Stuart Smith University of Massachusetts Lowell

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APL “Wish Lists”: the Nial Programming Language Revisited

Stuart Smith

University of Massachusetts Lowell

Lowell, MA 01854 USA


Main topics:APL, Nial, functional programming, function arrays, array theory

1. Abstract

Proposals for new language features have appeared regularly within the APL community. Some of these have been adopted, while others remain objects of discussion and controversy. This paper focuses on three recurring categories of proposed language features: function arrays, additional data types, and control structures. We examine these features from the point of view of Nial, a general-purpose programming language that has a long history within the APL community and whose array processing capabilities are similar to those of APL2 (both APL2 and Nial were strongly influenced by versions of Trenchard More’s array theory). Because of their utility in many common programming situations, function arrays, certain special data types, and conventional control structures were incorporated into Nial almost from the beginning of its development. The purpose of this paper is to show (1) how function arrays, the special data types, and control structures work in Nial and (2) what considerations went into their inclusion in the language. The presence of these features in an actual programming language provides a concrete basis for discussion of the pertinent language issues. Although no longer in serious contention with current commercial APL's, Nial is nonetheless worth another look for its outstanding design and implementation.

2. Introduction

Nial’s origins are in Trenchard More’s work in array theory [7,8,9,10]. Jenkins and Falster [3, p. 24] characterize array theory as follows:

What is unique about array theory is that it attempts to combine two different organizing principles for data: rectangular arrangement and nested collections. The former is observed in the vectors and matrices used by linear algebra and in tensors used in physics. The latter correspond to finite sets, nested lists, and various forms of hierarchy. They correspond to the two classical ways of looking at ordering. From the point of view of programming languages, the data structures of APL correspond to the first organization and those of Lisp correspond to the second.

Nial was created by More, who was then at the IBM Cambridge Scientific Center, and Michael Jenkins, now professor emeritus of Computer Science at Queen’s University in Kingston, Ontario. Colleagues at IBM, Queen's, and the Technical University of Denmark also contributed to the language. Beginning in 1972, IBM developed a succession of experimental interpreters for More's array theory. These interpreters became the starting point for the Nial programming language while Jenkins was working with More at IBM in 1979. In the period 1980 to 1983 Jenkins and More collaborated closely to refine the language definition and Jenkins led a team at Queen’s to produce Q’Nial, a portable interpreter for Nial written in C [5]. This effort was funded by Queen’s, IBM, and a Canadian research grant. Queen's subsequently spun off a company, N.I.A.L Systems Limited (NSL), and gave it exclusive rights to market Q'Nial for the University. By the late 1980's Q'Nial had spread to more than 60 universities, but it never attracted wide interest among commercial users. NSL now promotes Q'Nial primarily for educational purposes and supports a small cadre of loyal customers. Free versions of Q'Nial are currently available on request from NSL for Windows 9x and 2000, Solaris, and Linux[12].

3. Nial Terminology and Syntax

In Nial, functions are called “operations”, and what in APL are called “operators” (scan, reduce, inner and outer product, etc.) are called “transformers” in Nial. In general the Nial terminology will be used here.

In Nial, lists of items can be notated in either bracket-comma form, e.g.,

[ 1 , [ 2 , 3 ] , 4 ]

or “strand” form, which is simply a juxtaposition of two or more array expressions, e.g.:

1 ( 2 3 ) 4

Both examples result in a nested list of three items with the second item a list of two items.

The composition of operations is also accomplished by juxtaposition. No composition operator is required; the names of the desired operations are simply juxtaposed. For example, the expression

G IS ln sqrt abs

defines an operation G which takes the natural logarithm of the square root of the absolute value of its unnamed array argument. This form of function definition is similar to tacit definition in J. Nial also supports the lambda style of operation definition, which names its arguments. For example, G could be defined like this:


ln sqrt abs X


A Nial program “statement” is an expression terminated with a semicolon. The return value of an operation is indicated by an expression not terminated with a semicolon. This expression must be the last one on its execution path within the operation. Here, for example, is an operation that computes an approximation to using Rectangle Rule numeric integration:

pi_approx IS OPERATION n {

x := 0.5 + tell n / n ;

4 / n * sum recip ( 1 + ( x * x ) )


The first statement computes n equally spaced values in the interval [0,1] and assigns them to array x. The second statement, which is not terminated with a semicolon, computes the approximation and returns it as the value of pi_approx for the given n. The equivalent function in APL is


[1] X←(0.5+⍳n)÷N

[2] Z←(4÷N)×+/÷1+X⋆2

Either infix or prefix notation can be used in Nial. For example, both X + 1 and + X 1 are valid.

