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.

Abstract. The problem of finding an optimal solution in a constraint satisfaction problem with preferences has attracted a lot of researchers in Artificial Intelligence in general, and in the constraint programming community in particular. As a consequence, several approaches for expressing and reasoning about satisfiability problems with preferences have been proposed, and viable solutions exist for finding one optimal solution. However, in many cases, it is not desirable to find just one solution. Indeed, it might be desirable to be able to compute more, and possibly all, optimal solutions, e.g., for comparatively evaluate them on the basis of other criteria not captured by the preferences. In this paper we present a procedure for computing all optimal solutions of a satisfiability problem with preferences. The procedure is guaranteed to compute all and only the optimal solutions, i.e., models which are not optimal are not even computed.

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The ability to effectively reason in the presence of qualitative preferences on literals or formulas is a central issue in Artificial Intelligence. In the last few years, two procedures have been presented in order to reason with propositional satisfiability (SAT) problems in the presence of additional, partially ordered qualitative preferences on literals or formulas: the first requires a modification of the branching heuristic of the SAT solver in order to guarantee that the first solution is optimal, while the second computes a sequence of solutions, each guaranteed to be better than the previous one. The two approaches have their own advantages and disadvantages and when compared on specific classes of instances -each having an empty partial order -the second seems to have superior performance.In this paper we show that the above two approaches for reasoning with qualitative preferences can be combined yielding a new effective procedure. In particular, in the new procedure we modify the branching heuristic -as in the first approach -by possibly changing the polarity of the returned literal, and then we continue the search -as in the second approach -looking for better solutions. We extended the experimental analysis conducted in previous papers by considering a wide variety of problems, having both an empty and a non-empty partial order: the results show that the new procedure performs better than the two previous approaches on average, and especially on the "hard" problems. As a preliminary result, we show that the framework of qualitative preferences on literals is more general and expressive than the framework on quantitative preferences.

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