picking out of range returns a fault.

The concept of picking closely corresponds to subscript notation in mathematics. Q’Nial supports a direct notation for subscripting a variable: Var@I. This notation, called the at notation, can be used on both the left and right of an assignment expression. Its use on the right is equivalent to pick.

Equations

   I pick A = (list I) pick A
   I in grid A ==> I pick grid A = I
   I in grid A ==> I pick (EACH f A) = f (I pick A)
   A choose B = A EACHLEFT pick B
   I pick A = \[I] reach A

picture

The operation picture computes a character table that describes the array A.

An atom is displayed directly. A nonatomic array is displayed as a tableau of rows and columns arranged to allow sufficient space for the largest items. In most cases, the items of the array are displayed within a frame diagram.

The operation is affected by the setting of two internal switches: diagram/sketch and decor/nodecor. The first controls the use of frames in the pictures. The second controls the decoration of atoms, solitaries, strings and empty arrays.

In diagram mode, a nonatomic array is displayed with a frame and the upper left corner has an o if it is a single. In sketch mode, a simple array, one having atomic items, is displayed without a frame. A non-simple array has a frame and a nonatomic single has an o in the upper left corner of the frame.

The decor/nodecor switch pictures atoms in an undecorated manner if nodecor mode is set. If sketch mode is also set, strings and phrases appear the same. The decor mode pictures atoms so that they can be distinguished: characters are preceded by a grave symbol `, phrases by a double quote " and faults by a question mark ?. In sketch mode with decor set, a string is enclosed in quotes and a simple solitary is placed in brackets.

In sketch mode an empty array has an empty picture, whereas in diagram mode, the picture of an empty array is one side of a frame diagram, with no items.

An array of dimensionality higher than two is laid out as an arrangement of tables. If the array has valence 3, the tables are arranged horizontally. If it has valence 4, the first axis is placed down the page and the second axis across the page. Thus, an array of shape 3 4 5 6 is pictured as 3 rows of 4 tables each of which is a 5 by 6 tableau. A space is left between each of the tables. For higher dimensions, the axes continue to alternate across and down the page with greater spacing between arrangements.

The result of picture coincides with the output displayed by Q’Nial in the top level loop.

In the top level loop, the picture is wrapped around as a whole if it is too wide to fit on the screen.

     set "sketch; set "nodecor;
     P := picture l 1 7 7.0 `a "a 'a' (3 4 5) Null
+-+-+-+--+-+-+-+-----++
|l|1|7|7.|a|a|a|3 4 5||
+-+-+-+--+-+-+-+-----++

     set "diagram; set "nodecor; P
+-+-+-+--+-+-+---+-------+-+
|l|1|7|7.|a|a|+-+|+-+-+-+|+|
| | | |  | | ||a|||3|4|5||||
| | | |  | | |+-+|+-+-+-+|+|
+-+-+-+--+-+-+---+-------+-+

     set "sketch; set "decor; P
+-+-+-+--+--+--+---+-----++
|l|1|7|7.|`a|"a|'a'|3 4 5||
+-+-+-+--+--+--+---+-----++

     set "diagram; set "decor; P
+-+-+-+--+--+--+----+-------+-+
|l|1|7|7.|`a|"a|+--+|+-+-+-+|+|
| | | |  |  |  ||`a|||3|4|5||||
| | | |  |  |  |+--+|+-+-+-+|+|
+-+-+-+--+--+--+----+-------+-+

     set "diagram; set "decor;
     picture 3 4 5
+--+--+--+--+--+--+--+
|`+|`-|`+|`-|`+|`-|`+|
+--+--+--+--+--+--+--+
|`||`3|`||`4|`||`5|`||
+--+--+--+--+--+--+--+
|`+|`-|`+|`-|`+|`-|`+|
+--+--+--+--+--+--+--+

The first four examples illustrate the same array pictured with the four combinations of the mode settings. The final example illustrates that if the result of picture is displayed at top level in diagram - decor mode, the implicit use of picture on the resulting character table causes it to be decorated.

Equations

   write A = writescreen picture A
   valence picture A = 2

place

The operation place is the fundamental insertion operation for arrays. The normal use is that I is the address where B is to be placed in a modified version of A. The result is the modified array. If I is a solitary integer and A is a list, I is converted to an integer before being used. If I is not a valid address for A, the result is the fault ?address.

     "abc 2 place 2 3 4 5 6
2 3 abc 5 6
     place (`_ 2) 'my work'
my_work
     A := 4 5 6 7
4 5 6 7
     A := 27 2 place A; A
4 5 27 7
     50 100 place A
?address

The fourth example shows the use of place in updating the value of a variable A. The last example illustrates an out of range use of place.

Nial provides a simpler notation for updating a variable. A@I (pronounced A at I) can be used on the left of assignment to update the item at location I. If the address is out of range a fault occurs.

     A@1 := 25
4 25 27 7
     A@5 := 7
?address
     A
4 25 27 7

The semantics of using the at notation on the left in assignment can be achieved using the operation update. The at notation and the update operation are usually more efficient than place because they avoid doing a copy of the right argument of place.

Equations

   I in grid A ==> (I pick A) I place A = A
   B \[I] deepplace A = B I place A
   I in grid A ==>shape(B I place A) = shape A

placeall

The operation placeall generalizes the operation place to modify a collection of items of A. It returns a modified version of array A with the items at the addresses I replaced by the items of B. It corresponds to the sequential use of place with corresponding items from B and I. If an item of I is repeated, its last occurrence will determine the effect on A. If B has fewer items than I, its items are used cyclically (the first item is used after the last and the list repeated).

     ("abc "def) (2 3) placeall 2 3 4 5 6
2 3 abc def 6

     Text := 'my work is fun';
     placeall ('_' (`  findall Text)) Text
my_work_is_fun

     A := 4 5 6 7 ;
     A := (27 28) (3 2) placeall A; A
4 5 28 27

The last example shows the use of placeall to update the value of a variable A.

Q’Nial provides a simpler notation for updating a variable at several locations. The notation A#I (pronounced A at all I) can be used on the left of assignment to indicate the update of the items at the addresses given by I. If an item of I is out of range, the result is a fault but the variable is modified by the assignments done to locations that precede the invalid one.

     A#(0 1) := 25 26
25 26 27 7
     A#(2 100) := 50 51
?addresses
     A
25 26 50 7

The semantics of using the at all notation on the left in assignment can be achieved using the operation updateall.

The at all notation and the updateall operation are usually more efficient than placeall as they avoid doing a copy of the entire right argument of placeall.

Placeall is not the same as (pack B I) EACHBOTH place A. The latter results in an array of modifications of A, one for each item to be replaced.

Equations

   I allin grid A ==> (I choose A) I placeall A = A
   I allin grid A ==> shape (B I placeall A) = shape A

plus

The operation plus is the same as sum except that plus enforces the rule that it must be applied to a pair.

Sum is multi pervasive and can add up any number of items. The symbol

     7 plus 9
16

     (2 3 4) plus (12 22 33)
14 25 37

     plus 2 3 4
?plus expects a pair

positions

The operation positions is used in conjunction with picture on singles, lists or tables. For such arrays, it returns an array of the same shape with the items being the addresses at which the pictures of the corresponding items of A appear in the picture of A.

     A gets 2 3 reshape "cat 'fish' 34.5 "dog 'meat' 26.5
+---+----+----+
|cat|fish|34.5|
+---+----+----+
|dog|meat|26.5|
+---+----+----+

     positions A
+---+---+----+
|1 1|1 5|1 10|
+---+---+----+
|3 1|3 5|3 10|
+---+---+----+

The operation positions is useful in applications that do interactive editing of data from its display on the screen.

post

The operation post restructures an array A to be a table with one column holding the items of A in row major order.

     post 3 4 5
3
4
5

     post 'Now is the time'
          'for all good men'
          'to come to the aid'
          'of the party'
+------------------+
|Now is the time   |
+------------------+
|for all good men  |
+------------------+
|to come to the aid|
+------------------+
|of the party      |
+------------------+

The major purpose of post is to restructure data for display.

