# Systematic Syntax: Lisp vs Mathematica

The following is a (edited) newsgroup post with computer scientist Richard Fateman about computer language syntax, in particular, Mathematica's syntax, and describes my idea about a “systematic syntax”.

Dear Richard,

You are, perhaps the top 10 haters of Mathematica/Wolfram on this earth. Me, is a dedicated lover of Mathematica, and actually i admire Stephen Wolfram, his ideas, personality, too, in fact many of his views, philosophies, are in alignment with my own. (one other guy i could think of is Bertrand Russell.)

Richard Fateman wrote:

This is in contrast to “almost all computer languages” which DO have a syntax described in Backus-Naur Form.

Xah wrote:

No they don't. Show me some. Let's say the most popular ones: java, c, c++, python, perl, php, html, xml.

Richard Fateman wrote:

OK, I thought you had access to Google and could find these yourself..

the C grammar is in Kernighan and Ritchie, The C Programming Language, App. A, the Java grammar is in http://java.sun.com/docs/books/jls/second_edition/html/syntax.doc.html the Python grammar is in http://www.python.org/doc//current/reference/expressions.html C++ is probably in various places, including http://www.nongnu.org/hcb/

perl is, I think not amenable to systematic context-free description, in that it uses some odd tricks during lexical analysis. (As does Mathematica).

html is not a programming language but a markup language; it is described formally not by bnf but by a formal SGML DTD.

I'm not familiar enough with PHP to say anything about it except that from googling for its grammar, it seems to be a piece of malleable garbage..

Other languages, like Algol-60, Pascal, FORTRAN, Basic, … also have (annotated) formal syntax descriptions, typically available in any reference manual for them.

they have syntax spec in the same sense that there's syntax spec for english. Just read grammar books!

i didn't clarify myself before. Let me do now.

No major programing language in use has a formal grammar for their syntax. By formal grammar, i mean a “formal language” (in the context of symbolic logic. See Formal language . Parsing Expression Grammar would be a example of such formal language).

In particular, let's look at 2 examples. Java and Python.

Neither have a FORMAL LANGUAGE spec of their syntax. Worse, both their “spec” is not sufficient to implement the language, even just the syntax part. The vast majority of “language spec” are like that. They are more of “a guide to implementation”, and that's it.

Fateman wrote:

(The term “regular” wrt syntax has a particular formal distinction).

xah wrote: «Sure, and i would ask you some questions about syntax, but given your known antagonism against Mathematica and Wolfram, am not sure i'd get answers that's unbiased and useful to me.»

Google is your friend. Look up “regular grammar” to find out about right and left linear grammars, for example.

… i mentioned “regular syntax”, but actually, a better term is “systematic syntax”. (i don't know if there's proper a term for this idea.)

The “systematic syntax” i have in mind is this:

- a language with a formal language spec of the syntax.
- this formal language spec, is “regular” in the sense that it is very simple, perhaps just a handful of rules. As a example of opposite, you could have a formal language spec for Java syntax, but that would be tens or hundreds of pages.

Now, my claim about Mathematica is this: its syntax specified in a general formal language would be very small, yet the syntax is very rich. No other major lang comes close. (lisp would also have a very simple syntax spec in formal lang, but its syntax isn't rich).

Now, here's what i meant by “simple”. Let's give a formal lang spec in BNF for a simplified lisp syntax.

- The set of symbols are english letters a to z, and the parens “(” and “)” and space “ ”.
- let's call the letters a to z as atoms, and denote it by α.

starting strings:

α ()

transformation rules

α → (α) (α) → (α α …)

That's it. Very simple.

Now, a lang can have such simple syntax, but usually such simple ones are not useful, not expressive, hard to read. For example, assembly langs. Or think of arithmetic with just “+”.

What we want is a rich syntax, but still regular with simple. In some sense, such syntax grammar is a systematic one.

lisp do have very simple syntax, yet it is irregular, yet the typical lisp fans don't realize it, but drivel all day about sexp and macros and certain “code is data” meaninglessness. (lisp macros is a idiocy.)

as a example of the complexities of C-like syntax, let's have a peek show:

i++ ++i for(;;){} while(){} 0x123 expr1 ? expr2 : expr3 sprint(…%s,#@$#!%*&faak@#)

Further readings:

- Pattern Matching vs Grammar Specification
- What is Function, What is Operator?
- Concepts and Confusions of Prefix, Infix, Postfix and Lisp Notations
- Fundamental Problems of Lisp, Syntax Irregularity

Btw, a little extra tip for my readers:

There is a confusion of the word “formal”. When people say “formal”, as in “formal proof”, even mathematicians, they usually losely mean “rigorous”. Basically, they use the word “formal” as a synonym of “rigor”. (And what's “rigorous” changes with time. Usually it just means the current standard, accepted by other mathematicians.)

This is partly a abuse of language, partly a establish english usage habit (called “phraseme”). Still, many mathematicians are ignorant of “formal” in formal languages. In fact, many of them sneer at the idea of David Hilbert's formalism or Bertrand Russel logicism.

For more, see:

- Math Notation, Proof System, Computer Algebra, in One Language
- State of Theorem Proving Systems 2008

Happy new year.

RJF

Happy New Year to you too Richard!

Xah Lee

original newsgroup post source http://groups.google.com/group/comp.lang.lisp/msg/d58a82f4db51b862