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Stable-unstable semantics: Beyond NP with normal logic programs

Published online by Cambridge University Press:  14 October 2016

BART BOGAERTS ,
TOMI JANHUNEN  and
SHAHAB TASHARROFI
BART BOGAERTS
Affiliation:
Helsinki Institute for Information Technology HIIT * Department of Computer Science, Aalto University, FI-00076 AALTO, Finland, (e-mail: [email protected])
TOMI JANHUNEN
Affiliation:
Helsinki Institute for Information Technology HIIT * Department of Computer Science, Aalto University, FI-00076 AALTO, Finland, (e-mail: [email protected])
SHAHAB TASHARROFI
Affiliation:
Helsinki Institute for Information Technology HIIT * Department of Computer Science, Aalto University, FI-00076 AALTO, Finland, (e-mail: [email protected])
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Abstract

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Standard answer set programming (ASP) targets at solving search problems from the first level of the polynomial time hierarchy (PH). Tackling search problems beyond NP using ASP is less straightforward. The class of disjunctive logic programs offers the most prominent way of reaching the second level of the PH, but encoding respective hard problems as disjunctive programs typically requires sophisticated techniques such as saturation or meta-interpretation. The application of such techniques easily leads to encodings that are inaccessible to non-experts. Furthermore, while disjunctive ASP solvers often rely on calls to a (co-)NP oracle, it may be difficult to detect from the input program where the oracle is being accessed. In other formalisms, such as Quantified Boolean Formulas (QBFs), the interface to the underlying oracle is more transparent as it is explicitly recorded in the quantifier prefix of a formula. On the other hand, ASP has advantages over QBFs from the modeling perspective. The rich high-level languages such as ASP-Core-2 offer a wide variety of primitives that enable concise and natural encodings of search problems. In this paper, we present a novel logic programming–based modeling paradigm that combines the best features of ASP and QBFs. We develop so-called combined logic programs in which oracles are directly cast as (normal) logic programs themselves. Recursive incarnations of this construction enable logic programming on arbitrarily high levels of the PH. We develop a proof-of-concept implementation for our new paradigm.

Keywords

disjunctive logic programmingpolynomial hierarchyquantified Boolean formulas
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 16 , Special Issue 5-6: 32nd International Conference on Logic Programming , September 2016 , pp. 570 - 586
Copyright
Copyright © Cambridge University Press 2016 

