Thursday, 8 March 2007

Coq

A few months ago I started messing around with The Coq Proof Assistant to figure out what exactly it was, and proceeded to go through the tutorial.

Coq provides a nice interface for doing proofs in an interactive session. It was a lot of fun but didn't seem particularly useful for my research.

As I've been playing with the notion of Total Functional Programming, I ended up reading (and actually doing homework) from a book entitled "Type Theory and Functional Programming". This book is excellent by the way! I highly recommend it to anyone interested in dependent types/proof theory/type theory. It has a very lucid style and makes a lot of complex notions in type theory very easy to understand. I even got to find an error in one of the homework examples (now listed in the errata) which is always fun.

After working through the first 5 chapters I started looking around for a system that would let me apply some of the principles that I had learned.

As it turns out Coq is really a dependently typed functional programming language masquerading as a proof assistant! I've spent quite a lot of time over the past few weeks writing total functional programs in Coq and proving properties about them. I've done a bunch of simple things so far, including a proof of correctness for various properties of insertion sort. I started with merge sort, but stalled when it got too complicated. I'm starting to get a feel for using Coq however and large proofs are getting much easier.

In my research I've been working on the implementation of the distillation algorithm (a form of super-compilation) for logic programming. As it turns out
the distillation algorithm as described by Geoff Hamilton could be a real boon to total functional programming in Coq.

In Coq all functions and other values are terms that represent witnesses of a proof. Inhabitation of types is proved exactly by creating a term of the appropriate type. The "tactic" system in Coq is basically a library that helps you build appropriate terms. Alternately you can supply the proofs directly by writing in the functional programming language that acts as the witness terms of the types.

In order to avoid inconsistency functions are not capable of general recursion or errors or exceptions, as this can immediately lead to proofs of propositions which are not actually inhabited. A simple example (in psuedo-haskell) would be a function with the following structure.

loop :: int -> arbitrary
loop x = loop x

Clearly "int -> arbitrary" is not actually the type of this function. It doesn't terminate and so has type _|_. Types in languages like ML and haskell aren't actually just as they are written, but include the possibilities of non-termination or errors into the type implicitly. This (arguably) works out alright if you expect to run your program, but if you are trying to prove useful properties in your type system it turns out to be pretty worthless.

Syntactic restrictions are therefor necessary to avoid including non-terminating functions in Coq. The method chosen is to accept only structurally recursive functions, which can be checked with simple syntactic criterion. In fact the functions require you to specify *which* argument of a function is structurally recursive. This works out surprisingly well but can occasionally be a real pain (try working out a unification algorithm without using anything but structural recursion).

In Coq 8.1 they include a way to define functions that include a measure function which provably decreases or to use a well-founded relation (in conjunction with a proof) to show that the function terminates.

Distillation likes to take functions from general recursion to structural tail recursion. If you could define functions in Coq such that they were then processed by distillation, it would be very useful!

If you haven't tried Coq yet, you should. And if you haven't tried total functional programming yet, I suggest trying it in Coq.