Semantical Characterizations and Complexity of Equivalences in Answer Set Programming

In recent research on non-monotonic logic programming, repeatedly strong equivalence of logic programs P and Q has been considered, which holds if the programs P union R and Q union R have the same answer sets for any other program R. This property strengthens equivalence of P and Q with respect to answer sets (which is the particular case for R is the empty set), and has its applications in program optimization, verification, and modular logic programming. In this paper, we consider more liberal notions of strong equivalence, in which the actual form of R may be syntactically restricted. On the one hand, we consider uniform equivalence, where R is a set of facts rather than a set of rules. This notion, which is well known in the area of deductive databases, is particularly useful for assessing whether programs P and Q are equivalent as components of a logic program which is modularly structured. On the other hand, we consider relativized notions of equivalence, where R ranges over rules over a fixed alphabet, and thus generalize our results to relativized notions of strong and uniform equivalence. For all these notions, we consider disjunctive logic programs in the propositional (ground) case, as well as some restricted classes, provide semantical characterizations and analyze the computational complexity. Our results, which naturally extend to answer set semantics for programs with strong negation, complement the results on strong equivalence of logic programs and pave the way for optimizations in answer set solvers as a tool for input-based problem solving.

A Polynomial Translation of Logic Programs with Nested Expressions into Disjunctive Logic Programs: Preliminary Report

Nested logic programs have recently been introduced in order to allow for arbitrarily nested formulas in the heads and the bodies of logic program rules under the answer sets semantics. Nested expressions can be formed using conjunction, disjunction, as well as the negation as failure operator in an unrestricted fashion. This provides a very flexible and compact framework for knowledge representation and reasoning. Previous results show that nested logic programs can be transformed into standard (unnested) disjunctive logic programs in an elementary way, applying the negation as failure operator to body literals only. This is of great practical relevance since it allows us to evaluate nested logic programs by means of off-the-shelf disjunctive logic programming systems, like DLV. However, it turns out that this straightforward transformation results in an exponential blow-up in the worst-case, despite the fact that complexity results indicate that there is a polynomial translation among both formalisms. In this paper, we take up this challenge and provide a polynomial translation of logic programs with nested expressions into disjunctive logic programs. Moreover, we show that this translation is modular and (strongly) faithful. We have implemented both the straightforward as well as our advanced transformation; the resulting compiler serves as a front-end to DLV and is publicly available on the Web.

QUIP - A Tool for Computing Nonmonotonic Reasoning Tasks

In this paper, we outline the prototype of an automated inference tool, called QUIP, which provides a uniform implementation for several nonmonotonic reasoning formalisms. The theoretical basis of QUIP is derived from well-known results about the computational complexity of nonmonotonic logics and exploits a representation of the different reasoning tasks in terms of quantified boolean formulae.