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.
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For most abstract domains, it is possible to find a suitable invariant if one is known to exist, by enumerating all candidate invariants and stopping when an inductive separating one is found. This justifies the use of the existence decision problem for the purposes of theoretical computability studies.
References
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This work was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 306595 “STATOR”.
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Monniaux, D. On the decidability of the existence of polyhedral invariants in transition systems. Acta Informatica 56, 385–389 (2019). https://doi.org/10.1007/s00236-018-0324-y
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DOI: https://doi.org/10.1007/s00236-018-0324-y