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

## Tag: big step

### 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.

### 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!

### Strong reduction in big steps

It has been a long while since I updated this blog, let me finally revive it with an easy but (hopefully) fun post, well ehm… at least some usable reference material!

Strong reduction is the ability for the interpreter of a functional language to reduce under function declaration. Usually, in OCaml for example, evaluation stops at the λ boundary; we might like to reduce under them for various reasons:

• partial evaluation (e.g. λx. 2+2+x ⇒ λx. 4+x)
• conversion checking in dependent type systems (is λx. (λy. y) x the same as λx. x?)

I never remember the big-step semantics of strong reduction, so I’m going to write them down here in glorious Unicode-art. Over the years I accumulated many small variants of reduction machines, both weak and strong, call-by-name or call-by-value… Here is a first shot, starting with the canonical, CBV weak head reduction and incrementally building up strong reduction; I’ll be probably extending this list when I remember other variants.