Type Comments

I’ve taken to commenting the closing brace of my inner dfns with a home-grown type notation pinched from the Functional Programming community:

    dref←{                  ⍝ Value for name ⍵ in dictionary ⍺ 
        names values←⍺      ⍝ dictionary pair
        (names⍳⊂⍵)⊃values   ⍝ value corresponding to name ⍵
    }                       ⍝ :: Value ← Dict ∇ Name

I keep changing my mind about whether the result type should be to the left (Value ← ...) or to the right (... → Value). The FP crowd favours → Value but I’m coming around to Value ← because:

* In contrast to (say) Haskell, APL’s function/argument sequences associate right.
* Value ← mirrors the result pattern in a tradfn header and so looks familiar.
* The type of function composition f∘g is simpler this way round.

Such comments serve as an aide-mémoire when I later come to read the code though, with some ingenuity, the notation might possibly be extended to a more formal system, which could have value to a compiler or code-checker. We would need:

Glyphs for Dyalog’s three primitive atomic data types. For no particularly good reason, I’ve been using:

# number
' character
. ref

Glyphs for a few generic (polymorphic) types. These could be just regular lower-case letters a b c … though I currently prefer greek letters:

⍺ ∊ ⍳ ⍴ ⍵ ...

Some constructors for type expressions. This is the most contentious part. For what it’s worth, I’ve been using:

::  is of type ...
∇  function
∇∇  operator
←  returns
[⍺] vector of ⍺s
{⍺} optional left argument ⍺

For example:

foo :: ⍵ ← {⍺} ∇ ⍵

foo is an ambi-valent function whose
– result is of the same type () as its right argument and whose
– optional left argument may be of a different type ().

I can abstract/name type expressions with (capitalised) identifiers using :=. For example:

Dict   := [Name][Value]        ⍝ dictionary name and value vectors
Eval   := Expr ← Dict ∇ Expr   ⍝ expression reduction
List ⍵ := '∘' | ⍵ (List ⍵)     ⍝ recursive pairs. See
Name   := ⍞                    ⍝ primitive type: character vector

The type: character vector ['] is used so frequently that the three glyphs fuse into: . This means that a vector-of-character-vectors, also a common type, is [⍞].

Primitive and derived function types.
If we’re not too nit-picky and ignore issues such as single extension and rank conformability, we can give at least hints for the types of some primitive functions and operators.

 ⍳ :: # ← ⍺ ∇ ⍺              ⍝ dyadic index-of
 ⍴ :: ⍺ ← [#] ∇ ⍺            ⍝ reshape (also take, transpose, ...)

The three forms of primitive composition have interesting types:

∘ :: ⍴ ← {⍺} (⍴ ← {⍺} ∇ ⍳) ∇∇ (⍳ ← ∇ ⍵ ) ⍵     ⍝ {⍺}f∘g ⍵
:: ⍴ ←                 ⍺ ∇∇ (⍴ ← ⍺ ∇ ⍵ ) ⍵   ⍝ A∘g ⍵
:: ⍴ ←      ((⍴ ← ⍺ ∇ ⍵) ∇∇ ⍵ )⍺             ⍝ (f∘B)⍵

It follows that:

f :: ⍴ ← {⍺} ∇ ⍳
g :: ⍳ ←     ∇ ⍵
=> f∘g :: ⍴ ← {⍺} ∇ ⍵          ⍝ intermediate type ⍳ cancels out

and for trains:

A :: ⍳                  ⍝ A is an array of type ⍳
f :: ⍳ ← {⍺} ∇ ⍵
g :: ⍴ ← {⍳} ∇ ∊
h :: ∊ ← {⍺} ∇ ⍵
=> f g h :: ⍴ ← {⍺} ∇ ⍵        ⍝ fgh fork
=> A g h :: ⍴ ←     ∇ ⍵        ⍝ Agh fork
=>   g h :: ⍴ ← {⍺} ∇ ⍵        ⍝ gh atop

For a more substantial example, search function joy for :: and := in a recent download of dfns.dws.

