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Normalization by Evaluation, for combinators

  • combinator.agda defines the basic concepts. We work in a language with the combinators 𝕂 𝕊, and the natural numbers O S, together with a recursion combinator ℝ.

  • reduce.agda describes how to reduce combinators with reductions, basically a big-step semantics. Note that we have to add the Agda pragma {-# TERMINATING #-}, because it's not obvious that such a reduction terminates.

  • nbe.agda uses normalization by evaluation. Apart from being slightly faster (I cannot measure accurately, but it seems to be around 2x faster), it also convinces Agda that the process terminates.

  • nbe.py gives a quick implementation in python, stripped of all the proofs. It is basically just 10 lines!

Normalization by Evaluation, for simply typed lambda calculus

I eventually got around to implement NbE for STLC. Please note that since I'm working on a case-insensitive filesystem, you might need to adjust the file cases according to this Readme.

  • Equivalence.agda defines handy tools.
  • STLC.agda defines simply typed lambda calculus, demonstrates how to translate it into combinators, and defines relevant basic concepts.
  • Substitution.agda proves various substitution lemmas.
  • NbE.agda implements normalization by evaluation.
  • Category.agda packs up everything we proved in previous files step by step into a neat, categorical language, as described in Chapter 4, Sections 1-2 of Jonathan Sterling's thesis First Steps in Synthetic Tait Computability.

The files have plenty of comments, and are intended to be read in the order as listed.