A decidable subclass of finitary programs

Answer set programming - the most popular problem solving paradigm based on logic programs - has been recently extended to support uninterpreted function symbols. All of these approaches have some limitation. In this paper we propose a class of programs called FP2 that enjoys a different trade-off between expressiveness and complexity. FP2 programs enjoy the following unique combination of properties: (i) the ability of expressing predicates with infinite extensions; (ii) full support for predicates with arbitrary arity; (iii) decidability of FP2 membership checking; (iv) decidability of skeptical and credulous stable model reasoning for call-safe queries. Odd cycles are supported by composing FP2 programs with argument restricted programs

On finitely recursive programs

Disjunctive finitary programs are a class of logic programs admitting function symbols and hence infinite domains. They have very good computational properties, for example ground queries are decidable while in the general case the stable model semantics is highly undecidable. In this paper we prove that a larger class of programs, called finitely recursive programs, preserves most of the good properties of finitary programs under the stable model semantics, namely: (i) finitely recursive programs enjoy a compactness property; (ii) inconsistency checking and skeptical reasoning are semidecidable; (iii) skeptical resolution is complete for normal finitely recursive programs. Moreover, we show how to check inconsistency and answer skeptical queries using finite subsets of the ground program instantiation. We achieve this by extending the splitting sequence theorem by Lifschitz and Turner: We prove that if the input program P is finitely recursive, then the partial stable models determined by any smooth splitting omega-sequence converge to a stable model of P.Comment: 26 pages, Preliminary version in Proc. of ICLP 2007, Best paper awar

On program grounding in ASP

Answer set programming (ASP) is a declarative problem solving framework introduced by Michael Gelfond and Vladimir Lifschitz in the late ’80s. ASP has received much attention by researchers for its expressiveness and simpleness so that well-engineered and optimized implementations have been developed for it. However, state-of-the-art answer set solvers have still a strong limitation: they are not be able to reason on nonground programs and then the input program have to be instantiated before the solver can start to reason on it. Consequently, answer set solvers (i) cannot handle infinite domains and (ii) use huge amounts of memory even if domains are finite. This work wants to give some contribution for these two not trivial problems. First, I analyze finitary programs as a class of programs that can effectively deal with function symbols and recursion (hence infinite domains and models). Interestingly, even if finitary programs are computationally complete, their restrictions make it possible to keep complexity under control. I study the consequences of relaxing the restrictions on finitary programs and my results enforce a kind of minimality of the properties that characterize finitary programs. Next, I investigate what happens when we “compose” two programs P and Q belonging to some particular classes that imposing them some restrictions guarantee good computational properties, so obtaining a program P [ Q that, as a whole, might not be subject to the restrictions of P or Q but that again enjoys good computational properties. Finally, I study a new approach to tackle the problem (ii) of ASP. The idea is to integrate answer set generation and constraint solving to reduce the memory requirements for a class of multi-sorted logic programs with cardinality constraints: constrained programs. I prove some theoretical results, introduce provably sound and complete algorithms, and report experimental results on my prototype system for evaluating constrained programs, showing that my approach can solve problem instances with significantly larger domains

A Preliminary Report on Integrating of Answer Set and Constraint Solving

Abstract. Despite all efforts on intelligent grounding, state-of-the-art answer set solvers still have huge memory requirements, because they compute the ground instantiation of the input program before the actual reasoning starts. This prevents ASP to be effective on several classes of problems. In this paper we integrate answer set generation and constraint solving to reduce the memory requirements for a class of multi-sorted logic programs with cardinality constraints. We prove some theoretical results, introduce a provably sound and complete algorithm, and report experimental results showing that our approach can solve problem instances with significantly larger domains.