What's Decidable About Program Verification Modulo Axioms?
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We consider the decidability of the verification problem of programs 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 coherent uninterpreted programs, and showed that they admit decidable verification <cit.>. 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, 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.
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