Definition

     post IS OPERATION A {­ (tally A) 1 reshape A}

Equations

   shape post A = (tally A) 1
   post post A = post A

power

The operation power returns the result of raising the number A to the power B. The result is an integer if both A and B are boolean or integer and B is non-negative. If B is an integer, the power is done by multiplication. Otherwise it uses exp (B * ln A).

     R gets l 2 2.5 `a "abc ??error
l 2 2.5 a abc ?error

     R OUTER power R
     1        1        1.  ?A  ?A  ?error
     2        4   5.65685  ?A  ?A  ?error
   2.5     6.25   9.88212  ?A  ?A  ?error
?A      ?A       ?A        ?A  ?A  ?A
?A      ?A       ?A        ?A  ?A  ?A
?error  ?error   ?error    ?A  ?A  ?error

     2 power 3
8

     2 power -3
0.125

     16 power .5
4.

The examples show that power has different meanings depending on the right argument B. If B is a positive integer, the result is A multiplied by itself B times. If B is a negative integer, the result is the reciprocal of the result when B is positive. The last case illustrates that if B is 0.5, the result is a real number that is a square root of

Equations

   isreal B ==> A power B = exp (B * ln A)
   sqrt A = A power 0.5   (within roundoff error)

predicate

A predicate is an operation that tests a condition and returns true if the condition holds and false otherwise. A predicate is not pervasive.

The predefined predicate operations include:

Operation Test
allin set-like inclusion
atomic an array is an atom
diverse all items differ
empty an array has no items
equal all items are equal
isboolean arg is a boolean atom
ischar arg is a character atom
isfault arg is a fault symbol
isinteger arg is a integer number
isphrase arg is a symbol
isreal arg is a real number
isstring arg is a list of characters
like two arrays contain same items
simple all items are atoms
unequal items are not all equal
up lexical ordering

prefix notation

An operation-expression can precede an array-expression. This is called a prefix use of the operation expression.

        + 7 5
12

     reshape [2 3,1 2 3 4 5 6]
1 2 3
4 5 6

prelattice of atoms

The ordering sequence of the characters for sorting purposes is fixed for each Q’Nial version.

The binary and multi pervasive comparative operations of Nial organize the atoms of Nial in a prelattice or sorting sequence. The operation lte (or <=) does a less than or equal comparison between atoms. The numeric atoms are comparable, with a coercion being done if the arguments are of different numeric type. The nadir, represented by the fault ?O, is less than or equal to all atoms. The zenith, ?I , is greater than or equal to all atoms. Except for these two special cases, literal atoms are incomparable with atoms of different type. However, they can be compared within the same type using a character collating sequence that is version specific.

Phrases are compared lexicographically, such that "apple is after "ape but before "apt. Strings, being character lists, produce a list of results when compared.

Comparisons are of two forms: binary pervasive predicates which return boolean values and multi pervasive predicates that obtain the largest or smallest item in a list. If two atoms are incomparable, the predicates return false, whereas max returns the zenith and min returns the nadir.

These rules were chosen so that the following laws hold for all arrays A and B:

   max A <= B = and (A EACHLEFT <= B)
   A <= min B = and (A EACHRIGHT <= B)

See the entry for up for a discussion of the lexicographic ordering in Q’Nial.

product

The operation product multiplies the items of a simple array of numbers, reducing them to a single number that is their product. The type of the result is the highest type of the items. The operation is extended to non-simple arrays by the multi pervasive mechanism. The symbol * and prod are synonyms for product. The operation times is product restricted to use on pairs.

For an empty array, the result is 1.

     R gets l 2 2.5 `a "abc ??error
l 2 2.5 a abc ?error

     R OUTER product R
     1       2      2.5  ?A  ?A  ?error
     2       4       5.  ?A  ?A  ?error
   2.5      5.     6.25  ?A  ?A  ?error
?A      ?A      ?A       ?A  ?A  ?A
?A      ?A      ?A       ?A  ?A  ?A
?error  ?error  ?error   ?A  ?A  ?error

     product 3. 45. 23. 18. 3.5
195615.

     product (3 4 5)(2 3 2)(0 2 4)(1 1 1)
0 24 40

The first example shows the result of product on all combinations of types of atoms. The last example shows that pervasive extension of product multiples a list of triples in an item by item fashion.

Equations

   A product B = B product B
   product single A = EACH(product single) A
   product EACH product A =f= product link A
   shape cart A = product EACH tally A
   product shape A = tally A

profile

The operation profile is used to convert the profiling statistics into a displayable form. If Fnm is a file name, as a string or a phrase, then the output is written to the file. If it is the empty string, then the profiling information is displayed on the screen. If logging is on, it will be copied to the log file. The gathering of profile statistics must be turned off when profile is called.

A detailed example of the output is given in the help entry on profiling.

Examples of use:

     profile "myprofl

     profile ''

profiletable

The expression Profiletable is used to summarize the internal data structures that are used in the gathering of profiling statistics to produce the same information that is displayed by the operation profile. It produces a list of entries, one for each definition that has been encountered during execution with profiling on. Each entry includes the name, the number of direct calls, the number of recursive calls, the execution time, and a list of subentries for the definitions directly called by the definition. The subentries summarize the information for the called definitions in the same format except that no further breakdown is given on definitions they call.

A detailed example of the output of Profiletable is given in the help entry on profiling.

     Profiletable
+-----------------+-------------------------------------+---
|+----+--+-+----++|+-----+--+-+----+-------------------+|+--
||test|12|0|4.12||||tryit|11|0|4.12|+-----------------+|||ag
|+----+--+-+----++||     |  | |    ||+----+--+-+----++||||
|                 ||     |  | |    |||test|11|0|4.12||||||
|                 ||     |  | |    ||+----+--+-+----++||||
|                 ||     |  | |    |+-----------------+|||
|                 |+-----+--+-+----+-------------------+|+--
+-----------------+-------------------------------------+---

----------------------------------+
---+-+-+----+--------------------+|
ain|1|0|3.96|+------------------+||
   | | |    ||+-----+--+-+----++|||
   | | |    |||tryit|10|0|3.96|||||
   | | |    ||+-----+--+-+----++|||
   | | |    |+------------------+||
---+-+-+----+--------------------+|
----------------------------------+

profiletree

The expression Profiletree is used to display the detailed information from the internal data structures that are used in the gathering of profiling statistics. It produces a nested list of entries showing the calling dynamic calling sequence from top level, with the amount of time and number of calls at each level. Each entry includes the name, the number of direct calls, the execution time, and a list of subentries for the definitions directly called by the definition. The subentries provide the same information for the called definitions in the same format including entries on all the definitions that they call in turn.

A detailed example of the output of Profiletable is given in the help entry on profiling.

Example of use:

     Profiletree
+--------+-+----+-------------------------------------------
|TOPLEVEL|0|4.18|+------------+-----------------------------
|        | |    ||+----+-+--++|+-----+-+----+---------------
|        | |    |||test|1|0.||||tryit|1|0.16|+--------------
|        | |    ||+----+-+--++||     | |    ||+----+-+----++
|        | |    ||            ||     | |    |||test|1|0.16||
|        | |    ||            ||     | |    ||+----+-+----++
|        | |    ||            ||     | |    |+--------------
|        | |    ||            |+-----+-+----+---------------
|        | |    ||            |
|        | |    ||            |
|        | |    ||            |
|        | |    ||            |
|        | |    |+------------+-----------------------------
+--------+-+----+-------------------------------------------

------------------------------------------------------+
--+--------------------------------------------------+|
-+|+-----+-+----+-----------------------------------+||
+|||again|1|3.96|+---------------------------------+|||
||||     | |    ||+-----+--+----+-----------------+||||
||||     | |    |||tryit|10|3.96|+---------------+|||||
||||     | |    |||     |  |    ||+----+--+----++||||||
+|||     | |    |||     |  |    |||test|10|3.96||||||||
-+||     | |    |||     |  |    ||+----+--+----++||||||
  ||     | |    |||     |  |    |+---------------+|||||
  ||     | |    ||+-----+--+----+-----------------+||||
  ||     | |    |+---------------------------------+|||
  |+-----+-+----+-----------------------------------+||
--+--------------------------------------------------+|
------------------------------------------------------+

profiling

Q’Nial Profiling Facility

Q’Nial has a profiling capability that is used to gather relative execution times for operations, transformers and defined expressions written in Nial. The profiling capability has the following features:

Command Purpose
setprofile A turns on or off the internal routines that collect the statistics
profile Fnm that displays the profile data to the screen or writes it to a file
Clearprofile clears the current profile information and reinitializes the profiling system
Profiletable provides the profile information as a table
Profiletree provides the detailed profile information in terms of the call tree

The operation setprofile is called with argument l to turn on the collection of data for a profiling session. It is called with argument o to suspend gathering statistics for a session. Both calls to setprofile should be in the same scope, and cannot be nested. Only one profiling session can be underway at a time.

