Towards Correctness of Program Transformations Through Unification and Critical Pair Computation

TL;DR

This paper presents a novel method for verifying the correctness of program transformations in extended lambda calculi by using unification-based critical pair computation, demonstrated through a case study involving recursive let-expressions.

Contribution

It introduces a first-order unification approach for computing overlaps in program transformations, enhancing correctness proofs in lambda calculus with recursive constructs.

Findings

Effective unification algorithm for overlaps in lambda calculus

Application to recursive let-expressions case study

Improved correctness verification method for program transformations

Abstract

Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, and then of so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study we apply the method to a lambda calculus with recursive let-expressions and describe an effective unification algorithm to determine all overlaps of a set of transformations with all reduction rules. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modelling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions.