Formal neighbourhoods, combinatory Bohm trees, and untyped normalization by evaluation

Artikel i vetenskaplig tidskrift, 2012

We prove the correctness of an algorithm for normalizing untyped combinator terms by evaluation. The algorithm is written in the functional programming language Haskell, and we prove that it lazily computes the combinatory Bohm tree of the term. The notion of combinatory Bohm tree is analogous to the usual notion of Bohm tree for the untyped lambda calculus. It is defined operationally by repeated head reduction of terms to head normal forms. We use formal neighbourhoods to characterize finite, partial information about data, and define a Bohm tree as a filter of such formal neighbourhoods. We also define formal topology style denotational semantics of a fragment of Haskell following Martin-Lof, and let each closed program denote a filter of formal neighbourhoods. We prove that the denotation of the output of our algorithm is the Bohm tree of the input term. The key construction in the proof is a "glueing" relation between terms and semantic neighbourhoods which is defined by induction on the latter. This relation is related to the glueing relation which was earlier used for proving the correctness of normalization by evaluation algorithm for typed combinatory logic.