The operation profile is called with a text argument (string or phrase) that is used as the file name for the profile information. If an empty string is the argument, the output is sent directly to the screen. It computes the statistics on the profiling session underway, prepares the output, and writes it to the file.

The expression Clearprofile clears the two internal data structures that are built as execution is profiled so that a new profiling session can be started.

The expression Profiletree provides the summary data that is displayed in the output produced by profile, but stores it in a Nial array.

The expression Profiletree provides a dynamic call tree of the execution with related times. This provides a more detailed breakdown of the profile data. The following session illustrates the use of the profiling capability:

        setwidth 60;

     test is op n {­
       z := random n n;
       max abs (z - inv inv z) }

     tryit is  op N {­ test N }

     again is op M N {­ sum EACH tryit (M reshape N) }

     fact is op n {­
       IF n<=1 THEN 1 ELSE n * fact (n - 1) ENDIF }

# first profiling session

     setprofile l
o

     test 20
3.86843e-15

     tryit 50
2.83107e-14

     again 10 75
6.72483e-13

     setprofile o
l

     profile ''

Total execution time of profile session:         4.780000
Total execution time in top level calls:          4.420000

op name[.tr arg]                 calls[rec]    time time/call  % time
test..........................   12           4.42   0.3683    100.0<

tryit.........................   11           4.31   0.3918     97.5<
 test.........................   11           4.31   0.3918   100.00

again.........................    1           4.04   4.0400     91.4<
 tryit........................   10           4.04   0.4040   100.00



        profiletree
+--------+-+----+-------------------------------------------
|TOPLEVEL|0|4.78|+--------------+---------------------------
|        | |    ||+----+-+----++|+-----+-+----+-------------
|        | |    |||test|1|0.11||||tryit|1|0.27|+------------
|        | |    ||+----+-+----++||     | |    ||+----+-+----
|        | |    ||              ||     | |    |||test|1|0.27
|        | |    ||              ||     | |    ||+----+-+----
|        | |    ||              ||     | |    |+------------
|        | |    ||              |+-----+-+----+-------------
|        | |    ||              |
|        | |    ||              |
|        | |    ||              |
|        | |    ||              |
|        | |    |+--------------+---------------------------
+--------+-+----+-------------------------------------------

--------------------------------------------------------+
----+--------------------------------------------------+|
---+|+-----+-+----+-----------------------------------+||
--+|||again|1|4.04|+---------------------------------+|||
++||||     | |    ||+-----+--+----+-----------------+||||
||||||     | |    |||tryit|10|4.04|+---------------+|||||
++||||     | |    |||     |  |    ||+----+--+----++||||||
--+|||     | |    |||     |  |    |||test|10|4.04||||||||
---+||     | |    |||     |  |    ||+----+--+----++||||||
    ||     | |    |||     |  |    |+---------------+|||||
    ||     | |    ||+-----+--+----+-----------------+||||
    ||     | |    |+---------------------------------+|||
    |+-----+-+----+-----------------------------------+||
----+--------------------------------------------------+|
--------------------------------------------------------+

# second profiling session

     clearprofile

     setprofile l
o

     test 30
1.72085e-14

     fact 4
24

     setprofile o
l

     profile ''

Total execution time of profile session:         0.330000
Total execution time in top level calls:          0.050000

op name[.tr arg]                 calls[rec]    time time/call  % time
test..........................    1           0.01   0.0100     20.0<

fact..........................    1[    3]    0.04   0.0400     80.0<



     profiletree
+--------+-+----+-------------------------------------------
|TOPLEVEL|0|0.33|+--------------+---------------------------
|        | |    ||+----+-+----++|+----+-+----+--------------
|        | |    |||test|1|0.01||||fact|1|0.04|+-------------
|        | |    ||+----+-+----++||    | |    ||+----+-+--+--
|        | |    ||              ||    | |    |||fact|1|0.|+-
|        | |    ||              ||    | |    |||    | |  ||+
|        | |    ||              ||    | |    |||    | |  |||
|        | |    ||              ||    | |    |||    | |  |||
|        | |    ||              ||    | |    |||    | |  |||
|        | |    ||              ||    | |    |||    | |  |||
|        | |    ||              ||    | |    |||    | |  |||
|        | |    ||              ||    | |    |||    | |  ||+
|        | |    ||              ||    | |    |||    | |  |+-
|        | |    ||              ||    | |    ||+----+-+--+--
|        | |    ||              ||    | |    |+-------------
|        | |    ||              |+----+-+----+--------------
|        | |    |+--------------+---------------------------
+--------+-+----+-------------------------------------------

------------------------------+
-----------------------------+|
----------------------------+||
---------------------------+|||
--------------------------+||||
-------------------------+|||||
----+-+--+--------------+||||||
fact|1|0.|+------------+|||||||
    | |  ||+----+-+--++||||||||
    | |  |||fact|1|0.||||||||||
    | |  ||+----+-+--++||||||||
    | |  |+------------+|||||||
----+-+--+--------------+||||||
-------------------------+|||||
--------------------------+||||
---------------------------+|||
----------------------------+||
-----------------------------+|
------------------------------+

 % end of session

The profile statistics show the division of time between the various definitions. For each definition called during the profiling session there is a summary of the number of calls and time used. For each definition there is also a breakdown on the time spent in other definitions that were directly called from that definition.

Entries in the profile statistics that have a “<” to the right of the last value are top level calls. All other entries have been called by these top level calls.

The time spent in direct recursive calls is counted in the original call. The number of such calls is placed in brackets after the number of top level calls.

The statistics do not account for local definitions within global ones; their execution time is simply added to the time for the global definition.

A feature of the profiler not shown in the above example is that when timing a transformer definition, the time spent executing the argument operation(s) is computed in order to understand how much of the cost of the transformer is due to applying the argument operation(s).

The accuracy of the timing information is limited by the precision of the information available through system calls to the host system. For very small definitions the statistics may show no execution time. Usually their effect can be estimated by considering the execution time of the calling definition.

The Profiletable result provides the summarized profile statistics as a quintuple consisting of the name of the definition, the number of calls, the number of recursive calls, the time, and a list of records for each definition it directly calls. Each of the latter records has the same information provided for the routine itself, but does not report on the definitions it itself calls.

The Profiletree result starts with a node indicating the toplevel and the total execution time. It has a list of subnodes for each definition called from the top level. Each of these has the number of calls, the execution time and nodes for each definition it calls. A recursive call is reported directly in a subnode and hence a deeply recursive routine will produce a deeply nested array of profile information.

program fragment

Nial is a programming language specifically designed for use in an interactive environment. The formal description of the language describes the valid language constructs that can be entered in one interaction; and explains the meaning of one such entry in terms of the environment created by earlier interactions in the same session. The term program fragment is used to describe a meaningful piece of program text.

The rules for writing well formed program fragments in Nial are called the syntax rules of Nial. A program fragment that is well formed is said to be syntactically correct. The syntax rules are analogous to the rules of grammar that determine the correctness of English.

putfile

The operation putfile writes the rows of items of A as text records to the file with name Filename. It is used in conjunction with getfile. The file must not be open. Items of A are arrays of characters of any valence.