References

Andres, B., Rajaratnam, D., Sabuncu, O. and Schaub, T. 2015. Integrating ASP into ROS for reasoning in robots. In Proceedings of the 13th International Conference on Logic Programming and Non-monotonic Reasoning, LPNMR 2015, Lecture Notes in Computer Science, vol. 9345. Springer, Lexington, Kentucky, USA, 69–82.Google Scholar
Ben-Eliyahu, R. and Dechter, R. 1994. Propositional semantics for disjunctive logic programs. Ann. Math. Artif. Intell. 12, 1–2, 53–87.CrossRefGoogle Scholar
Bogaerts, B., Janhunen, T. and Tasharrofi, S. 2016a. Declarative solver development: Case studies. In Proceedings of the 15th International Conference on Principles of Knowledge Representation and Reasoning, KR 2016. AAAI Press, Cape Town, South Africa, 74–83.Google Scholar
Bogaerts, B., Janhunen, T. and Tasharrofi, S. 2016b. Solving QBF instances with nested SAT solvers. In Proceedings of the AAAI-16 Workshop on Beyond NP. AAAI Press, Phoenix, Arizona, 307–313.Google Scholar
Bogaerts, B., Jansen, J., Bruynooghe, M., De Cat, B., Vennekens, J. and Denecker, M. 2014. Simulating dynamic systems using linear time calculus theories. Theory and Practice of Logic Programming 14, 4–5 (7), 477–492.CrossRefGoogle Scholar
Bomanson, J. and Janhunen, T. 2013. Normalizing cardinality rules using merging and sorting constructions. In Proceedings of the 12th International Conference on Logic Programming and Non-monotonic Reasoning, LPNMR 2013, Lecture Notes in Computer Science, vol. 8148. Springer, Corunna, Spain, 187–199.Google Scholar
Brooks, D. R., Erdem, E., Erdogan, S. T., Minett, J. W. and Ringe, D. 2007. Inferring phylogenetic trees using answer set programming. J. Autom. Reasoning 39, 4, 471–511.CrossRefGoogle Scholar
Bruynooghe, M., Blockeel, H., Bogaerts, B., De Cat, B., De Pooter, S., Jansen, J., Labarre, A., Ramon, J., Denecker, M. and Verwer, S. 2015. Predicate logic as a modeling language: modeling and solving some machine learning and data mining problems with IDP3. Theory and Practice of Logic Programming 15, 6 (November), 783–817.CrossRefGoogle Scholar
Calimeri, F., Faber, W., Gebser, M., Ianni, G., Kaminski, R., Krennwallner, T., Leone, N., Ricca, F. and Schaub, T. 2013. ASP-Core-2 input language format. Tech. rep., ASP Standardization Working Group.Google Scholar
Calimeri, F., Gebser, M., Maratea, M. and Ricca, F. 2016. Design and results of the fifth answer set programming competition. Artif. Intell. 231, 151–181.CrossRefGoogle Scholar
Denecker, M., Lierler, Y., Truszczyński, M. and Vennekens, J. 2012. A Tarskian informal semantics for answer set programming. In Technical Communications of the 28th International Conference on Logic Programming, ICLP 2012. LIPIcs, vol. 17. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Budapest, Hungary, 277–289.Google Scholar
Denecker, M. and Vennekens, J. 2007. Well-founded semantics and the algebraic theory of non-monotone inductive definitions. In Proceedings of the 9th International Conference on Logic Programming and Non-monotonic Reasoning, LPNMR 2007, Lecture Notes in Computer Science, vol. 4483. Springer, Tempe, Arizona, USA, 84–96.Google Scholar
Drescher, C., Gebser, M., Grote, T., Kaufmann, B., König, A., Ostrowski, M. and Schaub, T. 2008. Conflict-driven disjunctive answer set solving. In Proceedings of the 11th International Conference on Principles of Knowledge Representation and Reasoning, KR 2008. AAAI Press, Sydney, Australia, 422–432.Google Scholar
Eiter, T. and Gottlob, G. 1995. On the computational cost of disjunctive logic programming: Propositional case. Ann. Math. Artif. Intell. 15, 3–4, 289–323.CrossRefGoogle Scholar
Eiter, T., Gottlob, G. and Veith, H. 1997. Modular logic programming and generalized quantifiers. In Proceedings of the 4th International Conference on Logic Programming and Non-monotonic Reasoning, LPNMR 1997, Lecture Notes in Computer Science, vol. 1265. Springer, Dagstuhl Castle, Germany, 289–308.Google Scholar
Eiter, T. and Polleres, A. 2006. Towards automated integration of guess and check programs in answer set programming: a meta-interpreter and applications. Theory and Practice of Logic Programming 6, 1–2, 23–60.CrossRefGoogle Scholar
Emerson, E. A. and Jutla, C. S. 1991. Tree automata, mu-calculus and determinacy. In Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, FOCS 1991. IEEE Computer Society, San Juan, Puerto Rico, 368–377.Google Scholar
Gebser, M., Harrison, A., Kaminski, R., Lifschitz, V. and Schaub, T. 2015. Abstract gringo. Theory and Practice of Logic Programming 15, 4–5, 449–463.CrossRefGoogle Scholar
Gebser, M., Kaminski, R., Kaufmann, B., Romero, J. and Schaub, T. 2015. Progress in clasp series 3. In Proceedings of the 13th International Conference on Logic Programming and Non-monotonic Reasoning, LPNMR 2015, Lecture Notes in Computer Science, vol. 9345. Springer, Lexington, Kentucky, USA, 368–383.Google Scholar
