Rigid E-unification

By replacing syntactical unification with rigid E-unification, equality handling can be added to rigid variable calculi for first-order logic, including free variable tableau (Fitting, 1996), the mating method (Andrews, 1981), the connection method (Bibel, 1982), and model elimination (Loveland, 1969); for an overview of these calculi, see Chapters I.1.1 and I.1.2. Rigid E-unification and its significance for automated theorem proving was first described in (Gallier et al., 1987). An earlier attempt to formulate the generalized unification problem that has to be solved for handling equality in rigid variable calculi can be found in (Bibel, 1982). Ground E-unification (i.e., E-unification with variable-free equalities) has long been known to be decidable (Sect. 2.3), and classical universal E-unification has long been known to be undecidable (Chap. I.2.7). Rigid E-unification is in between: It is decidable in the simple, non-simultaneous case (Sect. 2.4), but it is undecidable whether there is a simultaneous solution for several rigid E-unification problems (Sect. 3.2), which is unfortunate as simultaneous rigid E-unification is of great importance for handling equality in automated theorem proving (Sect. 5). In the remainder of this section, we describe the basic idea of rigid Eunification and its importance for adding equality to rigid variable calculi and introduce syntax and semantics of first-order logic with equality. In Section 2, we formally define (non-simultaneous) rigid E-unification and the notion of (minimal) complete sets of unifiers; and we briefly sketch proofs for the decidability of ground E-unification and—based on this—for rigid E-unification; methods for solving rigid E-unification problems are compared. In Section 3.3, the problem of finding a simultaneous solution for several rigid E-unification problems is discussed; and in Section 4, mixed E-unification is introduced, that is a combination of classical and rigid E-unification. Using the example of free variable semantic tableaux, we show in Section 5 how rigid E-unification can be used to handle equality in a rigid variable calcu-