This notation is not yet complete or rigorous enough to be of much use to a compiler but there may already be enough to allow the writing of a dfn, which checks its own and others internal consistency. In the long term, if a notation similar to this evolved into something useful, it might be attractive to allow optional type specification as part of the function definition: without the comment symbol:

    dref←{                  ⍝ Value for name ⍵ in dictionary ⍺ 
        names values←⍺      ⍝ dictionary pair
        (names⍳⊂⍵)⊃values   ⍝ value corresponding to name ⍵
    } :: Value ← Dict ∇ Name

Response to Name Colouring for Dfns

This post contains comments to John Scholes’ post on name colouring; please continue to post any further comments with the original post.

This is a very interesting topic – as the discussion has already showed, there are many different needs. It seems to me that some of the suggestions that have been made are forms of highlighting that you would want to turn on briefly in order to search for something specific or highlight structures while editing code. For example:

  • to verify the structure (highlight the matching parenthesis or bracket, or rainbow colouring of parentheses/brackets)
  • highlight all uses of a particular name or small group of names.

I don’t think I would really want any of these enabled for any length of time; some of them can probably work very well if they are turned on and off as the cursor moves through the text (when on a parenthesis, highlight the other half of the pair, when on a name, highlight all uses of that name, etc).

Making it easier to read statements through highlighting of “kinds” feels more like something that I might well want to have enabled all the time. However, I find colouring to be quite distracting – it breaks the flow for me. I would certainly want the colours to be as calm as possible for the most common kinds, getting more exciting for the “exotic ones”. So I would suggest:


For example:

   r←(foo dop 1 2 3) mat hoo mop vec

I find that syntax colouring is much more effective in a bold font:

   r←(foo dop 1 2 3) mat hoo mop vec 

But…looking at the above, I personally find the colours to be quite distracting; they make it hard for me to read the expression as a “sentence”. How about if we experiment with emphasis instead, for example we could make functions italic, monadic operators bold, and dyadic operators both.

   r←(foo dop 1 2 3) mat hoo mop vec

This is very subjective of course, but this is much more readable to me than the coloured version.

Further comments with the original post please.

Name Colouring for Dfns

APL is sometimes criticised because expressions that include names cannot, in general, be parsed without knowing whether the names represent functions or variables. For example, the name thing in the expression thing⍳3 could reference an array (in which case the is dyadic) or it could reference a function (making the monadic).

An APL expression becomes completely parsable if we distinguish each name with one of four colours, depending on the “kind” of its referent: array, function, monadic operator, dyadic operator. Now, with a bit of practice, we can at least parse thing⍳3 vs thing⍳3 without recourse to the definition of thing. Notice how kind differs slightly from APL’s name-class, which does not distinguish monadic from dyadic operators.

Names whose definitions are external to the code under consideration and so cannot be classified would be given a distinct colour, say red, which would at least draw attention to them. Colouring a number of interdependent functions at the same time should help with such issues.

Name-colouring can co-exist with the more superficial token-colouring ( green for comments and so forth) though we would probably want to configure the two schemes separately.

There’s a related argument for colouring parentheses to show the kind of value they contain: (~∘(⊂'')). This would mean that we could always determine the valency of a function call, or the kind of a hybrid token, such as /, by looking at the colour of the single token immediately to its left. Finally, we should probably kind-colour curly braces: {}, {⍺⍺ ⍵}, {⍵⍵ ⍵}.

Yeah but how?

In the following, where appropriate, “function” is a shorthand for “function or operator”.

Most generally, we want to process a number of definitions at the same time, so that inter-function references can be coloured. Such a set of functions may be considered as internal definitions in an anonymous outer function so, without loss of generality, we need consider only the case of a single multi-line nested function.

