Ground interpolation for the theory of equality

Abstract

Abstract. Given a theory T and two formulas A and B jointly unsatisfiable in T , a theory interpolant of A and B is a formula I such that (i) its non-theory symbols are shared by both A and B, (ii) it is entailed by A in T , and (iii) it is unsatisfiable with B in T . Theory interpolation has found several successful applications in model checking. We present a novel method for computing interpolants for ground formulas in the theory of equality. The method produces interpolants from colored congruence graphs representing derivations in that theory. These graphs can be produced by conventional congruence closure algorithms in a straightforward manner. By working with graphs, rather than at the level of individual proof steps, we are able to derive interpolants that are pleasingly simple (conjunctions of Horn clauses) and smaller than those generated by other tools. Our interpolation method can be seen as a theory-specific implementation of a cooperative interpolation game between two provers. We present a generic version of the interpolation game, parametrized by the theory T , and define a general method to extract runs of the game from proofs in T and then generate interpolants from these runs.

Controller synthesis for pipelined circuits using uninterpreted functions

“…An approach, with a refinement loop were transitivity constraints are added "on demand", i.e., only when they are needed to prove equivalence, could alleviate this problem. Moreover, since BDDs seem to be a suboptimal data structure for this kind of problem, we currently investigate how QBF-solving techniques and interpolation in equality logic with uninterpreted functions [15], [26] can be used as alternative methods to compute control functions. Preliminary experiments suggest that interpolation in equality logic with uninterpreted functions is a very efficient method for finding control functions.…”

“…Proof: "⇒": We assume validity of Equation 14 and prove validity of Equation 15. Letd x f ,d x f ,s, ands be arbitrary interpretations ford x f ,d x f ,s, ands , in Equation 15 respectively.…”

Quantifier-free interpolation in combinations of equality interpolating theories

“…For any signature , let EUF( ) be the pure theory of equality over . This theory has quantifier-free interpolation (see, e.g., [McMillan 2004;Fuchs et al 2009]). Indeed, it is easy to see that EUF( ) has the (strong) sub-amalgamation property by building a model M of EUF( ) from two models M 1 and M 2 sharing a substructure A as follows.…”