The Complexity of Reasoning for Fragments of Default Logic

Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as \SigmaPtwo-complete, and the complexity of the credulous and skeptical reasoning problem as SigmaP2-complete, resp. PiP2-complete. Additionally, he investigated restrictions on the default rules, i.e., semi-normal default rules. Selman made in 1992 a similar approach with disjunction-free and unary default rules. In this paper we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a hexachotomy (SigmaP2-, DeltaP2-, NP-, P-, NL-complete, trivial) for the extension existence problem, while for the credulous and skeptical reasoning problem we obtain similar classifications without trivial cases.Comment: Corrected versio

FlexiMine - A Flexible Platform for KDD Research and Application Development

FlexiMine is a KDD system designed as a testbed for data-mining research, as well as a generic knowledge discovery tool for varied database domains. Flexibility is achieved by an open-ended design that enables integration of existing data-mining algorithms, new locally developed algorithms, and utility functions such as visualization and preprocessing. The system interfaces an INFORMIX database server via SQL queries and thus support for new databases is simple and clean. With a view of serving remote, as well as local, users, internet availability was a design goal. Fleximine is implemented in Java and hence we can run the user-end of the system either as a Java applications or (with some limitations on the user) as a Java applet. This paper reviews the architecture, design and operation of FlexiMine and presents new ideas incorporated in the data-mining algorithms (Association rules, Decision trees, Bayesian knowledge-bases and Meta-queries). Keywords: Knowledge Discovery in Databas..

Minimal-model-guided approaches to solving polynomial constraints and extensions

In this paper we present new methods for deciding the satisfiability of formulas involving integer polynomial constraints. In previous work we proposed to solve SMT(NIA) problems by reducing them to SMT(LIA): non-linear monomials are linearized by abstracting them with fresh variables and by performing case splitting on integer variables with finite domain. When variables do not have finite domains, artificial ones can be introduced by imposing a lower and an upper bound, and made iteratively larger until a solution is found (or the procedure times out). For the approach to be practical, unsatisfiable cores are used to guide which domains have to be relaxed (i.e., enlarged) from one iteration to the following one. However, it is not clear then how large they have to be made, which is critical. Here we propose to guide the domain relaxation step by analyzing minimal models produced by the SMT(LIA) solver. Namely, we consider two different cost functions: the number of violated artificial domain bounds, and the distance with respect to the artificial domains. We compare these approaches with other techniques on benchmarks coming from constraint-based program analysis and show the potential of the method. Finally, we describe how one of these minimal-model-guided techniques can be smoothly adapted to deal with the extension Max-SMT of SMT(NIA) and then applied to program termination proving.Peer Reviewe