A research blog about programming languages, formal logics, software development and their interactions, by Matthias Puech.

Month: April, 2014

Typeful disjunctive normal form

This is the answer to last post’s puzzle. I gave an algorithm to put a formula in disjunctive normal form, and suggested to prove it correct in OCaml, thanks to GADTs. My solution happens to include a wealth of little exercises that could be reused I think, so here it is.

I put the code snippets in the order that I think is more pedagogical, and leave to the reader to reorganize them in the right one.

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Disjunctive normal forms in big steps

This is probably a second-semester functional programming exercise, but I found it surprisingly hard, and could not find a solution online. So at the risk of depriving a TA from a problem for its mid-term exam, here is my take on it, that I painfully put together yesterday.

Given a formula built out of conjunction, disjunction and atoms, return its disjunctive normal form, in big step or natural semantics, that is, not applying repetitively the distributivity and associativity rules, but in a single function run. Before you go any further, please give it a try and send me your solution!

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Representing pattern-matching with GADTs

Here is a little programming pearl. I’ve been wanting to work on pattern-matching for a while now, and it seems like I will finally have this opportunity here at my new (academic) home, McGill.

Encoding some simply-typed languages with GADTs is now routine for a lot of OCaml programmers. You can even take (kind of) advantage of (some form of) convenient binding representation, like (weak) HOAS; you then use OCaml variables to denote your language’s variables. But what about pattern-matching? Patterns are possibly “deep”, i.e. they bind several variables at a time, and they don’t respect the usual discipline that a variable is bound for exactly its subterm in the AST.

It turns out that there is an adequate encoding, that relies on two simple ideas. The first is to treat variables in patterns as nameless placeholders bound by λ-abstractions on the right side of the arrow (this is how e.g. Coq encodes matches: match E₁ with (y, z) -> E₂ actually is sugar for match E₁ with (_, _) -> fun x y -> E₂); the second is to thread and accumulate type arguments in a GADT, much like we demonstrated in our printf example recently.

The ideas probably extends seamlessly to De Bruijn indices, by threading an explicit environment throughout the term. It stemmed from a discussion on LF encodings of pattern-matching with Francisco over lunch yesterday: what I will show enables also to represent adequately pattern-matching in LF, which I do not think was ever done this way before.

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