In the following example, three strings would be written to the file Maillist.

     putfile "Maillist ['Now is the time' 'for all good
men','to come to the aid']

Definition

     putfile IS OPERATION File A {­
        IF not isfault (Filenum := open File "w) THEN
           ITERATE (Filenum writefile) (link EACH rows A) ;
           close Filenum ;
        ELSE
           Filenum
        ENDIF }

quiet_fault

The operation quiet_fault is a version of operation fault that turns off fault triggering before producing the fault and restores the triggering switch to its prior value after producing the fault.

Definition

     quiet_fault IS OPERATION Str {­
        Oldsetting := settrigger o;
        Res := fault Str;
        settrigger Oldsetting;
        Res }

quotient

The operation quotient returns the quotient of dividing integer A by integer B. If the divisor B is zero, the result is zero. If it is negative, the result is the fault ?negative divisor. The operation quotient converts boolean arguments to integer but otherwise produces a fault if either argument is not an integer.

     R gets l 2 2.5 `a "abc ??error;
     R OUTER quotient R
     1       0  ?A      ?A  ?A  ?error
     2       1  ?A      ?A  ?A  ?error
?A      ?A      ?A      ?A  ?A  ?error
?A      ?A      ?A      ?A  ?A  ?A
?A      ?A      ?A      ?A  ?A  ?A
?error  ?error  ?error  ?A  ?A  ?error

     (5 quotient 2) (-5 quotient 2)
2 -3

     (5 quotient -2) (5 quotient 0)
?negative divisor 0

The first example illustrates all combinations of atom types for the two arguments to quotient.

The quotient on division by a positive integer B is always an integer on the same side of the origin as A. The result of A quotient 0 is 0 rather than a fault since this rule is compatible with the choice that A mod 0 is A.

The operation quotient is used in place of divide in situations where the result must be an integer.

Equation

   isinteger A ==> A quotient 0 = 0
   floor (A - (B*(A quotient B))) =f= A mod B
   A quotient B =f=  floor (A / B)

raise

The operation raise is used to partition an array A along its axes by indicating that the first N axes are to be retained in the result. The remaining axes become axes of the items of the result. N must be an integer in the range from 0 to valence A. The result is an array of shape given by taking the first N items of shape A. The items of the result have the shape given by dropping the first N items of shape A. Thus, the 2 raise of an array of shape 3 4 2 is a 3 by 4 table of pairs. The 1 raise of the same array is a triple of 4 by 2 tables.

     A := 3 4 2 reshape 'ABCDEFGHIJKLMNOPQRSTUVWX'
AB  IJ  QR
CD  KL  ST
EF  MN  UV
GH  OP  WX
     2 raise A
+--+--+--+--+
|AB|CD|EF|GH|
+--+--+--+--+
|IJ|KL|MN|OP|
+--+--+--+--+
|QR|ST|UV|WX|
+--+--+--+--+
     1 raise A
+--+--+--+
|AB|IJ|QR|
|CD|KL|ST|
|EF|MN|UV|
|GH|OP|WX|
+--+--+--+

Definition

     raise IS OPERATION A {­ N drop axes A split A }

Equations

   N a nonnegative integer <= valence A and not empty A ==> shape (N raise A) = N take shape A
   shape first (N raise A) = N drop shape A ==> mix (N raise A) = A

random

The operation random is used to generate pseudo-random real numbers in the range from 0. to 1. The argument S is a shape and random generates product S numbers storing them in an array of shape S. Random uses a congruential method with period 2147483647. The random generator generates a sequence of integers using the formula: R := 16807 * R mod 214783647 starting with R = 314159262. The result is the real R/2147483647. The sequence can be initialized using the operation seed with an argument N, where 0. < N < 1.0.

Seed resets the random number generator to use `R = floor (R1

  • 214783647.)`.
     seed 0.314159; random 5
0.070 0.743 0.240 0.627 0.309

     floor ( 100. * random 2 5 )
98 19 94 47 36
88 44 47 82 38

The first example shows the five random numbers generated after a seed of 0.314159 is used. The second example illustrates how to use the operation random to generate 10 random integers between 0 and 99 and store them in an array of shape 2 5.

rank

The transformer RANK applies an operation f to arrays formed from the last N axes of A. The results of the applications of f must all be the same shape and are combined to form a result array using mix.

     2 RANK reverse (3 4 5 reshape count 60)
20 19 18 17 16   40 39 38 37 36   60 59 58 57 56
15 14 13 12 11   35 34 33 32 31   55 54 53 52 51
10  9  8  7  6   30 29 28 27 26   50 49 48 47 46
 5  4  3  2  1   25 24 23 22 21   45 44 43 42 41

     1 RANK sum (3 4 reshape count 12)
10 26 42

The first example reverses the planes of the generated array. The second example sums the rows of the generated matrix.

Definition

     RANK IS TRANSFORMER f OPERATION N A {­
        mix EACH f (N lower A) }

reach

The operation reach is used to select an array at an arbitrary path P within the nested structure of A. P is a list of addresses where the first item of P selects an item of A, the second selects an item of that item, etc. If P is empty, the result is A. The operation is implemented as an iterated pick and hence the addresses in the path are converted using first if they are solitary integers. If one of the picks attempts a selection out of bounds, the result is fault ?path.

     A := 3 4 reshape (pack 'ABCDEFGHIJKL' '1234567890ab')
+--+--+--+--+
|A1|B2|C3|D4|
+--+--+--+--+
|E5|F6|G7|H8|
+--+--+--+--+
|I9|J0|Ka|Lb|
+--+--+--+--+

     (2 0) 1 reach A
9

In the example, the path (2 0) 1 picks the item with address 1 of the location with address 2 0.

Q’Nial supports a direct notation for doing a reach into an array associated with a variable: Var@@I (pronounced Var at path I). The at path notation can be used on both the left and right of an assignment expression.

Definition

     reach IS P A {­
        IF empty P THEN
           A
        ELSE
           rest P reach (first P pick A)
        ENDIF }

Equations

   P reach A = (EACH list P) reach A
   items of P are valid addresses where used ==> (P reach A) P deepplace A = A

read

The operation read is used in an interaction with the user running a program to obtain an array value from the keyboard after issuing a prompt given by the string or character P. After receiving the prompt, the user types a sequence of characters and then presses the Return key. The sequence of characters, excluding the Return, becomes the string entered. The string is executed to give the result of read. If the string being executed contains an integer representation that would convert to a number outside the range of integers, the corresponding real number is returned.

     A := read 'Type an expression : ';
Type an expression : 3 + count 10

     A
4 5 6 7 8 9 10 11 12 13

The operation read is similar to operation readscreen. The difference is that in read the string received from the keyboard is evaluated, whereas in readscreen, the string is returned.

Definition

     read IS OPERATION P {­ execute readscreen P }

readarray

The operation readarray is used to read the component or components indicated by N from the direct access file with file port Fnum. The file must have been created using writearray and must have been opened as a direct file using open.

If N is an integer, the result is the array stored at record position N of the file.

If N is a list of integers, the result is a list of the corresponding records. If there is no record at position N, the result is the fault ?missing. If N is greater than filetally A, the result is the fault ?eof.

     readarray Fnum [23,24,25];

If N is a solitary integer, the result is a solitary holding the selected item.

readfield

The operation readfield is used to read a portion of an existing host file as a character string. The string or phrase Filename is the name of the file, the integer Start is the offset to the beginning of the field to be read and integer Size is the size of the field to be read. The result is character string of the items read. If the logical field extends across a line boundary the field will include the newline indicator appropriate for the host system (either linefeed or return-linefeed).

Example

     readfield "Myfile 80 20

The result is the string beginning at offset 80 in file Myfile of length 20.

readfile

The operation readfile is used to read the next record from the sequential text file designated by file port Fnum. The file port Fnum is the integer returned by open, which must have been called before executing the readfile.

A record of text in a text file is a sequence of characters up to but excluding an end-of-line indication. The end-of-line indication is system dependent and may be one or more characters. The interface between Q’Nial and the host system recognizes end-of-line and, on successive readfile requests, returns the records of the text file with the end-of-line indications removed.

If readfile is used with a second argument N, it reads the next N bytes of the text file and returns them as a string. In this usage, end-of-line characters are processed as any other character and are not removed. This second form is intended for reading information from device drivers or a communications port, where end-of-line indications may not be given.

