Mathematica Notation and syntax mods

Question

I am experimenting with syntax mods in Mathematica, using the Notation package.

I am not interested in mathematical notation for a specific field, but general purpose syntax modifications and extensions, especially notations that reduce the verbosity of Mathematica's VeryLongFunctionNames, clean up unwieldy constructs, or extend the language in a pleasing way.

An example modification is defining Fold[f, x] to evaluate as Fold[f, First@x, Rest@x]
This works well, and is quite convenient.

Another would be defining *{1,2} to evaluate as Sequence @@ {1,2} as inspired by Python; this may or may not work in Mathematica.

Please provide information or links addressing:

• Limits of notation and syntax modification

• Tips and tricks for implementation

• Existing packages, examples or experiments

• Why this is a good or bad idea

Answer 1

Not a really constructive answer, just a couple of thoughts. First, a disclaimer - I don't suggest any of the methods described below as good practices (perhaps generally they are not), they are just some possibilities which seem to address your specific question. Regarding the stated goal - I support the idea very much, being able to reduce verbosity is great (for personal needs of a solo developer, at least). As for the tools: I have very little experience with Notation package, but, whether or not one uses it or writes some custom box-manipulation preprocessor, my feeling is that the whole fact that the input expression must be parsed into boxes by Mathematica parser severely limits a number of things that can be done. Additionally, there will likely be difficulties with using it in packages, as was mentioned in the other reply already.

It would be easiest if there would be some hook like $PreRead, which would allow the user to intercept the input string and process it into another string before it is fed to the parser. That would allow one to write a custom preprocessor which operates on the string level - or you can call it a compiler if you wish - which will take a string of whatever syntax you design and generate Mathematica code from it. I am not aware of such hook (it may be my ignorance of course). Lacking that, one can use for example the program style cells and perhaps program some buttons which read the string from those cells and call such preprocessor to generate the Mathematica code and paste it into the cell next to the one where the original code is.

Such preprocessor approach would work best if the language you want is some simple language (in terms of its syntax and grammar, at least), so that it is easy to lexically analyze and parse. If you want the Mathematica language (with its full syntax modulo just a few elements that you want to change), in this approach you are out of luck in the sense that, regardless of how few and "lightweight" your changes are, you'd need to re-implement pretty much completely the Mathematica parser, just to make those changes, if you want them to work reliably. In other words, what I am saying is that IMO it is much easier to write a preprocessor that would generate Mathematica code from some Lisp-like language with little or no syntax, than try to implement a few syntactic modifications to otherwise the standard mma.

Technically, one way to write such a preprocessor is to use standard tools like Lex(Flex) and Yacc(Bison) to define your grammar and generate the parser (say in C). Such parser can be plugged back to Mathematica either through MathLink or LibraryLink (in the case of C). Its end result would be a string, which, when parsed, would become a valid Mathematica expression. This expression would represent the abstract syntax tree of your parsed code. For example, code like this (new syntax for Fold is introduced here)

"((1|+|{2,3,4,5}))"

could be parsed into something like

"functionCall[fold,{plus,1,{2,3,4,5}}]"

The second component for such a preprocessor would be written in Mathematica, perhaps in a rule-based style, to generate Mathematica code from the AST. The resulting code must be somehow held unevaluated. For the above code, the result might look like

Hold[Fold[Plus,1,{2,3,4,5}]]

It would be best if analogs of tools like Lex(Flex)/Yacc(Bison) were available within Mathematica ( I mean bindings, which would require one to only write code in Mathematica, and generate say C parser from that automatically, plugging it back to the kernel either through MathLink or LibraryLink). I may only hope that they will become available in some future versions. Lacking that, the approach I described would require a lot of low-level work (C, or Java if your prefer). I think it is still doable however. If you can write C (or Java), you may try to do some fairly simple (in terms of the syntax / grammar) language - this may be an interesting project and will give an idea of what it will be like for a more complex one. I'd start with a very basic calculator example, and perhaps change the standard arithmetic operators there to some more weird ones that Mathematica can not parse properly itself, to make it more interesting. To avoid MathLink / LibraryLink complexity at first and just test, you can call the resulting executable from Mathematica with Run, passing the code as one of the command line arguments, and write the result to a temporary file, that you will then import into Mathematica. For the calculator example, the entire thing can be done in a few hours.

Of course, if you only want to abbreviate certain long function names, there is a much simpler alternative - you can use With to do that. Here is a practical example of that - my port of Peter Norvig's spelling corrector, where I cheated in this way to reduce the line count:

Clear[makeCorrector];
makeCorrector[corrector_Symbol, trainingText_String] :=
Module[{model, listOr, keys, words, edits1, train, max, known, knownEdits2},
(* Proxies for some commands - just to play with syntax a bit*)
With[{fn = Function, join = StringJoin, lower = ToLowerCase, 
 rev = Reverse, smatches = StringCases, seq = Sequence, chars = Characters, 
 inter = Intersection, dv = DownValues, len = Length, ins = Insert,
 flat = Flatten, clr = Clear, rep = ReplacePart, hp = HoldPattern},
(* body *)
listOr = fn[Null, Scan[If[# =!= {}, Return[#]] &, Hold[##]], HoldAll];
keys[hash_] := keys[hash] = Union[Most[dv[hash][[All, 1, 1, 1]]]];
words[text_] := lower[smatches[text, LetterCharacter ..]];
With[{m = model}, 
 train[feats_] := (clr[m]; m[_] = 1; m[#]++ & /@ feats; m)];
 With[{nwords = train[words[trainingText]], 
  alphabet = CharacterRange["a", "z"]},
  edits1[word_] := With[{c = chars[word]}, join @@@ Join[
     Table[
      rep[c, c, #, rev[#]] &@{{i}, {i + 1}}, {i, len[c] - 1}], 
     Table[Delete[c, i], {i, len[c]}], 
     flat[Outer[#1[c, ##2] &, {ins[#1, #2, #3 + 1] &, rep}, 
       alphabet, Range[len[c]], 1], 2]]];
  max[set_] := Sort[Map[{nwords[#], #} &, set]][[-1, -1]];
  known[words_] := inter[words, keys[nwords]]]; 
 knownEdits2[word_] := known[flat[Nest[Map[edits1, #, {-1}] &, word, 2]]];
 corrector[word_] := max[listOr[known[{word}], known[edits1[word]],
   knownEdits2[word], {word}]];]];

You need some training text with a large number of words as a string to pass as a second argument, and the first argument is the function name for a corrector. Here is the one that Norvig used:

text = Import["http://norvig.com/big.txt", "Text"];

You call it once, say

In[7]:= makeCorrector[correct, text]

And then use it any number of times on some words

In[8]:= correct["coputer"] // Timing

Out[8]= {0.125, "computer"}

You can make your custom With-like control structure, where you hard-code the short names for some long mma names that annoy you the most, and then wrap that around your piece of code ( you'll lose the code highlighting however). Note, that I don't generally advocate this method - I did it just for fun and to reduce the line count a bit. But at least, this is universal in the sense that it will work both interactively and in packages. Can not do infix operators, can not change precedences, etc, etc, but almost zero work.