4. Operation (Function) Arrays

In Nial there are several ways to create new operations from existing ones: by composition, by transformation (similar to APL’s operators), by left currying, by parameterized expression, and by a list of operations called an “atlas.” The atlas provides the capabilities desired in a function array. An atlas applies each component operation to the argument, producing a list of results. Here, for example, the square root, reciprocal, and opposite (negation) operations are applied as an atlas to the argument 2, giving as a result a list of three values:

[ sqrt , recip , opp ] 2

1.41421 0.5 –2

This is equivalent to

[ sqrt 2 , recip 2 , opp 2 ]

1.41421 0.5 –2

An atlas can be given a name by the Nial definition mechanism:

my_atlas IS [ sqrt , recip , opp ]

my_atlas 2

1.41421 0.5 –2

The atlas notation serves several important purposes. According to the Q’Nial Manual [11] its primary use is as a shorthand for describing operations without a need to explicitly name their arguments. It is therefore one of the key supports for functional programming styles like the FP notation described by Backus in his 1977 Turing Award paper [1]. For example, we can define an operation to compute the arithmetic mean of its array argument using the conventional lambda form, which names its arguments:

average IS OPERATION A {

sum A / tally A


or, with the use of an atlas, we can define the same operation like this:

average IS /[sum,tally]

This is a functional programming-style definition, which does not name its arguments.

A second use of the atlas is to form an operation argument for a transformer that uses two or more operations. For example, without the atlas the equivalent of +.× in APL would have to be denoted in Nial as


which, as an infix form, can take only two operations as arguments. With the atlas, it is denoted as

INNER [+,*]

The atlas provides a way for all transformers to take one argument, namely, an operation or a single atlas. This use of the atlas simplifies the syntax for transformers and makes the visual parsing of transformer expressions easier. Because an atlas can contain any number of component operations, it allows the creation of programmer-defined transformers that can take any number of operation arguments. For example, we can define a transformer my_fork that simulates J’s fork construct. The J fork involves three functions:
( f g h ) y is equivalent to ( f y ) g ( h y )
i.e., the result is obtained by applying the dyadic function g to the results of f and h applied individually to argument y. my_fork gives the same behavior in Nial. Here is its definition:
my_fork IS TRANSFORMER f g h (g[f,h])

As with J’s fork, my_fork applies the middle operation to the results of the first and last operations. If we call my_fork with an atlas argument whose component operation arguments are sum, /, and tally, the result is an operation that computes the arithmetic mean of its argument:

my_fork[sum,/,tally] 1 2 3 4


Finally, the atlas allows a list of operations to be applied element-wise to a list the same length as the atlas. The component operations of the atlas are applied to the corresponding items of the argument. In these cases the TEAM transformer must be used. Here is an example using the my_atlas operation defined above:
TEAM my_atlas 2 3 4

1.41421 0.33333 –4

In this expression, each operation of my_atlas (sqrt, recip, and opp, respectively) is applied to the corresponding item of the list argument.
Although an atlas is specified using Nial's bracket-comma array notation, it is syntactically an operation, not an array. As such it is not a first-class object in Nial. It cannot be passed to an operation as an argument and it cannot be returned by an operation as a result. The inability to pass an operation as an argument to another operation is not a serious restriction because—as shown above—programmers can define their own transformers, which do take operation arguments.

5. Additional Data Types


As Lucas [6] points out, “in APL, an expression like ÷A could result in a DOMAIN ERROR if some elements of A are zero. Yet taken individually, only some of the elements of  would generate errors, and frequently one would only want to identify those elements, but still get the results for the others.” This capability would obviously be very useful during the debugging phase of program development.

To handle such situations, Nial provides a data type called fault. Faults are special values to mark exceptional conditions such as errors or the end of a file. A fault is designated by a leading question mark (?). For example, ?div denotes a division-by-zero error and ?address denotes an out-of-bounds array reference. A set of standard faults is predefined in the Nial interpreter. Programmers can also define their own faults.

Users can set fault triggering if they want execution to stop on the occurrence of a fault. If fault triggering is not set, a fault is propagated as the result of any operation that has the fault as an argument. Because faults propagate in characteristic ways through element-wise operations, inner and outer products, reductions, and scans, the ability to turn fault triggering off can help the programmer to identify the source of an error and also to examine the potentially good results of a computation, which would otherwise simply be discarded.

