Case Study: Saturations as Explicit Models in Equational Theories

TL;DR

This paper reveals that saturated sets from ATPs in the equational fragment can serve as explicit, checkable models, and introduces a method to extract such models, enhancing trustworthiness of automated reasoning.

Contribution

It introduces a novel approach to extract explicit, checkable models from saturated sets in equational theories, implemented in Vampire and E.

Findings

Explicit models can be derived from saturated sets in equational theories.

The extracted rewrite systems are checkable for confluence and termination.

The method was applied successfully to hundreds of implications without finite countermodels.

Abstract

Automated theorem provers (ATPs) can disprove conjectures by saturating a set of clauses, but the resulting saturated sets are opaque certificates. In the unit equational fragment, a saturated set can in fact be read as a convergent rewrite system defining an explicit, possibly infinite, model -- but this is not widely known, even amongst frequent users of ATPs. Moreover, ATPs do not emit these explicit certificates for infinite (counter-)models. We present such a certificate construction in full, implement it in Vampire and E, and apply it to the recent Equational Theories Project, where hundreds of implications do not admit finite countermodels. The resulting rewrite systems can be checked for confluence and termination by existing certified tools, yielding trustworthy countermodels.