Free Variables and Theories: Revisiting Rigid E-unification

Abstract

The efficient integration of theory reasoning in first-order calculi with free variables (such as sequent calculi or tableaux) is a long-standing challenge. For the case of the theory of equality, an approach that has been extensively studied in the 90s is rigid E-unification, a variant of equational unification in which the assumption is made that every variable denotes exactly one term (rigid semantics). The fact that simultaneous rigid E-unification is undecidable, however, has hampered practical adoption of the method, and today there are few theorem provers that use rigid E-unification.

One solution is to consider incomplete algorithms for computing (simultaneous) rigid E-unifiers, which can still be sufficient to create sound and complete theorem provers for first-order logic with equality; such algorithms include rigid basic superposition proposed by Degtyarev and Voronkov, but also the much older subterm instantiation approach introduced by Kanger in 1963 (later also termed minus-normalisation). We introduce bounded rigid E-unification (BREU) as a new variant of E-unification corresponding to subterm instantiation. In contrast to general rigid E-unification, BREU is NP-complete for individual and simultaneous unification problems, and can be solved efficiently with the help of SAT; BREU can be combined with techniques like congruence closure for ground reasoning, and be used to construct theorem provers that are competitive with state-of-the-art tableau systems. We outline ongoing research how BREU can be generalised to other theories than equality.

This work was partly supported by the Microsoft PhD Scholarship Programme and the Swedish Research Council.