Here are some examples of the ?A (arithmetic) fault being transmitted through various arithmetic operations on lists. In each case, the non-numeric phrase "three in array x causes the fault (the Nial phrase data type is discussed in the next section).

x := 1 2 "three 4

1 2 three 4

y := 1 2 3 4

1 2 3 4

In any element-wise operation, the fault simply appears in the result at the same location as the bad data in an operand:

x + y

2 4 ?A 8
In a reduction, a bad value anywhere in the argument produces the ?A fault as the result:

reduce + x


In a scan, a bad value at the i'th position in the argument causes the ?A fault to appear at all positions from the i'th to last in the result:

accumulate + x

1 3 ?A ?A

With outer product, a bad value in the i'th position of the left argument causes the ?A fault to occupy the entire i'th row of the result:

x outer * y

1 2 3 4

2 4 6 8

?A ?A ?A ?A

4 8 12 16

A bad value at the i'th position of the right argument causes the ?A fault to occupy the entire i'th column of the result:

y outer * x

1 2 ?A 4

2 4 ?A 8

3 6 ?A 12

4 8 ?A 16

With inner product, a single bad value will cause the ?A fault to occupy an entire row or column of the result. Here, for example, matrix x contains the non-numeric phrase “six, while matrix y consists entirely of integers:

x y

0 1 2 3 0 4 8 12

4 5 six 7 1 5 9 13

8 9 10 11 2 6 10 14

12 13 14 15 3 7 11 15

When we attempt to form the inner product of these two matrices, the single bad value in the second row of the left argument x causes the ?A fault to occupy the entire second row of the result:

x inner [+,*] y

14 38 62 86

?A ?A ?A ?A

62 214 366 518

86 302 518 734

Similarly, the single bad value in the third column of the right argument x causes the ?A fault to occupy the entire third column of the result:

y inner [+,*] x

224 248 ?A 296

248 276 ?A 332

272 304 ?A 368

296 332 ?A 404

In all the cases above, the occurrence (or first occurrence) of the ?A fault helps to pinpoint the location of the bad data value causing the error. At the same time, all of the good values of the computation are preserved.
The original motivation for incorporating faults into Nial is to be found in More's array theory, the theoretical foundation of Nial. In array theory there is only one kind of object: the nested rectangular array. As a consequence, each item of an array must also be an array. All primitive operations in the theory are defined for every array, and every primitive operation applied to any array produces an array as its result. These requirements eliminate boundary cases, which the programmer would otherwise have to know and keep in mind while coding. However, another problem then arises: when array theory is made the theoretical foundation of a programming language which manipulates numbers, truth values, and characters, it is necessary that, for example, 3.0 / 0.0 give an array result, as must 3 + `Z, where `Z is a literal character constant. The fault data type is the array-theoretic solution to this problem.

Faults have other important uses. In database applications, for example, it is often the case that one wants to know why missing data are missing, not simply that they are missing. A single value to indicate missing data is uninformative. As Lucas [6] suggests, there should be different values for “not applicable”, “not available”, “pending input”, “pending validation”, etc. These special values can be created as Nial faults. For example

MD := ??missing


Nial provides a conversion operation, fault, to convert a character string to a fault. This is handy if one wants to convert a string containing blanks into a fault, as with Lucas' examples above. For example

NA := fault '?not available'

?not available

Because fault triggering can be set dynamically in Nial, programs can be written to control “on the fly” whether or not missing data trigger an interruption of computation.


Because of the heavy use of symbolic data in AI applications, another special data type, phrase, was included in Nial to facilitate the processing of such data. A phrase is an atomic array consisting of zero or more characters taken as a unit. A phrase is designated by a leading double quote mark ("). Here are some examples of phrases: "dog, "X37, "this_is_a_phrase.
A list of phrases is easily created, e.g.

Lis := “cat “bird “ant

cat bird ant

Such lists can be manipulated with Lisp-like list processing operations provided by Nial. For example, the head of the list can be accessed with first:

first Lis


and the remainder of the list can be accessed with rest:
rest Lis

bird ant

A new item can be inserted at the head of the list with the hitch operation:
hitch “dog Lis

dog cat bird ant

Lisp programmers will recognize that first, rest, and hitch correspond to CAR, CDR, and CONS, respectively, in Lisp. Nial also provides equivalents to ATOM, EQ, NULL and the other fundamental Lisp list processing operations. Because of Nial’s foundation in array theory, these list processing capabilities harmonize nicely with Nial’s facilities for manipulating rectangular arrays.

Nial provides a conversion operation, phrase, to convert a character string to a phrase. This is useful if one wants to work with symbols whose representations contain embedded blanks. For example

Foo := phrase 'a multi-word symbol'

a multi-word symbol

By having phrases as unique atomic objects, equality-testing operations and set theory-like operations can be performed more efficiently. Nial also provides standard operations for efficient sorting of lists of phrases.

