Abstract
In answer set programming (ASP), one does not allow the use of function symbols. Disallowing function symbols avoids the problem of having logic programs which have stable models of excessively high complexity. For example, Marek, Nerode, and Remmel showed that there exist finite predicate logic programs which have stable models but which have no hyperarithmetic stable model. Of course, by eliminating function symbols, one loses a lot of expressive power in the language. In particular, it is difficult to directly reason about infinite sets in ASP.
Blair, Marek, and Remmel [BMR08] developed an extension of logic programming called set based logic programming. In the theory of set based logic programming, the atoms represent subsets of a fixed universe X and one is allowed to compose the one-step consequence operator with a monotonic idempotent operator O so as to ensure that the analogue of stable models are always closed under O. We show that if the sets represented by the atoms in a finite set based program P are languages accepted by finite automaton, and the operators involved in the construction have a certain natural property, then all the stable models of P are languages accepted by finite automaton and one can effectively check whether a language accepted by a finite automaton is a stable model of the set based logic program. Thus in this setting, one can effectively reason about certain classes of infinite sets.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Apt, K., Blair, H.A.: Arithmetic Classification of Perfect Models of Stratified Programs. Fundamenta Informaticae 13, 1–17 (1990)
Babovich, Y., Lifschitz, V.: Cmodels (2002), http://www.cs.utexas.edu/users/tag/cmodels.html
Baral, C.: Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, Cambridge (2003)
Blair, H.A., Marek, V.W., Remmel, J.B.: Spatial Logic Programming. In: Proceedings SCI 2001, Orlando, FL (July 2001)
Blair, H.A., Marek, V.W., Remmel, J.B.: Set Based Logic Programming (in print)
Blair, H.A., Marek, V.W., Schlipf, J.S.: The expressivness of locally stratified programs. Annals of Mathematics and Artificial Intelligence 15, 209–229 (1995)
Blumensath, A., Grädel, E.: Automatic Structures. In: Proceedings of the 15th Symposium on Logic in Computer Science LICS 2000, pp. 51–62 (2000)
Denecker, M.: Extending classical logic with inductive definitions. In: Palamidessi, C., Moniz Pereira, L., Lloyd, J.W., Dahl, V., Furbach, U., Kerber, M., Lau, K.-K., Sagiv, Y., Stuckey, P.J. (eds.) CL 2000. LNCS, vol. 1861, pp. 703–717. Springer, Heidelberg (2000)
Dowling, W.F., Gallier, J.H.: Linear-time algorithms for testing satisfiability of propositional Horn formulae. Journal of Logic Programming 3, 267–284 (1984)
Gebser, M., Kaufmann, B., Neumann, A., Schaub, T.: Clasp – a Conflict-driven Answer Set Solver. In: Baral, C., Brewka, G., Schlipf, J. (eds.) LPNMR 2007. LNCS, vol. 4483, pp. 260–265. Springer, Heidelberg (2007)
Gelfond, M.: Logic Programming and Knowledge Representation – A-Prolog perspective. Artificial Intelligence Journal 138, 3–38 (2002)
Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of the International Joint Conference and Symposium on Logic Programming, pp. 1070–1080. MIT Press, Cambridge (1988)
Giunchiglia, E., Lierer, Y., Maratea, M.: Answer Set Programming Based on Propositional Satisfiability. Journal of Automated Reasoning 36, 345–377 (2006)
Khoussainov, B., Nerode, A.: Automatic Presentations of Structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)
Khoussainov, B., Nies, A., Rubin, S., Stephan, F.: Automatic structures: richness and limitations. Logical Methods of Computer Science 3(2), 18 (2007) (electronic)
Leone, N., Pfeifer, G., Faber, W., Eiter, T., Gottlob, G., Perri, S., Scarcello, F.: The dlv system for knowledge representation and reasoning. ACM Transactions on Computational Logic (2006)
Lifschitz, V.: Minimal belief and negation as failure. Artificial Intelligence 70, 53–72 (1994)
Lin, F., Zhao, Y.: ASSAT: Computing answer sets of a logic program by SAT solvers. In: Proceedings of the 18th National Conference on Artificial Intelligence (AAAI 2002), pp. 112–117. AAAI Press, Menlo Park (2002)
Marek, W., Nerode, A., Remmel, J.B.: The stable models of predicate logic programs. Journal of Logic Programming 21(3), 129–154 (1994)
Marek, W., Truszczyński, M.: Nonmonotonic Logic – Context-Dependent Reasoning. Springer, Heidelberg (1993)
Marek, V.W., Truszczyński, M.: Stable Models and an Alternative Logic Programming Paradigm. In: The Logic Programming Paradigm. Series Artificial Intelligence, pp. 375–398. Springer, Heidelberg (1999)
Niemelä, I.: Logic programs with stable model semantics as a constraint programming paradigm. Annals of Mathematics and Artificial Intelligence 25, 3,4, 241–273 (1999)
Simons, P., Niemelä, I., Soininen, T.: Extending and implementing stable semantics of logic programs. Artificial Intelligence 138, 181–234 (2002)
Smullyan, R.: First-order Logic. Springer, Heidelberg (1968)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Marek, V., Remmel, J.B. (2008). Automata and Answer Set Programming. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2009. Lecture Notes in Computer Science, vol 5407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92687-0_22
Download citation
DOI: https://doi.org/10.1007/978-3-540-92687-0_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92686-3
Online ISBN: 978-3-540-92687-0
eBook Packages: Computer ScienceComputer Science (R0)