Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T12:06:54.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Knowledge compilation of logic programs using approximation fixpoint theory

Published online by Cambridge University Press:  03 September 2015

BART BOGAERTS  and
GUY VAN DEN BROECK
BART BOGAERTS
Affiliation:
Department of Computer Science, KU Leuven, Belgium (e-mail: [email protected], [email protected])
GUY VAN DEN BROECK
Affiliation:
Department of Computer Science, KU Leuven, Belgium (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent advances in knowledge compilation introduced techniques to compile positive logic programs into propositional logic, essentially exploiting the constructive nature of the least fixpoint computation. This approach has several advantages over existing approaches: it maintains logical equivalence, does not require (expensive) loop-breaking preprocessing or the introduction of auxiliary variables, and significantly outperforms existing algorithms. Unfortunately, this technique is limited to negation-free programs. In this paper, we show how to extend it to general logic programs under the well-founded semantics.

We develop our work in approximation fixpoint theory, an algebraical framework that unifies semantics of different logics. As such, our algebraical results are also applicable to autoepistemic logic, default logic and abstract dialectical frameworks.

Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 15 , Special Issue 4-5: 31st International Conference on Logic Programming , July 2015 , pp. 464 - 480
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abiteboul, S. and Vianu, V. 1991. Datalog extensions for database queries and updates. J. Comput. Syst. Sci. 43, 1, 62–124.Google Scholar
Antic, C., Eiter, T. and Fink, M. 2013. Hex semantics via approximation fixpoint theory. In Proceedings of LPNMR. 102–115.CrossRefGoogle Scholar
Asuncion, V., Lin, F., Zhang, Y. and Zhou, Y. 2012. Ordered completion for first-order logic programs on finite structures. Artif. Intell. 177–179, 1–24.CrossRefGoogle Scholar
Ben-Eliyahu, R. and Dechter, R. 1994. Propositional semantics for disjunctive logic programs. Ann. Math. Artif. Intell. 12, 1–2, 53–87.Google Scholar
Bryant, R. E. 1986. Graph-based algorithms for Boolean function manipulation. IEEE Transactions on Computers 35, 677–691.CrossRefGoogle Scholar
Cadoli, M. and Donini, F. M. 1997. A survey on knowledge compilation. AI Commun. 10, 3–4, 137–150.Google Scholar
Chavira, M. and Darwiche, A. 2005. Compiling bayesian networks with local structure. In Proceedings of IJCAI. 1306–1312.Google Scholar
Chavira, M. and Darwiche, A. 2008. On probabilistic inference by weighted model counting. Artif. Intell. 172, 6–7, 772–799.Google Scholar
Darwiche, A. 2011. SDD: A new canonical representation of propositional knowledge bases. In Proceedings of IJCAI. 819–826.Google Scholar
Darwiche, A. and Marquis, P. 2002. A knowledge compilation map. J. Artif. Intell. Res. (JAIR) 17, 229–264.Google Scholar
Denecker, M., Lierler, Y., Truszczyński, M. and Vennekens, J. 2012. A Tarskian informal semantics for answer set programming. In ICLP (Technical Communications). 277–289.Google Scholar
Denecker, M., Marek, V. and Truszczyński, M. 2000. Approximations, stable operators, well-founded fixpoints and applications in nonmonotonic reasoning. In Logic-Based Artificial Intelligence, Springer. Vol. 597. 127–144.Google Scholar
Denecker, M., Marek, V. and Truszczyński, M. 2004. Ultimate approximation and its application in nonmonotonic knowledge representation systems. Information and Computation 192, 1 (July), 84–121.Google Scholar
Denecker, M. and Vennekens, J. 2007. Well-founded semantics and the algebraic theory of non-monotone inductive definitions. In LPNMR. 84–96.Google Scholar
