What's Decidable About Program Verification Modulo Axioms?

TL;DR

This paper investigates the decidability of program verification when functions and relations are interpreted under various axioms, revealing a complex landscape where some classes are decidable and others remain undecidable.

Contribution

It systematically analyzes the decidability of verifying programs modulo different axioms, extending the understanding of coherent programs and their limitations.

Findings

Some axioms lead to decidable verification problems.

Other axioms still result in undecidable verification.

Coherence is not sufficient for decidability in all cases.

Abstract

We consider the decidability of the verification problem of programs \emph{modulo axioms} --- that is, verifying whether programs satisfy their assertions, when the functions and relations it uses are assumed to interpreted by arbitrary functions and relations that satisfy a set of first-order axioms. Unfortunately, verification of entirely uninterpreted programs (with the empty set of axioms) is already undecidable. A recent work introduced a subclass of \emph{coherent} uninterpreted programs, and showed that they admit decidable verification \cite{coherence2019}. We undertake a systematic study of various natural axioms for relations and functions, and study the decidability of the coherent verification problem. Axioms include relations being reflexive, symmetric, transitive, or total order relations, %and their combinations, functions restricted to being associative, idempotent or commutative, and combinations of such axioms as well. Our comprehensive results unearth a rich landscape that shows that though several axiom classes admit decidability for coherent programs, coherence is not a panacea as several others continue to be undecidable.