On Tackling the Limits of Resolution in SAT Solving

The practical success of Boolean Satisfiability (SAT) solvers stems from the CDCL (Conflict-Driven Clause Learning) approach to SAT solving. However, from a propositional proof complexity perspective, CDCL is no more powerful than the resolution proof system, for which many hard examples exist. This paper proposes a new problem transformation, which enables reducing the decision problem for formulas in conjunctive normal form (CNF) to the problem of solving maximum satisfiability over Horn formulas. Given the new transformation, the paper proves a polynomial bound on the number of MaxSAT resolution steps for pigeonhole formulas. This result is in clear contrast with earlier results on the length of proofs of MaxSAT resolution for pigeonhole formulas. The paper also establishes the same polynomial bound in the case of modern core-guided MaxSAT solvers. Experimental results, obtained on CNF formulas known to be hard for CDCL SAT solvers, show that these can be efficiently solved with modern MaxSAT solvers

Modelling Combinatorial Auctions in Linear Logic

We show that linear logic can serve as an expressive framework in which to model a rich variety of combinatorial auction mechanisms. Due to its resource-sensitive nature, linear logic can easily represent bids in combinatorial auctions in which goods may be sold in multiple units, and we show how it naturally generalises several bidding languages familiar from the literature. Moreover, the winner determination problem, i.e., the problem of computing an allocation of goods to bidders producing a certain amount of revenue for the auctioneer, can be modelled as the problem of finding a proof for a particular linear logic sequent

ON SIMPLE BUT HARD RANDOM INSTANCES OF PROPOSITIONAL THEORIES AND LOGIC PROGRAMS

In the last decade, Answer Set Programming (ASP) and Satisfiability (SAT) have been used to solve combinatorial search problems and practical applications in which they arise. In each of these formalisms, a tool called a solver is used to solve problems. A solver takes as input a specification of the problem – a logic program in the case of ASP, and a CNF theory for SAT – and produces as output a solution to the problem. Designing fast solvers is important for the success of this general-purpose approach to solving search problems. Classes of instances that pose challenges to solvers can help in this task. In this dissertation we create challenging yet simple benchmarks for existing solvers in ASP and SAT.We do so by providing models of simple logic programs as well as models of simple CNF theories. We then randomly generate logic programs as well as CNF theories from these models. Our experimental results show that computing answer sets of random logic programs as well as models of random CNF theories with carefully chosen parameters is hard for existing solvers. We generate random logic programs with 2-literals, and our experiments show that it is hard for ASP solvers to obtain answer sets of purely negative and constraint-free programs, indicating the importance of these programs in the development of ASP solvers. An easy-hard-easy pattern emerges as we compute the average number of choice points generated by ASP solvers on randomly generated 2-literal programs with an increasing number of rules. We provide an explanation for the emergence of this pattern in these programs. We also theoretically study the probability of existence of an answer set for sparse and dense 2-literal programs. We consider simple classes of mixed Horn formulas with purely positive 2- literal clauses and purely negated Horn clauses. First we consider a class of mixed Horn formulas wherein each formula has m 2-literal clauses and k-literal negated Horn clauses. We show that formulas that are generated from the phase transition region of this class are hard for complete SAT solvers. The second class of Mixed Horn Formulas we consider are obtained from completion of a certain class of random logic programs. We show the appearance of an easy-hard-easy pattern as we generate formulas from this class with increasing numbers of clauses, and that the formulas generated in the hard region can be used as benchmarks for testing incomplete SAT solvers

Probabilistic Analysis of Random Mixed Horn Formulas

We present a probabilistic analysis of random mixed Horn formulas (MHF), i.e., formulas in conjunctive normal form consisting of a positive monotone part of quadratic clauses and a part of Horn clauses, with m clauses, n variables, and up to n literals per Horn clause. For MHFs parameterized by n and m with uniform distribution of instances and for large n, we derive upper bounds for the expected number of models. For the class of random negative MHFs, where only monotone negative Horn clauses are allowed to occur, we give a lower bound for the probability that formulas from this class are satisfiable. We expect that the model studied theoretically here may be of interest for the determination of hard instances, which are conjectured to be found in the transition area from satisfiability to unsatisfiability of the instances from the parameterized classes of formulas

Probabilistic Analysis of Random Mixed Horn Formulas

We present a probabilistic analysis of random mixed Horn formulas (MHF), i.e., formulas in conjunctive normal form consisting of a positive monotone part of quadratic clauses and a part of Horn clauses, with m clauses, n variables, and up to n literals per Horn clause. For MHFs parameterized by n and m with uniform distribution of instances and for large n, we derive upper bounds for the expected number of models. For the class of random negative MHFs, where only monotone negative Horn clauses are allowed to occur, we give a lower bound for the probability that formulas from this class are satisfiable. We expect that the model studied theoretically here may be of interest for the determination of hard instances, which are conjectured to be found in the transition area from satisfiability to unsatisfiability of the instances from the parameterized classes of formulas

Algorithms for Variable-Weighted 2-SAT and Dual Problems

In this paper we study NP-hard weighted satisfiability optimization problems for the class 2-CNF providing worst-case upper time bounds. Moreover we consider the monotone dual class consisting of clause sets where all variables occur at most twice. We show that weighted SAT, XSAT and NAESAT optimization problems for this class are polynomial time solvable using appropriate reductions to specific polynomial time solvable graph problems