Satisfiability Modulo the Theory of Costs: Foundations and Applications

Abstract

Abstract. We extend the setting of Satisfiability Modulo Theories (SMT) by introducing a theory of costs C, where it is possible to model and reason about resource consumption and multiple cost functions, e.g., battery, time, and space. We define a decision procedure that has all the features required for the integration withint the lazy SMT schema: incrementality, backtrackability, construction of conflict sets, and deduction. This naturally results in an SMT solver for the disjoint union of C and any other theory T .This framework has two important applications. First, we tackle the problem of Optimization Modulo Theories: rather than checking the existence of a satisfying assignment, as in SMT, we require a satisfying assignment that minimizes a given cost function. We build on the decision problem for SMT with costs, i.e., finding a satisfying assigniment with cost within an admissibility range, and propose two algorithms for optimization. Second, we use multiple cost functions to deal with PseudoBoolean constraints. Within the SMT(C) framework, the effectively PseudoBoolean constraints are dealt with by the cost solver, while the other constraints are reduced to pure boolean reasoning.We implemented the proposed approach within the MathSAT SMT solver, and we experimentally evaluated it on a large set of benchmarks, also from industrial applications. The results clearly demonstrate the potential of the approach. Motivations and GoalsImportant verification problems are naturally encoded as Satisfiability Modulo Theory (SMT) problems, i.e. as satisfiability problems for decidable fragments of first order logic. Efficient SMT solvers have been developed, that combine the power of SAT solvers with dedicated decision procedures for several theories of practical interest.In many practical domains, problems require modeling and reasoning about resource consumption and multiple cost functions, e.g., battery, time, and space. In this paper, we extend SMT by introducing a theory of costs C. The language of the theory of costs is very expressive: it allows for multiple cost functions and, for each of these, arbitrary Boolean conditions may be stated to result in a given cost increase; costs may be be both lower-and upper-bounded, also depending on Boolean conditions.

Symbolic optimization with SMT solvers

“…In [17], Cimatti et al proposed a theory of costs for augmenting an SMT solver with pseudo-boolean (PB) constraints [9]. At a high level, their theory allows associating a cost with individual Boolean constraints.…”

MaxSAT-Based MCS Enumeration

“…In 2010, Cimatti et al [6] proposed a new theory called the theory of Costs C that allows modeling multiple cost functions, and they developed a decision procedure for C. Using the theory C, Cimatti et al [6] showed how to address the problem of minimizing the value of one cost function subject to the satisfaction of a SMT(T ) formula, which they called the Boolean Optimization Modulo Theory (BOMT) problem. The optimization itself is obtained by linear search or binary search, asserting atoms of C that bound the cost, and using an incremental SMT solver.…”

“…The optimization itself is obtained by linear search or binary search, asserting atoms of C that bound the cost, and using an incremental SMT solver. Cimatti et al [6] encoded the weighted partial MaxSMT into BOMT by adding a new Boolean variable A i j to each soft clause. Then, the cost function is the sum of the weights of the soft clauses, whose variable A i j is assigned true.…”

Minimal-Model-Guided Approaches to Solving Polynomial Constraints and Extensions

“…We propose heuristics for guiding the domain relaxation step by means of the analysis of minimal models generated by the SMT(LIA) solver. More specifically, we consider two different cost functions: first, the number of violated artificial domain bounds, which leads to Maximum Satisfiability Modulo Theories (Max-SMT, [28,29]) problems; and second, the distance with respect to the artificial domains, which boils down to Optimization Modulo Theories (OMT, [30,31]) problems. The results of comparing these approaches with other techniques show the potential of the method.…”