We investigate the application of inconsistency measures to the problem of analysing business rule bases. Due to some i ntri-cacies of the domain of business rule bases, a straightforwa rd application is not feasible. We therefore develop some new rat ionality postulates for this setting as well as adapt and modify exist ing inconsistency measures. We further adapt the notion of inconsistency values (or culpability measures) for this setting and give a comprehensive feasibility study.

In this paper, we propose a general framework, both parameterized and parameter-free, for defining a family of fine-grained inconsistency measures for propositional knowledge bases. The parameterized approach allows to encompass several existing inconsistency mea- sures as specific cases, by properly setting its parameter. And the parameter-free approach is defined to avoid the difficulty in choosing a suitable parameter in practice but still keeps a desired ranking for knowledge bases by their inconsistency degrees. The fine granularity of our framework is based on the notion of MIS partition that considers the inner structure of all the minimal inconsistent subsets of a knowledge base. Moreover, MinCostSAT-based encodings are provided, which enable the use of efficient SAT solvers for the computation of the proposed measures. We implement these algo- rithms and test them on some real-world datasets. The preliminary experimental results for a variety of inputs show that the proposed framework gives a wide range of possibilities for evaluating large knowledge bases.

Recently, MaxSAT reasoning is shown very effective in computing a tight upper bound for a Maximum Clique (MC) of a (unweighted) graph. In this paper, we apply MaxSAT reasoning to compute a tight upper bound for a Maximum Weight Clique (MWC) of a wighted graph. We first study three usual encodings of MWC into weighted partial MaxSAT dealing with hard clauses, which must be satisfied in all solutions, and soft clauses, which are weighted and can be falsified. The drawbacks of these encodings motivate us to propose an encoding of MWC into a special weighted partial MaxSAT formalism, called LW (Literal-Weighted) encoding and dedicated for upper bounding an MWC, in which both soft clauses and literals in soft clauses are weighted. An optimal solution of the LW MaxSAT instance gives an upper bound for an MWC, instead of an optimal solution for MWC. We then introduce two notions called the Top-k literal failed clause and the Top-k empty clause to extend classical MaxSAT reasoning techniques, as well as two sound transformation rules to transform an LW MaxSAT instance. Successive transformations of an LW MaxSAT instance driven by MaxSAT reasoning give a tight upper bound for the encoded MWC. The approach is implemented in a branch-and-bound algorithm called MWCLQ. Experimental evaluations on the broadly used DIMACS benchmark, BHOSLIB benchmark, random graphs and the benchmark from the winner determination problem show that our approach allows MWCLQ to reduce the search space significantly and to solve MWC instances effectively. Consequently, MWCLQ outperforms state-of-the-art exact algorithms on the vast majority of instances. Moreover, it is surprisingly effective in solving hard and dense instances.

At each node of the search tree, Branch and Bound solvers for Max-SAT compute the lower bound (LB) by estimating the number of disjoint inconsistent subsets (IS) of the formula. IS are detected thanks to unit propagation (UP) then transformed by max-resolution to ensure that they are counted only once. However, it has been observed experimentally that the max-resolution transformations impact the capability of UP to detect further IS. Consequently, few transformations are learned and the LB computation is redundant. In this paper, we study the effect of the transformations on the UP mechanism. We introduce the notion of UP-resiliency of a transformation, which quantifies its impact on UP. It provides, from a theoretical point of view, an explanation to the empirical efficiency of the learning scheme developed in the last ten years. The experimental results we present give evidences of UP-resiliency relevance and insights on the behavior of the learning mechanism.

We analyse the computational complexity of three problems in judgment aggregation: (1) computing a collective judgment from a profile of individual judgments (the winner determination problem); (2) deciding whether a given agent can influence the outcome of a judgment aggregation procedure in her favour by reporting insincere judgments (the strategic manipulation problem); and (3) deciding whether a given judgment aggregation scenario is guaranteed to result in a logically consistent outcome, independently from what the judgments supplied by the individuals are (the problem of the safety of the agenda). We provide results both for specific aggregation procedures (the quota rules, the premise-based procedure, and a distance-based procedure) and for classes of aggregation procedures characterised in terms of fundamental axioms.

Exact Max-SAT solvers, compared with SAT solvers, apply little inference at each node of the proof tree. Commonly used SAT inference rules like unit propagation produce a simplified formula that preserves satisfiability but, unfortunately, solving the Max-SAT problem for the simplified formula is not equivalent to solving it for the original formula. In this paper, we define a number of original inference rules that, besides being applied efficiently, transform Max-SAT instances into equivalent Max-SAT instances which are easier to solve. The soundness of the rules, that can be seen as refinements of unit resolution adapted to Max-SAT, are proved in a novel and simple way via an integer programming transformation. With the aim of finding out how powerful the inference rules are in practice, we have developed a new Max-SAT solver, called MaxSatz, which incorporates those rules, and performed an experimental investigation. The results provide empirical evidence that MaxSatz is very competitive, at least, on random Max-2SAT, random Max-3SAT, Max-Cut, and Graph 3-coloring instances, as well as on the benchmarks from the Max-SAT Evaluation 2006.

State-of-the-art branch-and-bound algorithms for the maximum clique problem (Maxclique) frequently use an upper bound based on a partition P of a graph into independent sets for a maximum clique of the graph, which cannot be very tight for imperfect graphs. In this paper we propose a new encoding from Maxclique into MaxSAT and use MaxSAT technology to improve the upper bound based on the partition P. In this way, the strength of specific algorithms for Maxclique in partitioning a graph and the strength of MaxSAT technology in propositional reasoning are naturally combined to solve Maxclique. Experimental results show that the approach is very effective on hard random graphs and on DIMACS Maxclique benchmarks, and allows to close an open DIMACS problem.

Exact Max-SAT solvers, compared with SAT solvers, apply little inference at each node of the proof tree. Commonly used SAT inference rules like unit propagation produce a simplified formula that preserves satisfiability but, unfortunately, solving the Max-SAT problem for the simplified formula is not equivalent to solving it for the original formula. In this paper, we define a number of original inference rules that, besides being applied efficiently, transform Max-SAT instances into equivalent Max-SAT instances which are easier to solve. The soundness of the rules, that can be seen as refinements of unit resolution adapted to Max-SAT, are proved in a novel and simple way via an integer programming transformation. With the aim of finding out how powerful the inference rules are in practice, we have developed a new Max-SAT solver, called MaxSatz, which incorporates those rules, and performed an experimental investigation. The results provide empirical evidence that MaxSatz is very competitive, at least, on random Max-2SAT, random Max-3SAT, Max-Cut, and Graph 3-coloring instances, as well as on the benchmarks from the Max-SAT Evaluation 2006.