On the decidability of the existence of polyhedral invariants in transition systems

TL;DR

This paper proves that determining the existence of convex polyhedral invariants in transition systems is undecidable, highlighting fundamental limitations in automated program verification for numerical properties.

Contribution

It establishes the undecidability of the existence of convex polyhedral invariants, a problem previously unresolved in formal verification.

Findings

Existence of convex polyhedral invariants is undecidable.

Undecidability holds even with a single control state.

The problem remains open without nonlinear constraints.

Abstract

Automated program verification often proceeds by exhibiting inductive invariants entailing the desired properties.For numerical properties, a classical class of invariants is convex polyhedra: solution sets of system of linear (in)equalities.Forty years of research on convex polyhedral invariants have focused, on the one hand, on identifying "easier" subclasses, on the other hand on heuristics for finding general convex polyhedra.These heuristics are however not guaranteed to find polyhedral inductive invariants when they exist.To our best knowledge, the existence of polyhedral inductive invariants has never been proved to be undecidable.In this article, we show that the existence of convex polyhedral invariants is undecidable, even if there is only one control state in addition to the "bad" one.The question is still open if one is not allowed any nonlinear constraint.