The following example shows a text file, Myfile being opened and its records being gathered as a list of strings. The example has the same effect as the expression getfile "Myfile.

     File_number := open "Myfile "r ;
     Records := '' ;
     Record := readfile File_number ;
     WHILE Record ~= ??eof DO
          Records gets Records append Record ;
          Record := readfile File_number ;
     ENDWHILE ;
     close File_number ;

readrecord

The operation readrecord is used to read the component or components indicated by N from the direct access file with file port Fnum. If N is an integer, the result is the string stored at component position N of the file. The file must have been opened as a direct file and must have been created using writerecord.

If N is a list of integers, the result is a list of the corresponding records. If there is no component at N, the result is the fault ?missing. If N is greater than filetally A, the result is the fault ?eof.

A direct access file is stored as two host system files: a .ndx file holding index information and a .rec file holding the records.

     readrecord Fnum [23,24,25];

     convert_records IS OP Filein Fileout {­
        Fin := open Filein "d ;
        Fout := open Fileout "d ;
        Num := filetally Fin ;
        writerecord Fout (tell Num)
        (EACH convert_op (readrecord Fin (tell Num)));
        EACH close Fin Fout ; }

The first example shows a call of readrecord that returns three records. The second example shows an operation convert_records that reads all the records in direct file Filein, modifies them by convert_op and writes them to direct file Fileout. For a very large file, it may be necessary to replace the use of EACH with a loop.

The readrecord operation can access fields of a fixed format file created external to Q’Nial. If the component number used in a readrecord is beyond the end of file as indicated by the filetally Fnum, stored in the .ndx file, the .rec file is checked to see if the host file information indicates that the .rec file extends beyond the length expected for a file created by the Q’Nial operations. If so, the .rec file is assumed to consist of a sequence of blocks of records, with each block having fields of the sizes indicated by the .ndx file. The index information for the record at (N mod filetally Fnum) is used to find the requested component within the block of records found from (N quotient filetally Fnum).

To use this feature, a separate process must be followed to create the .ndx file corresponding to one block of records in the external file. This is done by writing a sequence of strings of the lengths of the records to a dummy direct access file and then renaming the resulting index file and the external file to be names expected by the Q’Nial direct access facility.

readscreen

The operation readscreen is used to obtain a string from the keyboard after issuing the prompt P. The prompt P must be a character or a string. After receiving the prompt, the user types a sequence of characters and then presses the Return key. The sequence of characters, excluding the Return, is the string returned as the result of readscreen.

     Name := readscreen 'Your name? : ' ;
Your name? : Mike

     Name
Mike

     Answer := first 'Do you wish to continue? y/n: ' ;
     IF Answer in 'yY' THEN ...

The operation readscreen is similar to the operation read, the difference being that in read, the string received from the keyboard is evaluated, whereas with readscreen, the string is returned.

reciprocal

The operation reciprocal or recip for short, returns a real number which is the reciprocal value of a numeric atom.

Reciprocal produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean reciprocal of the corresponding real
integer reciprocal of the corresponding real
real 1.0 / A if A ~= 0.; ?div if A = 0.
character fault ?A
phrase fault ?A
fault argument A
     reciprocal l -2 3.5 `a "abc ??error
1. -0.5 0.285714 ?A ?A ?error

     reciprocal -2 (3 -4.5)
+----+------------------+
|-0.5|0.333333 -0.222222|
+----+------------------+

Definition

     reciprocal IS OPERATION A (1 divide A)

Equations

   A divide B = A * reciprocal B
   reciprocal reciprocal reciprocal A = reciprocal A   (within roundoff error)

recur

RECUR is a general recursion transformer with five operation arguments: test checks that the argument meets an end condition, endf is applied to the end argument before starting to build the result, parta computes the left value from the argument, which is stacked, joinf combines the left and right values as the recursion unwinds, and partb gives the value to be recurred on to produce the right value.The recursion terminates provided the repeated application of the operation partb results in an array that satisfies test.

     RECUR [ 0 =, 1 first, pass, product, -1 +] 4
24
     RECUR [ empty, 0 first, first, plus, rest ] 3 4 5
12

Definition

 RECUR is TR test endf parta joinf partb OP A {­
    Elements := Null;
    WHILE not test A DO
      Elements := Elements append parta A;
      A := partb A;
    ENDWHILE;
    Res := endf A;
    FOR E WITH reverse Elements DO
      Res := E joinf Res;
    ENDFOR;
    Res }

Equations

   RECUR [test, endf, parta, joinf, partb] A
         = FORK [test, endf, joinf [parta, RECUR [test, endf, parta, joinf, partb] partb]] A
   RECUR [ 0 =, 1 first, pass, product, -1 +] N = factorial N
   RECUR [ empty, 0 first, first, plus, rest ] A = sum A

reduce

The transformer REDUCE transforms an operation f into an operation which, when applied to an array A, has the same effect as evaluating an expression in which f is placed between the items of the array, with grouping done in a right-to-left manner. If f is an operation that maps a pair of atoms to an atom, REDUCE f reduces a simple array to an atom. For the built-in reductive operations: sum, product, and, or, max, min , and link, REDUCE f is the same as f.

     REDUCE divide 3 4 5 6
.625

     3 divide (4 divide ( 5 divide 6 ) )
.625

     REDUCE plus (count 10)
55

     REDUCE hitch 'abcde'
abcde

     REDUCE append 'abcde'
+-+----------+
|a|+-+------+|
| ||b|+-+--+||
| || ||c|de|||
| || |+-+--+||
| |+-+------+|
+-+----------+

     REDUCE pass 73 45 39 97
+--+----------+
|73|+--+-----+|
|  ||45|39 97||
|  |+--+-----+|
+--+----------+

Definition

     REDUCE IS TRANSFORMER f OPERATION A {­
        % if f is reductive, apply f directly;
        IF empty A THEN
           Res := ??identity;
        ELSE
           Res := last A;
           FOR B WITH reverse front A DO
              Res := B f Res;
           ENDFOR;
        ENDIF;
        Res }

Equations

   REDUCE f solitary A = A
   REDUCE f single A = A
   REDUCE f A B C = A f (B f C)

reducecols

The transformer REDUCECOLS does the reduction of the columns of A with the operation f.

     REDUCECOLS sum (5 6 reshape count 30)
65 70 75 80 85 90

The example returns the sum of the columns of the generated table.

Definition

   REDUCECOLS IS TRANSFORMER f OPERATION A {­
      [valence A - 2 max 0,axes first A] PARTITION REDUCE f A }

reducerows

The transformer REDUCEROWS does the reduction of the rows of A with the operation f.

     REDUCEROWS product (3 4 reshape count 12)
24 1680 11880

The example returns the product of the rows of the generated table.

Definition

   REDUCEROWS IS TRANSFORMER f OPERATION A {­
      BYROWS (REDUCE f) A }

reductive

A reductive operation is one that extends a functional capability normally defined on a pair to a list, reducing the result by pairwise application of the function.

The predefined reductive operations include:

Operation Function
and logical “and” of a boolean array
link the list of all the items of the items in the array
max highest item in the array
min lowest item in the array
or logical “or” of a boolean array
product arithmetic product of an array of numbers
sum arithmetic sum of an array of numbers

All of the above operations except link are also multi pervasive.

The transformer REDUCE can be used to produce a reductive operation from a binary one.

repeat-loop

The REPEAT-loop notation is used when executing an expression sequence repeatedly until a conditional expression returns true.

     F := open Filenm "r;
     Lines := '';
     Done := o;
     REPEAT
       Line := readfile F;
       IF isfault Line THEN
         Done := l;
       ELSE
         Lines := Lines append Line;
       ENDIF;
     UNTIL Done ENDREPEAT;

reserved words

A reserved word or keyword is one that has a special usage in a Nial construct and must be used only for that purpose. A block delimits a local environment. It allows new uses of names which do not interfere with uses of those names outside the block. For example, within a block, a predefined operation name can be redefined and used for a different purpose. Only the reserved words of Q’Nial cannot be reused in this fashion.

