Combining Data Structures with Nonstably Infinite Theories Using Many-Sorted Logic

Abstract

Abstract. Most computer programs store elements of a given nature into container-based data structures such as lists, arrays, sets, and multisets. To verify the correctness of these programs, one needs to combine a theory S modeling the data structure with a theory T modeling the elements. This combination can be achieved using the classic Nelson-Oppen method only if both S and T are stably infinite. The goal of this repot is to relax the stable infiniteness requirement. To achieve this goal, we introduce the notion of polite theories, and we show that natural examples of polite theories include those modeling data structures such as lists, arrays, sets, and multisets. Furthemore, we provide a method that is able to combine a polite theory S with any theory T of the elements, regardless of whether T is stably infinite or not. The results of this report generalize to many-sorted logic those recently obtained by Tinelli and Zarba concerning the combination of shiny theories with nonstably infinite theories in one-sorted logic.

Combined Satisfiability Modulo Parametric Theories

“…However, as we argued, many-sorted logic is not well-suited for working with elaborate combinations of theories, while in a logic with parametric types such combinations are straightforward. In particular, our main result about combination of multiple pairwise disjoint parametric theories, would be difficult even to state in the language of [15]. Yet, the important insight that it is parametricity and not stable infiniteness that justifies Nelson-Oppen cooperation of common solvers is already in [15]; we have given it full expression.…”

“…This corresponds to our requiring that the data structure be a parametric type with flexibility conditions. (More specifically, the "smoothness" and "finite witnessability" parts of politeness correspond to our up-flexibility and downflexibility, the latter being significantly weaker than its counterpart in [15].) The results in [15] can be extended in principle to more than two theories by incremental pairwise combinations.…”

“…(More specifically, the "smoothness" and "finite witnessability" parts of politeness correspond to our up-flexibility and downflexibility, the latter being significantly weaker than its counterpart in [15].) The results in [15] can be extended in principle to more than two theories by incremental pairwise combinations. However, as we argued, many-sorted logic is not well-suited for working with elaborate combinations of theories, while in a logic with parametric types such combinations are straightforward.…”

“…In this, our approach is closely related to the recent work of Ranise et al [15]. They present an extension of the Nelson-Oppen method in which a many-sorted theory S modeling a data structure like lists or arrays can be combined with an arbitrary theory T modeling the elements of the data structure.…”

“…It unnecessarily complicates correctness arguments for combination methods and restricts the applicability of sufficient conditions for their completeness. Thus, researchers have recently begun to frame SMT problems directly in terms of richer typed logics and to develop combination results for these logics [21,4,24,15,3,6]. Ahead of the theory, solvers supporting the PVS system [19], solvers of the CVC family [2], and some others adopted a typed setting early on.…”