Modular Termination and Combinability for Superposition Modulo Counter Arithmetic

Abstract

Abstract. Modularity is a highly desirable property in the development of satisfiability procedures. In this paper we are interested in using a dedicated superposition calculus to develop satisfiability procedures for (unions of) theories sharing counter arithmetic. In the first place, we are concerned with the termination of this calculus for theories representing data structures and their extensions. To this purpose, we prove a modularity result for termination which allows us to use our superposition calculus as a satisfiability procedure for combinations of data structures. In addition, we present a general combinability result that permits us to use our satisfiability procedures into a non-disjoint combination method a la Nelson-Oppen without loss of completeness. This latter result is useful whenever data structures are combined with theories for which superposition is not applicable, like theories of arithmetic.

Theory Combination: Beyond Equality Sharing

“…Another way to go beyond equality sharing is to admit combinations of non-disjoint theories (e.g., [65,67,101,125,118]). Work on this direction has begun for methods based on gentleness and politeness [43,44], as well as for superposition-based decision procedures [111], while it is a direction of future work for CDSAT.…”

A rule-based system for automatic decidability and combinability

“…Example 1 The schematic saturation of Ax(LLI) ∪ G + 0 consists of Ax(LLI), G + 0 and the following schematic literals: Example 2 Let RII be the theory of records of length 3 with increment whose signature is Σ RII = 3 i=1 {rstore i : rec × int → rec, rselect i : rec → int, incr : rec → rec, s : int → int} and whose set of axioms Ax(RII) consists of 3 i {rselect i (rstore i (X, Y )) = Y, rselect i (incr(X)) = s(rselect i (X))} and {rselect i ( rstore j (X, Y )) = rselect i (X)} for all i, j ∈ {1, 2, 3}, i = j. The schematic saturation of Ax(RII) ∪ G + 0 consists of Ax(RII), G + 0 and the following schematic literals, for all i ∈ {1, 2, 3}: In [18], Lemma 1 has been lifted to the case of Integer Offsets: a specific condition on the form of the schematic saturation computed by SUPC I is sufficient to show that a theory can be combined by using UPC I , along the lines of [19]. This condition is indeed satisfied for both theories LLI and RII as shown in [18], which in particular means that UPC I is also a satisfiability procedure for LLI ∪ RII.…”

“…A superposition between (33) and (17) generates the new constrained clause (26) that subsumes (33). Superposition between ( 11) and ( 21) yields a renamed copy of (19) which is immediately removed by Subsumption. Similarly, Superposition between ( 12) and ( 21) yields a renamed copy of (20) also removed by Subsumption.…”