On finitely recursive programs

📝 Abstract

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.

💡 Analysis

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.

📄 Content

arXiv:0901.2850v1 [cs.AI] 19 Jan 2009 To appear in Theory and Practice of Logic Programming (TPLP) 1 On Finitely Recursive Programs∗ Sabrina Baselice, Piero A. Bonatti, Giovanni Criscuolo Universit`a di Napoli “Federico II”, Italy submitted 7 April 2008; revised 30 December 2008; accepted 16 January 2009 Abstract 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 de- cidable while in the general case the stable model semantics is Π1 1-hard. 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 instan- tiation. 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 ω-sequence converge to a stable model of P. KEYWORDS: Answer set programming with infinite domains, Infinite stable models, Finitary pro- grams, Compactness, Skeptical resolution. 1 Introduction Answer Set Programming (ASP) (Marek and Truszczynski 1998; Niemel¨a 1999) is one of the most interesting achievements in the area of Logic Programming and Nonmono- tonic Reasoning. It is a declarative problem solving paradigm, mainly centered around some well-engineered implementations of the stable model semantics of logic programs (Gelfond and Lifschitz 1988; Gelfond and Lifschitz 1991), such as SMODELS (Niemel¨a and Simons 1997) and DLV (Eiter et al. 1997). The most popular ASP languages are extensions of Datalog, namely, function-free, pos- sibly disjunctive logic programs with negation as failure. The lack of function symbols has several drawbacks, related to expressiveness and encoding style (Bonatti 2004). In order to overcome such limitations and reduce the memory requirements of current im- plementations, a class of logic programs called finitary programs has been introduced (Bonatti 2004). In finitary programs function symbols (hence infinite domains) and recursion are al- lowed. However, recursion is restricted by requiring each ground atom to depend on finitely many ground atoms; such programs are called finitely recursive. Moreover, only finitely ∗This paper extends and refines (Baselice et al. 2007) 2 S. Baselice, P.A. Bonatti, G. Criscuolo many ground atoms must occur in odd-cycles—that is, cycles of recursive calls involving an odd number of negative subgoals—which means that there should be only finitely many potential sources of inconsistencies. These two restrictions bring a number of nice seman- tical and computational properties (Bonatti 2004). In general, function symbols make the stable model semantics highly undecidable (Marek and Remmel 2001). On the contrary, if the given program is finitary, then consistency checking, ground credulous queries, and ground skeptical queries are decidable. Nonground queries were proved to be r.e.-complete. Moreover, a form of compactness holds: an inconsistent finitary program has always a fi- nite unstable kernel, i.e. a finite subset of the ground instantiation of the program with no stable models. All of these properties are quite unusual for a nonmonotonic logic. As function symbols are being integrated in state-of-the-art reasoners such as DLV (Calimeri et al. 2008), it is interesting to extend these good properties to larger program classes. This goal requires a better understanding of the role of each restriction in the def- inition of finitary programs. It has already been noted (Bonatti 2004) that by dropping the first condition (i.e., if the program is not finitely recursive) one obtains a superclass of strat- ified programs, whose complexity is then far beyond computability. In the same paper, it is argued that the second restriction (on odd-cycles) is needed for the decidability of ground queries. However, if a program is only finitely recursive (and infinitely many odd-cycles are allowed), then the results of (Bonatti 2004) do not characterize the exact complexity of reasoning and say nothing about compactness, nor about the completeness of the skeptical resolution calculus (Bonatti 2001b). In this paper we extend and refine those results, and prove that several important proper- ties of finitary programs carry over to all disjunctive finitely recursive programs. We prove that for all such programs the compactness property still holds, and that inconsistency checking and skeptical reasoning are semidecidable. Moreover, we extend the complete- ness of skeptical reso

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