Automated Termination Proofs for Logic Programs by Term Rewriting

There are two kinds of approaches for termination analysis of logic programs: "transformational" and "direct" ones. Direct approaches prove termination directly on the basis of the logic program. Transformational approaches transform a logic program into a term rewrite system (TRS) and then analyze termination of the resulting TRS instead. Thus, transformational approaches make all methods previously developed for TRSs available for logic programs as well. However, the applicability of most existing transformations is quite restricted, as they can only be used for certain subclasses of logic programs. (Most of them are restricted to well-moded programs.) In this paper we improve these transformations such that they become applicable for any definite logic program. To simulate the behavior of logic programs by TRSs, we slightly modify the notion of rewriting by permitting infinite terms. We show that our transformation results in TRSs which are indeed suitable for automated termination analysis. In contrast to most other methods for termination of logic programs, our technique is also sound for logic programming without occur check, which is typically used in practice. We implemented our approach in the termination prover AProVE and successfully evaluated it on a large collection of examples.Comment: 49 page

Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms

In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (c-atoms). These approaches generalize the fixpoint-based and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of c-atoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpoint-based semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complement. These definitions, which differ from each other in the treatment of negation-as-failure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of c-atoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly well-supported models, are generalizations of the notion of well-supported models of normal logic programs to the case of programs with c-atoms. As for the case of fixpoint-based semantics, the difference between these two definitions is rooted in the treatment of naf atoms. We prove that answer sets by reduct (resp. by complement) are equivalent to weakly (resp. strongly) well-supported models of a program, thus generalizing the theorem on the correspondence between stable models and well-supported models of a normal logic program to the class of programs with c-atoms. We show that the newly defined semantics coincide with previously introduced semantics for logic programs with monotone c-atoms, and they extend the original answer set semantics of normal logic programs. We also study some properties of answer sets of programs with c-atoms, and relate our definitions to several semantics for logic programs with aggregates presented in the literature