A function can be viewed as a tree, with nested subfunctions as subtrees. The definitions at each lexical level in the function comprise the information stored at each node of the tree.

The colouring process traverses the tree, accumulating a dictionary of (name kind) pairs at each level. At a particular level, definitions are processed in the same order as they would be executed, so that the (name kind) entries from earlier definitions are available to inform later expressions. Visiting each node before traversing its subtrees ensures that outer dictionary entries are available to lexically inner functions. Prefixing dictionary entries for inner functions ensures that a left-to-right search (à la dyadic iota) finds those names first, thus modelling lexical name-shadowing.

Each assignment expression adds one or more names to the dictionary. The kind of the assignment is inferred by looking at the defining expression to the right of the assignment arrow. This sounds heavy, but if we examine the expression from right-to-left then we can often stop after examining very few tokens. For example, an arbitrarily long expression ending in (… function array) must, if it is syntactically correct, reduce to an array value, while (… function function) must be a function (train). A sequence in braces {…} resolves to a function, monadic operator or dyadic operator, depending only on whether there are occurrences of ⍺⍺ or ⍵⍵ at its outermost level.

This approach, while relatively simple, is not perfect. It takes no account of the order in which names are defined relative to their being referenced. The assumption is that all definitions at an outer level are available to its inner levels. The following small function illustrates the problem:

        M{A}      ⍝ this A correctly coloured function
        N{A}⍵     ⍝ this A should be coloured unclassified
        A←÷          ⍝ A defined as a function
        M N          ⍝ application of function to array

It remains to be seen whether this would be a problem in practice or whether, if name-colouring proved popular, people might adjust their code to avoid such anomalies. The problem does not occur if we avoid re-using the same name for items of different kinds at the same lexical level – which IMHO is not a bad rule, anyway.

Implementation Status

I would like to experiment with name-kind colouring for the dfns code samples on the Dyalog website.

This project is under way, but has rather a low priority and so keeps stalling. In addition, I keep wavering between processing the tokens as a list with tail calling and the dictionary as an accumulating left argument, or in more classic APL style as a vector, developing parallel vectors for lexical depth and masks for assignment arrow positions and expression boundaries, etc.

In the meantime, here is an artist’s impression of what name- and bracket- coloured code might look like, with colours: array, function, dyadic-operator.

        stk I(op ops)←⍵
        op in⊃⍺:⍺ prt stk I((dref op)cat ops)
        f(fr rr)f rstk
        c{op≡⍵~' '}
        c'dip ':⍺('*⌷'rgs{prt rr I(f cat fr ops)})

Notice how functions stand out from their arguments. For example, at the start of the third line: op in⊃⍺:, it is clear that in is a function taking ⊃⍺ (first of alpha) as right argument and op as left argument, rather than a monadic function op taking in⊃⍺ (in pick of alpha) as right argument. Other interpretations of the uncoloured expression op in⊃⍺: include:

    op in⊃⍺:    op in is a vector (array), so is pick.
    op in⊃⍺:    both op and in are monadic function calls, so is first.
    op in⊃⍺:    in is a monadic operator with array operand op.
    op in⊃⍺:    in is a dyadic operator with function operands op and .

I’m keen to hear any feedback and suggestions about name-colouring for dfns. What d’ya think?

Do Functions Know Their Own Names?

Going back a long way when John Scholes and I were writing version 0 of Dyalog there was a big discussion about whether functions knew their own names. This discussion still surfaces, with John taking the side that they don’t and me taking the side that they do.

Essentially, John would argue that after A←2, the “2” does not know that it is called “A”. So after (in modern parlance):


the part in {} does not know that it is called “add”.

The real question here can be put in different terms: Is the symbol + representing the addition function itself or is it one of the possible names of the addition function.

From an APL perspective, does this matter? Most of the time it makes no difference. However, when you view your SI stack it does. Consider:

          ⍺ ⍺⍺.⍵⍵ ⍵

Now if we trace into

      1 2 add inner times 3 4

and stop on inner[1] what do we want to see when we type ⍺⍺ in the session. There are two possibilities:

Either you see:


or you see:


Which of these is more useful?

