Solving satisfiability problems with preferences

Abstract

Abstract. Propositional satisfiability (SAT) is a success story in Computer Science and Artificial Intelligence: SAT solvers are currently used to solve problems in many different application domains, including planning and formal verification. The main reason for this success is that modern SAT solvers can successfully deal with problems having millions of variables. All these solvers are based on the Davis-Logemann-Loveland procedure (DLL). In its original version, DLL is a decision procedure, but it can be very easily modified in order to return one or all assignments satisfying the input set of clauses, assuming at least one exists. However, in many cases it is not enough to compute assignments satisfying all the input clauses: Indeed, the returned assignments have also to be "optimal" in some sense, e.g., they have to satisfy as many other constraints -expressed as preferences-as possible. In this paper we start with qualitative preferences on literals, defined as a partially ordered set (poset) of literals. Such a poset induces a poset on total assignments and leads to the definition of optimal model for a formula ψ as a minimal element of the poset on the models of ψ. We show (i) how DLL can be extended in order to return one or all optimal models of ψ (once converted in clauses and assuming ψ is satisfiable), and (ii) how the same procedures can be used to compute optimal models wrt a qualitative preference on formulas and/or wrt a quantitative preference on literals or formulas. We implemented our ideas and we tested the resulting system on a variety of very challenging structured benchmarks. The results indicate that our implementation has comparable performances with other state-of-the-art systems, tailored for the specific problems we consider.

Implementing Preferences with asprin

“…In [8], qualitative preferences are modeled as a strict partially ordered set (Φ, <) of literals. The literals in Φ represent propositions that are preferably satisfied and the strict partial order < on Φ gives their relative importance.…”

“…To this end, it seems advantageous to direct the solving process towards putative optimal solutions by supplying heuristic information. Although this bears the risk of search degradation [12], it has already indicated great prospects by boosting regular optimization in ASP [13] as well as qualitative preferences [8]. While the latter had to be realized by modifications to a SAT solver, in asprin we draw upon the integration with clingo 4's declarative heuristic framework [13] .…”

“…#minimize statements or weak constraints [2,3]). On the other hand, such quantitative ways of optimization are often insufficient for applications and in stark contrast to the vast literature on qualitative and hybrid means of optimization [4][5][6][7][8]8].…”

“…The primary contribution of this paper is to substantiate this claim by showing how easily selected approaches from the literature can be realized with asprin. In particular, we detail how answer set optimization [5], minimization directives [2], strict partially ordered set [8], and the non-temporal part of the preference language in [7] can be implemented with asprin. Moreover, we sketch how the implementations of ordered disjunctions [4] and penalty-based answer set optimization [6] are obtained.…”

The Top-k Frequent Closed Itemset Mining Using Top-k SAT Problem

“…Preferences in propositional satisfiability (SAT) has not received a lot of attention. In [15], a new approach for solving satisfiability problems in the presence of qualitative preferences on literals (defined as partial ordered set) is proposed. The authors particularly show how DPLL procedure can be easily adapted for computing optimal models induced by the partial order.…”