6. Control Structures

It is often said that control structures are not “array-oriented” and that this can lead to execution inefficiencies. If and when this is true, the severity of the inefficiencies actually encountered will of course depend on the particular application for which the programming language is used, the quality and appropriateness of the algorithms and programming techniques employed, and the implementation of the language. These are all pragmatic concerns. In the current debate on programming language design, the question of whether or not control structures should be included appears rather to be more a philosophical issue.

Nial includes IF THEN ELSE, CASE, FOR loops, WHILE DO loops, and REPEAT UNTIL loops. Their presence in the language is the result of a quite definite view of the nature of programming languages. Jenkins and Falster [3, p. 9] state that “Nial can be viewed as having two distinct components: the mathematical expression language that describes array-theoretic computations, and the linguistic mechanisms added to make it a programming language.” In response to the author’s query about this statement, Jenkins wrote [4] :

This was a very deliberate statement. It has two implications:

1. that a programming language needs more than just mathematical evaluation. It must have choice e.g. if-then-else or CASE or something to choose between alternate paths. Needed for example to terminate a user-defined recursion. APL has conditional branch. It also must have indefinite iteration e.g. WHILE DO or equivalent so that a loop can continue until a process converges. Actually you can replace this with recursion if the right capabilities are provided.

2. that making this separation a conscious one gives clarity on the design issues of the expression language. It can be based on a clean mathematical theory independent of the needs of the first point [of my answer].

This approach to computing has a history that reaches back to ancient times. In a discussion of the conventional, or “von Neumann,” architecture of digital computers and the languages used to program them, Halpern [2, p. 43] observes that

the control statements in von Neumann languages, and the jump and test instructions in von Neumann machines, are based on the instructions that scientists have long given to those who performed long, monotonous computations for them on papyrus, parchment, paper, or Friden, Marchant, and Monroe calculators long before “computer” came to mean a machine.

The separation of expression evaluation from control is thus quite traditional, at least within the context of mathematical computation. Control structures naturally took their place in the world of digital computing and became part of the practice of ordinary programmers. The Nial design respects this practice while also providing mechanisms to support other programming styles.

7. Discussion

The language features discussed here were all integral parts of the initial design of Nial, and their interactions with other features of the language were carefully monitored from the beginning of the development of the language. Any change to an existing programming language to accommodate new features such as those discussed here is likely to have repercussions throughout that language. Because the type system in particular is one of the most fundamental aspects of any programming language, one must change it only with great care. Adding new types to a language may necessitate a complete redesign of the language. Omitting conventional control structures from a programming language forces programmers to learn new techniques (e.g., recursive programming) which may be quite elegant from a theoretical point of view but which are often difficult for ordinary programmers to master. The Nial approach of making a clean separation between expression evaluation and control allows the former to be implemented according to rigorous axiomatic methods while the latter can be implemented in a form familiar to the practical programmer.

8. Acknowledgements

Thanks to Mike Jenkins, the creator of Q’Nial, for reviewing the manuscript and making many helpful suggestions. Mike also provided details of the history of Nial.

9. References

[1] Backus, John R. “Can Programming be Liberated from the Von Neumann Style? A Functional Style and its Algebra of Programs,” Communications of the ACM, August 1978, pp. 613-641.

[2] Halpern, M. Binding Time: 6 Studies in Programming Technology and Milieu. Norwood, NJ: Ablex Publishing Corporation, 1990.

[3] Jenkins, M. and P. Falster. Array Theory and Nial. Nial Systems Limited,, 1999.

[4] Jenkins, M. Private communication. 2001.

[5] Jenkins, M. “Q'Nial: A Portable Interpreter for the Nested Interactive Array Language, Nial”. Software Practice & Experience (19)2, (1989), 111-126.

[6] Lucas, J. An Array-Oriented (APL) Wish List. APL Quote-Quad (31)2, (2001), 37-43.

[7] More, T. and M. Jenkins. The APL Workshop Session on General Arrays. APL Quote-Quad (8)2, (1977), 12-13.

[8] More, T. The Nested Rectangular Array as a Model of Data. APL Quote-Quad (9)4, (1979), 55-73.

[9] More, T. Nested Rectangular Arrays for Measures, Addresses, and Paths. APL Quote-Quad (9)4, (1979), 156-163.

[10] More, T. Rectangularly Arranged Collections of Collections. APL Quote-Quad (13)1, (1982), 219-228.

[11] Nial Systems Limited. The Q’Nial Manual., (1998).

[12] Nial Systems Limited. Send e-mail to to download a copy of Q’Nial or the documents referenced above.

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