An identifier, which is spelled the same, ignoring case, as any of the reserved words given below cannot be used to name a variable or a definition. In a local environment, an identifier can be chosen that is the same as a predefined or user-defined global definition name. Such a choice makes the global use of the name unavailable in the local context.

A reserved word is displayed in upper case in canonical form.

Reserved Word Construct
BEGIN Synonym for {­ in block
CASE case-expression
DO for-expression, while-expression
ELSE if-expression, case-expression
ELSEIF if-expression
END case-expression, synonym for } in block
ENDCASE case_expression
ENDFOR for-expression
ENDIF if-expression
ENDREPEAT repeat-expression
ENDWHILE while-expression
EXIT exit-expression
EXPRESSION declaration
EXTERNAL declaration
FOR for-expression
FROM case-expression
GETS assign-expression
IF if-expression
IS definition
LOCAL declaration
NONLOCAL declaration
OP synonym for operation
OPERATION operation-form, declaration
REPEAT repeat-expression
THEN if-expression
TR synonym for transformer
TRANSFORMER transformer-form, declaration
UNTIL repeat-expression
VARIABLE declaration
WHILE while-expression
WITH for-expression

reshape

The operation reshape is the major mechanism in Nial for creating multivalent arrays. The operation requires A to be a shape, either Null or a list of integers. If A is an integer, it is converted to the solitary holding the integer. If A is not a shape, the result is the fault ?shape.

The result of reshape is an array of shape A with items chosen from the list of items of B. If the number of items to be used to fill the result is less than the tally of B, the remaining items of B are ignored. If the number is more than the tally of B, the items of B are used cyclically. If B has no items, fault ?fill is used as the items of the result.

     2 3 reshape 4 6 (4 5) 3 "abc
+-+---+---+
|4|  6|4 5|
+-+---+---+
|3|abc|  4|
+-+---+---+
     2 3 reshape 'abcdefghij'
abc
def
     5 reshape solitary 3 4
+---+---+---+---+---+
|3 4|3 4|3 4|3 4|3 4|
+---+---+---+---+---+

The last example illustrates that to replicate an arbitrary array, in this case the pair 3 4, reshape is applied to the solitary holding the array.

Equations

   shape A reshape A = A
   shape A reshape list A = A
   shape A reshape (first A hitch rest A) =f= A
   A a shape implies ==> shape (A reshape B) = A
   A reshape list B = A reshape B
   A is a shape ==> EACH f (A reshape B) = A reshape EACH f B
   tally A reshape A = list A
   (tally A) 1 reshape A = post A
   single A = Null reshape solitary A
   solitary A = [ 1 ] reshape single A
   Null = [ 0 ] reshape A

rest

The operation rest returns the list of A after dropping the first item of the list. If the argument of rest is not a list, it is treated as a list.

     rest 3 4 5 6
4 5 6
     rest tell 2 3
+---+---+---+---+---+
|0 1|0 2|1 0|1 1|1 2|
+---+---+---+---+---+
     rest 7 = Null
l

The last example shows that the rest of an atom is the empty list Null.

Definition

     rest IS OPERATION A (1 drop list A)

Equations

   rest A = rest list A
   not empty A ==> first A hitch rest A = list A
   tally rest A = 0 max (tally A - 1)
   first rest A = second A
   rest Null = Null

resume

When break debug mode is initiated by means of the Break expression or the operation breakin (or by using <Ctrl b> while awaiting input in window mode on a console version), a break loop is initiated with output like:

--------------------------------------------------------------
    Break debug loop: enter debug commands, expressions or
      type: resume    to exit debug loop
      <Return> executes the indicated debug command
    current call stack :
foo
-------------------------------------------------------------
?.. C := A + ( + A + A )
-->[stepv]

In the above example the break loop awaits input with the default command stepv assumed. If resume is typed then the break loop is exited and control returns to the expression following the break point.

     Break
--------------------------------------------------------------
    Break debug loop: enter debug commands, expressions or
      type: resume    to exit debug loop
      <Return> executes the indicated debug command
    current call stack :
-------------------------------------------------------------
?.. C := A + ( + A + A )
-->[stepv] resume

reverse

The operation reverse returns an array of the same shape as A having the items in reverse order.

     reverse 4 5 6 7
7 6 5 4

     reverse 'able was I ere I saw Melba'
ableM was I ere I saw elba
     EACH reverse 'This' 'seems' 'too' 'wonderful'
+----+-----+---+---------+
|sihT|smees|oot|lufrednow|
+----+-----+---+---------+

     reverse count 3 4
+---+---+---+---+
|3 4|3 3|3 2|3 1|
+---+---+---+---+
|2 4|2 3|2 2|2 1|
+---+---+---+---+
|1 4|1 3|1 2|1 1|
+---+---+---+---+

     reverse (10 take) reverse sketch 1.23
      1.23

The fourth example illustrates that the reverse of a 3 by 4 table is a 3 by 4 table with the list of items reversed. The last example shows the use of take and reverse to right-justify text in a field.

Definition

     reverse IS OPERATION A {­
        shape A reshape (tally A - count tally A choose A) }

Equations

   shape reverse A = shape A
   reverse reverse A = A
   reverse A = shape A reshape reverse list A
   front A = reverse rest reverse A
   last A = first reverse A

role

The term role is used to describe the class of object associated with an identifier in Nial. The possible roles are: reserved word, variable, expression, operation or transformer. An identifier that corresponds to a reserved word can play no other role in a workspace. An identifier that corresponds to a predefined object in Nial cannot be changed in the global environment, but can take on a different object association in a local environment. In the global environment, once the role of an identifier is established, it must keep the same role, but may have its association changed to another object of the same role.

An external-declaration assigns a role to a name in the global environment so that the name can be used in other definitions before it is completely specified. This mechanism is useful for creating mutually recursive definitions. An external declaration is made only in the global environment.

If the name is already defined with the same role, the declaration has no effect. If the name has another role, a fault is reported.

If the expression on the right of IS in a definition uses the name being defined, the definition is assumed to be recursive. The name is assigned a role compatible with its use on the right if it does not already have a role.

If a definition appears within a block, the association between the name of the identifier and its meaning is made in the local environment. Otherwise, the association is made in the global environment and the definition assigns a role to the name as representing that kind of expression.

If the name being associated in a definition is already in use, the new definition must be for a construct of the same role and the earlier definition is replaced. The use of a defined name always refers to its most recent definition.

rotate

The operation rotate shifts the items of array B to the left A places, inserting the items that drop off the front of the list on the back. If A is a negative number, the shift is to the right.

     2 rotate 24 35 46 57 12 24 39
46 57 12 24 39 24 35

     -5 rotate tell 3 5
+---+---+---+---+---+
|2 0|2 1|2 2|2 3|2 4|
+---+---+---+---+---+
|0 0|0 1|0 2|0 3|0 4|
+---+---+---+---+---+
|1 0|1 1|1 2|1 3|1 4|
+---+---+---+---+---+

The second example shows that on a table rotate works on the list of items in row major order.

Definition

     rotate IS OPERATION N A {­
        Newlist :=  tally A + N + grid A mod tally A choose list A;
        shape A reshape Newlist }

Equations

   N an integer ==> shape (N rotate A) = shape A
   N rotate A = shape A reshape (N rotate list A)
   N rotate A = N mod tally A rotate A
   N rotate A = opposite(tally A-N) rotate A

rows

The operation rows is normally used on a table A. In this case, it returns a list of lists, with each item being a list of items from one row of the table. For a multidimensional array, the result is an array of lists of length equal to the length of the last axis of A. The valence of the result is one less than the valence of A.

     A := 2 3 reshape "Bill "Tom "Mary "Jack "Dawn "Alice
Bill Tom  Mary
Jack Dawn Alice

     rows A
+-------------+---------------+
|Bill Tom Mary|Jack Dawn Alice|
+-------------+---------------+

     A :=  2 2 2 3 reshape count 32
1 2 3       7  8  9
4 5 6      10 11 12

13 14 15   19 20 21
16 17 18   22 23 24

     rows A
+-----+--------+  +--------+--------+
|1 2 3|4 5 6   |  |13 14 15|16 17 18|
+-----+--------+  +--------+--------+
|7 8 9|10 11 12|  |19 20 21|22 23 24|
+-----+--------+  +--------+--------+

Definition

     rows IS OPERATION A ( valence A - 1 max 0 raise A)

Equations

   shape rows A = front shape A not empty A ==> shape first rows A = last shape A
   not empty A ==> mix rows A = A
   T a variable associated with a table ==> I pick rows T = T|[I,]

save

The operation save is used to save a workspace with the name given by the phrase or string Wsname. The convention in Q’Nial is to name workspaces with the extension .nws so that they are easy to find in the directory. When saving or loading a workspace, the extension may be omitted.