Denecker, M. and Vennekens, J. 2014. The well-founded semantics is the principle of inductive definition, revisited. In Proceedings of KR. 22–31.Google Scholar
Fierens, D., Van den Broeck, G., Renkens, J., Shterionov, D. S., Gutmann, B., Thon, I., Janssens, G. and De Raedt, L. 2015. Inference and learning in probabilistic logic programs using weighted boolean formulas. TPLP 15, 3, 358–401.Google Scholar
Fitting, M. 2002. Fixpoint semantics for logic programming –- A survey. Theoretical Computer Science 278, 1–2, 25–51.Google Scholar
Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Proceedings of ICLP/SLP. 1070–1080.Google Scholar
Huang, J. and Darwiche, A. 2005. On compiling system models for faster and more scalable diagnosis. In Proceedings of AAAI. 300–306.Google Scholar
Janhunen, T. 2004. Representing normal programs with clauses. In Proceedings of ECAI. 358–362.Google Scholar
Janhunen, T. 2006. Some (in)translatability results for normal logic programs and propositional theories. Journal of Applied Non-Classical Logics 16, 1–2, 35–86.CrossRefGoogle Scholar
Janhunen, T., Niemelä, I. and Sevalnev, M. 2009. Computing stable models via reductions to difference logic. In LPNMR, Erdem, E., Lin, F., and Schaub, T., Eds. LNCS, vol. 5753. Springer, 142–154.Google Scholar
Kleene, S. C. 1938. On notation for ordinal numbers. The Journal of Symbolic Logic 3, 4, 150–155.CrossRefGoogle Scholar
Lifschitz, V. and Razborov, A. A. 2006. Why are there so many loop formulas? ACM Trans. Comput. Log. 7, 2, 261–268.Google Scholar
Lin, F. and Zhao, J. 2003. On tight logic programs and yet another translation from normal logic programs to propositional logic. In Proceedings of IJCAI. 853–858.Google Scholar
Lin, F. and Zhao, Y. 2004. ASSAT: Computing answer sets of a logic program by SAT solvers. AIJ 157, 1–2, 115–137.Google Scholar
Lowd, D. and Domingos, P. 2008. Learning arithmetic circuits. In Proceedings of UAI. 383–392.Google 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, 375–398.Google Scholar
Palacios, H., Bonet, B., Darwiche, A. and Geffner, H. 2005. Pruning conformant plans by counting models on compiled d-dnnf representations. In Proceedings of ICAPS. 141–150.Google Scholar
Pelov, N., Denecker, M. and Bruynooghe, M. 2007. Well-founded and stable semantics of logic programs with aggregates. TPLP 7, 3, 301–353.Google Scholar
Przymusinski, T. C. 1988. On the declarative semantics of deductive databases and logic programs. In Foundations of Deductive Databases and Logic Programming. Morgan Kaufmann, 193–216.Google Scholar
Selman, B. and Kautz, H. A. 1996. Knowledge compilation and theory approximation. J. ACM 43, 2, 193–224.Google Scholar
Strass, H. 2013. Approximating operators and semantics for abstract dialectical frameworks. AIJ 205, 39–70.Google Scholar
Suciu, D., Olteanu, D., Ré, C. and Koch, C. 2011. Probabilistic databases.Google Scholar
Van den Broeck, G. and Darwiche, A. 2015. On the role of canonicity in knowledge compilation. In Proceedings of AAAI.Google Scholar
van Emden, M. H. and Kowalski, R. A. 1976. The semantics of predicate logic as a programming language. J. ACM 23, 4, 733–742.Google Scholar
Van Gelder, A., Ross, K. A. and Schlipf, J. S. 1991. The well-founded semantics for general logic programs. J. ACM 38, 3, 620–650.Google Scholar
Vlasselaer, J., Van den Broeck, G., Kimmig, A., Meert, W. and De Raedt, L. 2015. Anytime inference in probabilistic logic programs with -compilation. In Proceedings of IJCAI. Available on https://lirias.kuleuven.be/handle/123456789/494681.Google Scholar
Supplementary material: PDF

Bogaerts supplementary material

Online Appendix

Download Bogaerts supplementary material(PDF)
PDF 344.4 KB