Being more provocative, try putting the functions in a capsule:

[0] foo
[1] 1 2{
[2]     ⍺+⍵
[3] }{
[4]     ⍺ ⍺⍺.⍵⍵ ⍵
[5] }{
[6]     ⍺×⍵
[7] }3 4

and repeatedly trace until [6]:


Compare this with the following:

[0] goo
[1] add←{
[2]     ⍺+⍵
[3] }
[4] inner←{
[5]     ⍺ ⍺⍺.⍵⍵ ⍵
[6] }
[7] times←{
[8]     ⍺×⍵
[9] }
[10] 1 2 add inner times 3 4

In my view, the latter is much more communicative in a debugging environment.

Going back to the version 0 discussion: We didn’t have dfns or dops, so everything was traditional. The discussion was centred around:

∇r←a add b
[1] r←a+b

∇r←a times b
[1] r←a×b

∇ r←a (f inner g) b
[1] r←a f.g b

Now trace this:

      1 2 add inner times 3 4

until at times[1]

The key question at the time was whether )SI should show this:


or this:


We choose the first of these options as more informative.

So naming things is good and using those names when reporting state information is also good. When the issue was disputed, David Crossley (who was managing the development of Dyalog) resolved it using the argument about the )SI output.

These days it might not be so obvious. In those days we were essentially thinking in terms of a scrolling paper terminal. It pre-dates the full screen experience that even the tty version gives you. We had to wait for Adam Curtis to join the team before we got that. With the context display whilst tracing there is a stronger argument that the eyes using the debugging information do not need the names. Whilst I admit to the weakening I don’t think it actually changes the balance of the case.

We use a lot of C macros in the interpreter. On Linux, gdb gives us access to those macros when we debug the C code – lldb on MAC, dbx on AIX and Visual Studio on Windows all do not have that information and are, therefore, far less helpful.

Unforgettable Numbers

Professor Kip Murray asked on the J Forum for the “unforgettable” times seen on a 24-hour digital clock.

The problem is that every number has something notable about it, so that each number is “unforgettable” and consequently it’s hard to remember any single one of them.

0000 all zeros
0001 first counting number
0002 first prime number
0003 first odd prime number
0004 first composite number

      ⎕rl←7*5 ⋄ 24 60⊤¯1+?×/24 60
6 59

0659 the time I wake up if the alarm was set at 0700 ☺

You may have heard of the following story about Hardy and Ramanujan. One day Hardy took a taxi to visit Ramanujan. On arriving, Hardy told Ramanujan that the taxi had the thoroughly unremarkable 4-digit number n on its licence plate. Ramanujan immediately remarked that n is the first number that … . I forget what n or the property was, something like n is the first number that can be written as the sum of two perfect cubes in two different ways, something typically Ramanujanian.

Yes, that was it:

      c←3*⍨⍳200           ⍝ cubes of numbers 1..200
      t←(∘.≤⍨⍳≢c) × ∘.+⍨c  
           ⍝ upper triangle of the addition table of these cubes

      e←(,t){⍺⍵}⌸,⍳⍴t     ⍝ unique sums and their indices
      e←(2=≢¨e[;2])⌿e     ⍝ sums that derive two different ways
│1729   │┌────┬────┐      │
│       ││1 12│9 10│      │
│       │└────┴────┘      │
│1092728│┌─────┬─────┐    │
│       ││1 103│64 94│    │
│       │└─────┴─────┘    │
│       ││102 164│128 150││
│       │└───────┴───────┘│

      e⌷⍨(⊢⍳⌊/)e[;1]      ⍝ the row with the smallest sum
│    ││1 12│9 10││
│    │└────┴────┘│
      +/ 1 12 * 3         ⍝ check the sums
      +/ 9 10 * 3

Now that I have worked out the number I can find the story on the net. On hearing the story, J.E. Littlewood remarked that “every positive integer is one of Ramanujan’s personal friends”.