The effect of saving a workspace is to place the contents of the current workspace in the named file in an internal format. The contents of the current workspace are unchanged by doing a save.

     save "mywork

A save operation may be executed within an operation or expression. The effect is to interrupt the execution and do the save at the top level.

If the saved workspace contains an expression with the name Latent, whenever the workspace is loaded, the load operation will execute the Latent expression. This can be used to have a workspace automatically begin an application, or to set up desired default values for various switch settings.

scan

The operation scan translates Nial program text to internal form. This process is called tokenizing as it results in a list containing tokens. The argument S is a string. The result is a list beginning with 99 and followed by an alternating sequence of integer codes and phrases.

     scan 'A := 3 * 5.2'
99 2 A 1 := 16 3 2 * 18 5.2

     scan 'sum 2 (35 + 42.7)'
99 2 SUM 16 2 1 ( 16 35 2 + 18 42.7 1 )

     scan 'foo IS first rest'
99 2 FOO 1 IS 2 FIRST 2 REST

The first example has 5 tokens, the second has 7 and the third has 4. In a token stream, the token for an identifier is in upper case.

The scan token codes are given below:

Code Meaning
1 reserved word or delimiter
2 identifier
14 string
15 phrase
16 integer
18 real number
22 fault
42 atomic character
40 atomic boolean or bitstring

Equations

   S a string ==> execute S = eval parse scan S
   S a string ==> scan descan scan A = scan A

scope of a variable

The use of an assign-expression indicates that a name (identifier) is to be treated as a variable in the context surrounding the assign-expression. This context is called the scope of the variable. The context may be global, in which case the variable may be visible at all levels; or it may be local to some region of program text. A local scope is created for the parameters of operation forms and for variables created within a block.

Because operation forms or blocks may appear within other operation forms or blocks, it is possible to have one scope for a name nested within another. A name is said to be visible at a point in a program text if it has a local meaning at that point or has a meaning in some surrounding scope or is a global name. When a name is used in a local scope, it is the local association in the innermost scope that is used, instead of an association with the same name in a surrounding scope.

A block is a scope-creating mechanism that permits an expression-sequence to be created so that it has local definitions and variables which are visible only inside the block. A block may appear as a primary-expression or as the body of an operation-form.

Definitions that appear within the block have local scope. That is, the definitions can be referenced only in the body of the block. Variables assigned within the block may or may not have local scope, depending on the appearance of a local and/or a nonlocal declaration. If there is no declaration, all assigned variables have local scope. Declaring some variables as local does not change the effect on undeclared variables that are used on the left of assignment. They are automatically localized.

If a nonlocal declaration is used, an assigned name that is on the nonlocal list is sought in surrounding scopes. If the name is not found, a variable is created in the global environment.

second

The operation second returns the second of the items of A. If A has only one item or it is empty, it returns the fault ?address.

     second 4 5 6
5

     second tell 3 4
0 1

     second Null
?address

Definition

     second IS OPERATION A (1 pick list A)

Equations

   second list B = second B
   second B = first rest B

see

The operation see displays the definition of the user defined object named by the phrase or string Defname. The canonical form of the definition appears directly on the screen. The operation cannot be used on predefined names. The display is in sketch/nodecor mode regardless of the current settings.

     foo IS first rest;
     see "foo
foo IS first rest

It is possible to capture the display given by see using the composition of operations descan deparse getdef Defname, which returns the list of lines of the display. This technique is used in the operation defedit.

Definition

     see IS OPERATION A {­
        Settings gets EACH set "sketch "decor;
        ITERATE writescreen descan deparse getdef A;
        EACH set Settings; }

seed

The operation seed sets the initial number for the random number generator used by random. The argument Num is a positive decimal fraction less than 1. Seed is used to get predictable results from a program that uses random. Seed returns the value that would have been used to generate the next random number.

     seed .25374;
     floor (100. * random 3)
60 73 83

seek

The operation seek returns a pair of values. The first is a boolean indicating whether or not A is an item of B. The second is the address of the first occurrence of A as an item of B, searching B in row major order. If A does not occur in B, the second item of the result is the gage of the shape of B. Seek is the combination of in and find in one operation.

     3 seek 56 34 3 23 57 3
l 2

     `a seek 'hello world'
o 11

Seek is useful when determining whether or not A is in B and, if it is, obtaining its position in B. Both results are obtained in one internal computation rather than two separate ones.

Definition

     seek IS OPERATION A B {­ A [in,find] B }

Equations

   first (A seek B) = A in B
   second (A seek B) = A find B

Pragmatics

The operation seek uses a linear search on the items of B if the array has not been sorted, or uses a binary search algorithm if it has. The latter fact suggests that an array that is searched frequently should be kept in lexicographical order by applying sortup to it when it is created or changed.

seeprimcalls

The operation seeprimcalls sets an internal flag to the setting of the boolean argument Mode. If Mode is true, subsequent execution will monitor the use of all primitive defined expressions, operations and transformers and will print out a message indicating when the named primitive has completed execution. If Mode is false it turns off the flag that controls monitoring.

This mode of execution is useful for following the execution flow during debugging.

     seeprimcalls True
o

seeusercalls

The operation seeusercalls sets an internal flag to the setting of the boolean argument Mod. If Mod is true, subsequent execution will monitor the use of all user defined expressions, operations and transformers and will print out a message indicating when definitions are entered and executed. If Mod is false it turns off the flag that controls monitoring.

This mode of execution is useful for following the execution flow during debugging.

     seeusercalls True
o

set

The operation set changes internal switches in the Q’Nial interpreter. The switches are used to control the behaviour of a variety of optional features of Q’Nial. The argument Sw is a string or phrase in either case. The result of set is a phrase giving the setting of the switch as it was prior to the execution of the set operation. The result can be used subsequently to restore the switch to its original value. The valid phrases and their purposes are tabulated below:

Setting Description of Result
diagram set default display mode to diagram
sketch set default display mode to sketch
decor turn on decoration of atoms and empty arrays
nodecor turn off decoration of atoms and empty arrays
trace turn on tracing of expressions at the top level
notrace turn off tracing of expressions at the top level
log turn on the automatic logging of the session
nolog turn off the automatic logging of the session
     set "diagram ;
     displayinsketchmode IS OPERATION Result {­
        A := set "sketch; write picture Result; set A ; }

In the above definition, sketch mode is set at the beginning, saving the previous setting in the variable A. When the definition is ended, the mode is reset to the value in A.

The default settings are sketch, nodecor, notrace and nolog.

setdeftrace

The operation setdeftrace changes the trace mode for the user defined expression, operation or transformer definition named by the string or phrase Defname. Mode is an optional argument. If it is omitted, the trace mode is toggled. If Mode is present and has the value 1, it turns trace on. Otherwise it turns trace off. The effect when the trace mode is on is to trace the execution of the body of the operation whenever the operation is executed.

The trace mechanism shows intermediate values in the evaluation of expressions.

The operation returns the previous setting. If the result of setdeftrace is assigned to a variable, the previous setting can be restored later.

     library "average

     setdeftrace "average
0

     average 3 4 5
...trace call to operation
...average
...the arguments for the opform are
...A
...A
3 4 5
...A
3 4 5
...sum A
12
...sum A / tally A
4.
...end of operation call
4.

