Call-by-value Termination in the Untyped lambda-calculus

TL;DR

This paper presents a fully-automated, sound algorithm that determines whether untyped lambda-expressions terminate under call-by-value evaluation, extending the size-change principle to handle complex recursive and higher-order functions.

Contribution

It introduces a novel extension of the size-change principle for untyped lambda-expressions, enabling automatic termination proofs for a broad class of programs including those with recursion and the Y combinator.

Findings

Algorithm is sound and can certify termination of complex recursive programs.

Successfully applies to programs with mutual recursion and parameter exchanges.

Identifies a substantial subset of PCF as terminating.

Abstract

A fully-automated algorithm is developed able to show that evaluation of a given untyped lambda-expression will terminate under CBV (call-by-value). The ``size-change principle'' from first-order programs is extended to arbitrary untyped lambda-expressions in two steps. The first step suffices to show CBV termination of a single, stand-alone lambda;-expression. The second suffices to show CBV termination of any member of a regular set of lambda-expressions, defined by a tree grammar. (A simple example is a minimum function, when applied to arbitrary Church numerals.) The algorithm is sound and proven so in this paper. The Halting Problem's undecidability implies that any sound algorithm is necessarily incomplete: some lambda-expressions may in fact terminate under CBV evaluation, but not be recognised as terminating. The intensional power of the termination algorithm is reasonably high. It certifies as terminating many interesting and useful general recursive algorithms including programs with mutual recursion and parameter exchanges, and Colson's ``minimum'' algorithm. Further, our type-free approach allows use of the Y combinator, and so can identify as terminating a substantial subset of PCF.