Gebser, M., Kaminski, R. and Schaub, T. 2011. Complex optimization in answer set programming. Theory and Practice of Logic Programming 11, 4–5, 821–839.CrossRefGoogle Scholar
Gebser, M., Schaub, T. and Thiele, S. 2007. GrinGo: A new grounder for Answer Set Programming. In Proceedings of the 9th International Conference on Logic Programming and Non-monotonic Reasoning, LPNMR 2007, Lecture Notes in Computer Science, vol. 4483. Springer, Tempe, Arizona, USA, 266–271.Google Scholar
Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Proceedings of the Fifth International Conference on Logic Programming, ICLP 1988. MIT Press, Seattle, Washington, USA, 1070–1080.Google Scholar
Gelfond, M. and Lifschitz, V. 1991. Classical negation in logic programs and disjunctive databases. New Generation Computing 9, 3/4, 365–385.CrossRefGoogle Scholar
Gelfond, M. and Przymusinska, H. 1996. Towards a theory of elaboration tolerance: Logic programming approach. Int. J. of Soft. Eng. and Knowl. Eng. 6, 1, 89–112.CrossRefGoogle Scholar
Grasso, G., Iiritano, S., Leone, N. and Ricca, F. 2009. Some DLV applications for knowledge management. In Proceedings of the 10th International Conference on Logic Programming and Non-monotonic Reasoning, LPNMR 2009, Lecture Notes in Computer Science, vol. 5753. Springer, Potsdam, Germany, 591–597.Google Scholar
Heljanko, K., Keinänen, M., Lange, M. and Niemelä, I. 2012. Solving parity games by a reduction to SAT. J. Comput. Syst. Sci. 78, 2, 430–440.CrossRefGoogle Scholar
Janhunen, T., Gebser, M., Rintanen, J., Nyman, H., Pensar, J. and Corander, J. 2015. Learning discrete decomposable graphical models via constraint optimization. Statistics and Computing. Advance access.CrossRefGoogle Scholar
Janhunen, T., Niemelä, I., Seipel, D., Simons, P. and You, J. 2006. Unfolding partiality and disjunctions in stable model semantics. ACM Trans. Comput. Log. 7, 1, 1–37.CrossRefGoogle Scholar
Janhunen, T., Tasharrofi, S. and Ternovska, E. 2016. sat-to-sat: Declarative extension of SAT solvers with new propagators. In Proceedings of the 30th AAAI Conference on Artificial Intelligence, AAAI 2016. AAAI Press, Phoenix, Arizona, USA, 978–984.Google Scholar
Koponen, L., Oikarinen, E., Janhunen, T. and Säilä, L. 2015. Optimizing phylogenetic supertrees using answer set programming. Theory and Practice of Logic Programming 15, 4–5, 604–619.CrossRefGoogle Scholar
Leone, N., Pfeifer, G., Faber, W., Eiter, T., Gottlob, G., Perri, S. and Scarcello, F. 2006. The DLV system for knowledge representation and reasoning. ACM Trans. Comput. Log. 7, 3, 499–562.CrossRefGoogle Scholar
Leone, N., Rosati, R. and Scarcello, F. 2001. Enhancing answer set planning. In Proceedings of the IJCAI-01 Workshop on Planning under Uncertainty and Incomplete Information. Seattle, Washington, USA.Google Scholar
Lifschitz, V. 1999. Answer set planning. In Proceedings of the 16th International Conference on Logic Programming, ICLP 1999. MIT Press, Las Cruses, New Mexico, USA, 23–37.Google Scholar
Lindström, P. 1966. First order predicate logic with generalized quantifiers. Theoria 32, 3, 186–195.CrossRefGoogle Scholar
Marek, V. and Truszczyński, M. 1999. Stable models and an alternative logic programming paradigm. In The Logic Programming Paradigm: A 25-Year Perspective. Springer-Verlag, Berlin Heidelberg, 375–398.CrossRefGoogle Scholar
Mostowski, A. 1957. On a generalization of quantifiers. Fundamenta Mathematicae 44, 1, 12–36.CrossRefGoogle Scholar
Niemelä, I. 1999. Logic programs with stable model semantics as a constraint programming paradigm. Ann. Math. Artif. Intell. 25, 3–4, 241–273.CrossRefGoogle Scholar
Nogueira, M., Balduccini, M., Gelfond, M., Watson, R. and Barry, M. 2001. An A-Prolog decision support system for the space shuttle. In Proceedings of the 3rd International Symposium on Practical Aspects of Declarative Languages, PADL 2001, Lecture Notes in Computer Science, vol. 1990. Springer, Las Vegas, Nevada, USA, 169–183.Google Scholar
Oikarinen, E. and Janhunen, T. 2006. Modular equivalence for normal logic programs. In Proceedings of the 17th European Conference on Artificial Intelligence, ECAI 2006. IOS Press, Riva del Garda, Italy, 412–416.Google Scholar
Ricca, F., Dimasi, A., Grasso, G., Ielpa, S. M., Iiritano, S., Manna, M. and Leone, N. 2010. A logic-based system for e-tourism. Fundam. Inform. 105, 1–2, 35–55.CrossRefGoogle Scholar
Stockmeyer, L. J. and Meyer, A. R. 1973. Word problems requiring exponential time: Preliminary report. In Proceedings of the 5th Annual ACM Symposium on Theory of Computing, STOC 1973. ACM, Austin, Texas, USA, 1–9.Google Scholar
Tiihonen, J., Soininen, T., Niemelä, I. and Sulonen, R. 2003. A practical tool for mass-customising configurable products. In Proceedings of the 14th International Conference on Engineering Design, ICED 2003. Design Society, Stockholm, 1290–1299.Google Scholar