The tracing facility for expressions does not trace all intermediate computations. Parentheses around an expression will force its result to be shown during a trace.

setformat

The operation setformat controls the format used for picturing real numbers. String is a format specification for real numbers using the conventions for the C library routine printf. There are three styles of format:

Control Effect
%f displays a fixed number of places after the decimal point in a fixed size space with no scaling of the number
%e displays the number in scientific notation with an exponent scaling the number to have one digit before the decimal point
%g displays the number in f format if possible but defaults to e format if the number is not within a suitable range

A code can be inserted between the percent sign and the letter. Format '%15.5f' uses a field of width 15 to display a number in f format with 5 decimal places. The first digit is the width of the field. For f format, the second digit gives the number of places after the decimal while for e and g it indicates the number of significant digits to be displayed. Either digit can be omitted.

Q’Nial requires that the display of a real number includes a decimal point and that the display of an integer does not include a decimal point. In a %g format, printf does not include a decimal point if the real number matches an integer to the specified precision. In this case, Q’Nial makes the field one space wider than that specified in order to accommodate the decimal point. If an f format is not wide enough to accommodate the number, it is widened so that the number is displayed.

If the result of an f format requires more than 40 digits to the left of the decimal point, it is converted to e format.

The format '%.17g' is used by operation display in depicting real numbers as this format is accurate enough to reproduce the same number when executed on most systems. The default format is '%g'. It works as in C except that a period may be added to distinguish the result from an integer constant. If the argument to setformat is '' then it resets the formatting to the default value.

     setformat '%.5f' ;
     post exp 10. 80. 90. 100.
                                  22026.46579
    55406223843935098300000000000000000.00000
    1220403294317840830000000000000000000000.
2.688117141816135610000000000000000000000e+43

     X := 2 2 reshape  9777.985 4464784.692 4070333.511 0.079;

     setformat '%15.7f' ; [pass,sqrt] X
+-------------------------------+-------------------------------+
|   9777.9850000 4464784.6920000|     98.8836943    2113.0037132|
|4070333.5110000       0.0790000|   2017.5067561       0.2810694|
+-------------------------------+-------------------------------+
     setformat  '%15.7g' ; [pass,sqrt] X
+---------------------------------+-------------------------------+
|        9777.985         4464785.|       98.88369        2113.004|
|        4070334.            0.079|       2017.507       0.2810694|
+---------------------------------+-------------------------------+
     setformat '%15.7e' ; [pass,sqrt] X
+-------------------------------+-------------------------------+
|  9.7779850e+03   4.4647847e+06|  9.8883694e+01   2.1130037e+03|
|  4.0703335e+06   7.9000000e-02|  2.0175068e+03   2.8106939e-01|
+-------------------------------+-------------------------------+

setinterrupts

The operation setinterrupts permits or prevents the interruption of computation by the use of <Ctrl C>. If the argument is true, interrupts are permitted. If it is false, they are blocked. The default value is true.

Setinterrupts is used in a closed application to prevent the user from interfering with a computation in process. This may be critical if a file is being updated in a sequence of steps, which must all be completed for the data base to remain consistent.

setlogname

The operation setlogname is used to set the name of the host file in which a log of the session is recorded. The current log file is closed and a new one opened. The argument Filename must be a phrase or a string.

     setlogname "dec17pm

The main purpose of setlogname is to capture the history of part of a session in a particular file for later use. The log file can produce documentation, for example.

setprofile

The operation setprofile is used to turn on or turn off the gathering of profiling statistics. The calls must be made at the same scope level in order for profiling to work correctly. This can either be at the top level, or within the body of some expression or operation.

The argument is a boolean value: true starts the profiling process and false stops it. Profiling can be turned on and off several times within one profiling session.

A detailed explanation of the profiling mechanism is given in the help entry on profiling.

Example of use:

     setprofile True
o

setprompt

The operation setprompt sets the Q’Nial prompt to be the string or phrase S. The default prompt is the string with 5 blank spaces. The maximum size of a prompt is forty characters. Setprompt provides the facility to set a visible prompt which can be distinguished from prompts issued by other software.

     setprompt 'qnial>'
qnial>

settrigger

The operation settrigger sets the action taken when a fault is generated in an operation. If Switch is true, whenever a fault is generated, computation is interrupted, a message is displayed indicating the fault that has occurred and the expression where it occurred. Then the Callstack is displayed and a loop is entered that allows you to explore the cause of the fault. If you press Return at the prompt, control is sent to the top level loop.

If Switch is false, fault triggering is turned off.

The default setting at the start of an interactive session is fault triggering on. The triggering of an interrupt by fault generation is turned off for a direct execution of Q’Nial.

     settrigger o
l

setwidth

The operation setwidth sets the width of the display and log lines. The argument is an integer N. The result is the previous setting. The default setting is 80.

Setwidth is useful in controlling the format of display on output saved for documentation purposes. For example, a setting of 130 allows wide Nial diagrams to be printed on a line printer even though they may not display properly on the screen. Narrower settings are convenient for use in reports.

     setwidth 50
80

shape

The operation shape returns an array that describes the rectangular structure of array A. Every array has a shape. For a single, including atoms, the shape is the empty list Null. For a list, it is a solitary holding the integer giving the length of the list. For a table or higher valence array, it is a list of integers giving the extent (number of items) along each axis.

     shape 5 6 7
3
     A := 2 3 reshape 4 5 6 7 8 9;
     shape A
2 3
     B := 2 3 4 5 0 2 reshape 2 ;
     shape B
2 3 4 5 0 2

The operation shape always returns a list whereas tally returns an integer. An empty array such as B is one with a zero in its shape.

Equations

   product shape A = tally A
   tally shape A = valence A
   list shape A = shape A
   shape A reshape list A = A

simple

The operation simple tests whether or not an array has all atomic items or is an empty array. The result is true if the array is simple and false if it is not.

     (simple 2 3 4) (simple "abc)
ll

     simple Null
l

     simple (2 3) (4 5 6)
o

     (simple tell 2 3)
o

The first two examples show that a list of integers, an atom and the empty list Null are all simple arrays. The next two examples show that a pair of lists and the result of a tell are not simple.

Definition

     simple IS OPERATION A (EACH single A = A)

Equation

   simple A = and EACH atomic A

sin

The operation sin implements the sine function of mathematics. It produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean sine of the corresponding real
integer sine of the corresponding real
real sine of angle A given in radians
character fault ?sin
phrase fault ?sin
fault argument A
     sin  l  -1  0.5  `a  "abc  ??error
0.841 -0.841 0.479 ?sin ?sin ?error

Equations

   sin opposite A = opposite sin A
   (sin A power 2) + (cos A power 2) = 1.0   (within roundoff error)

single

The operation single returns an array with no axes holding A as its only item. The result is an array with shape Null and is said to be a single .

     single 2 3
o---+
|2 3|
+---+
     3 = single 3
l

The first example illustrates that a single of a pair contains the pair as its item. In both diagram and sketch modes, the display of a nonatomic single is decorated with an o in the upper left corner. The second example illustrates that the single of an atom is the atom itself.

Definition

     single IS OPERATION A {­ Null reshape solitary A }

Equations

   first single A = A
   shape single A = Null
   atomic A = single A equal A
   EACH f single A =  single f A
   A EACHLEFT f B = A EACHBOTH f single B
   A EACHRIGHT f B = single A EACHBOTH f B

sinh

The operation sinh implements the hyperbolic sine function of mathematics. It produces the following results when applied to atoms of the six types:

Atomic Type Result
boolean hyperbolic sine of the corresponding real
integer hyperbolic sine of the corresponding real
real hyperbolic sine of angle A given in radians
character fault ?sinh
phrase fault ?sinh
fault argument A
     sinh  l  -1  0.5  `a  "abc  ??error
1.1752 -1.1752 0.521095 ?sinh ?sinh ?error

Equation

   sinh opposite A = opposite sinh A

sketch

The operation sketch computes a character table that gives the picture of the array A as it is displayed in sketch mode. The details of the display of atoms and empty arrays is affected by the setting of the decor/nodecor switch.

     sketch tell 1 3

The sketch of an array provides an abbreviated display that often serves as an adequate output format for data. The entry for picture has a more complete description of the picturing mechanism.

Definition

     sketch IS OPERATION A {­
        Old_setting := set "sketch;
        Result := picture A;
        set Old_setting;
        Result }

Equation

   sketch sketch A = sketch A

solitary