P.S. In my youth, when I needed to remember a (5-digit) number for a time, I computed its largest prime factor by mental calculation. Try it and you’ll see why it works.

Translated and updated from a J Wiki essay which first appeared on 2009-08-22.

Foo, Shmoo

“We do not realize what tremendous power the structure of an habitual language has. It is not an exaggeration to say that it enslaves us through the mechanism of s[emantic] r[eactions] and that the structure which a language exhibits, and impresses upon us unconsciously, is automatically projected upon the world around us.”
—Korzybski (1930) in Science & Sanity p. 90

I’m trying to train myself to choose nouns as names for my functions, in the hope that doing so will help to free me from the procedural mindset, which was pervasive when I learned to program. In those days, coding concentrated more or less exclusively on the manipulation of state and only rarely on the construction of definitions. In saying this, I’m distinguishing procedures, which legitimately deal in state, from functions, which don’t. Transitive verbs: print, init, reverse(vt), … make good names for procedures.

Many of APL’s primitive monadic functions have nouns as names: floor, reciprocal, reverse(n), shape, …

Except this doesn’t quite work. Each of these names seems to need an “of” when applied to an argument, where the of seems to bind to the word that follows it:

   ⌊÷blah   ⍝ "floor of reciprocal of blah"

I’ve seen dyadic iota documented as index-of, though this grates a bit as “index of” isn’t a noun phrase in English. In fact, as far as I know, all of the romance languages would parse Richard of York as Richard (of York), where (of York) is a noun phrase in the genitive case: York’s Richard; Ricardus Eboracum. Nobody seems to parse it (Richard of) York.

So much for that idea for a blog post; ho hum; back to real work.

Then Nick Nickolov found a Wikipedia entry that suggested: “Some languages, such as the Cariban languages, can be said to have a possessed case, used to indicate the other party (the thing possessed) in a possession relationship”. So the (Richard of) parsing would work in one of the Cariban languages and isn’t therefore a property of any deep structure of language, per se.

Now when I see ceiling, I can still think of it as a noun, but in the possessed case. To emphasise this, for the rest of this posting, I’ll write my function names with a back-tick inflection to represent the case: reverse` meaning reverse-of.

   ⌊÷blah   ⍝ floor` reciprocal` blah

… and when reading my code, I can vocalise the ending with a slight [əv] or [ə] sound as in “Land’v plenty”, “Bagguh chips, please”, “Floor’v reciprocal’v blah”.

But what about dyadic functions? Glad you asked. The left argument of a function (arrays are naturally nouns) can be seen as a noun acting as determiner, technically a noun adjunctcoffee tablebicycle thiefcube root, noun adjunct, … Thus: 2-residue`, N-take`, zilde-reshape`⎕AV-index`

Practising this discipline has been quite illuminating: it has pushed me towards left argument currying. I now mentally parse ¯2⌽V as (¯2⌽)V the ((neg-two)-rotation`) V, where neg qualifies two, which in turn qualifies rotation`.

But surely 2+3 is just a symmetrical expression with respect to its arguments? Well yes, but we could also parse it as follows: Starting from the monadic successor` function 1∘+, the curried dyadic version 1+ would be the one-successor` and so 2+3 is the two-successor` three.

Now, how should we incorporate operators and their operands into this scheme? “The reverse` each` V“, where each` is functioning as a determiner?

PS: Choosing nouns for the names of functions shouldn’t be confused with J’s excellent taxonomy of language components using parts of speech, in which functions are labelled “verbs”.

PPS: Functions reverse`, successor`, shape`, … could also be seen as properties of their argument. A number N has many properties: its successor, its square-root, its floor, ceiling, prime-factorisation vector, … This is beginning to sound a bit like OO…