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Interdefinability of defeasible logic and logic programming under the well-founded semantics

Published online by Cambridge University Press:  09 August 2011

FREDERICK MAIER
FREDERICK MAIER*
Affiliation:
Kno.e.sis Center, Department of Computer Science & Engineering, Wright State University, 3640 Colonel Glenn Hwy, Dayton, OH 45435, USA (e-mail: fred@knoesis.org, fmaier@uga.edu)
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Abstract

We provide a method of translating theories of Nute's defeasible logic into logic programs, and a corresponding translation in the opposite direction. Under certain natural restrictions, the conclusions of defeasible theories under the ambiguity propagating defeasible logic ADL correspond to those of the well-founded semantics for normal logic programs, and so it turns out that the two formalisms are closely related. Using the same translation of logic programs into defeasible theories, the semantics for the ambiguity blocking defeasible logic NDL can be seen as indirectly providing an ambiguity blocking semantics for logic programs. We also provide antimonotone operators for both ADL and NDL, each based on the Gelfond–Lifschitz (GL) operator for logic programs. For defeasible theories without defeaters or priorities on rules, the operator for ADL corresponds to the GL operator and so can be seen as partially capturing the consequences according to ADL. Similarly, the operator for NDL captures the consequences according to NDL, though in this case no restrictions on theories apply. Both operators can be used to define stable model semantics for defeasible theories.

Keywords

defeasible logiclogic programmingwell-founded semanticsstable model semanticsambiguity blocking and propagation
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 13 , Issue 1 , January 2013 , pp. 107 - 142
Copyright
Copyright © Cambridge University Press 2011

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References

Antoniou, G. 2006. Defeasible reasoning: A discussion of some intuitions. International Journal of Intelligent Systems 21 (6), 545–558.CrossRefGoogle Scholar
Antoniou, G. and Billington, D. 2001. Relating defeasible and default logic. In Proc. of 14th Australian Joint Conference on Artificial Intelligence (AUS-AI '01). Lecture Notes in Computer Science, vol. 2256. Springer, Berlin, 13–24.Google Scholar
Antoniou, G., Billington, D., Governatori, G. and Maher, M. J. 2000a. A flexible framework for defeasible logics. In Proc. of 17th National Conference on Artificial Intelligence (AAAI '00). AAAI Press, Menlo Park, CA, 405–410.Google Scholar
Antoniou, G., Billington, D., Governatori, G. and Maher, M. J. 2001. Representation results for defeasible logic. ACM Transactions on Computational Logic 2 (2), 255–287.Google Scholar
Antoniou, G., Billington, D., Governatori, G. and Maher, M. J. 2006. Embedding defeasible logic into logic programming. Theory and Practice of Logic Programming 6 (6), 703–735.CrossRefGoogle Scholar
Antoniou, G., Billington, D., Governatori, G., Maher, M. J. and Rock, A. 2000b. A family of defeasible logics and its implementation. In Proc. of 14th European Conference on Artificial Intelligence (ECAI '00). IOS Press, Amsterdam, 459–463.Google Scholar
Baral, C. and Subrahmanian, V. S. 1993. Dualities between alternative semantics for logic programming and nonmonotonic reasoning. Journal of Automated Reasoning 10 (3), 399–420.Google Scholar
Billington, D. 1993. Defeasible logic is stable. Journal of Logic and Computation 3 (4), 379–400.CrossRefGoogle Scholar
Billington, D. 2004. A plausible logic which detects loops. In Proc. of 10th International Workshop on Nonmonotonic Reasoning (NMR '04), 65–71.Google Scholar
Billington, D. 2005a. The proof algorithms of plausible logic form a hierarchy. In Proc. of 18th Australian Joint Conference on Artificial Intelligence (AUS–AI '05). Lecture Notes in Computer Science, vol. 3809. Springer, Berlin, 796–799.Google Scholar
Billington, D. 2005b. A fixed-point semantics for plausible logic. In Proc. of 18th Australian Joint Conference on Artificial Intelligence (AUS–AI '05). Lecture Notes in Computer Science, vol. 3809. Springer, Berlin, 812–815.Google Scholar
Billington, D. 2007. Entailment semantics for rules with priorities. In Proc. of 20th International Joint Conference on Artifical Intelligence (IJCAI '07). Morgan Kaufmann Publishers, San Francisco, CA, USA, 256–261.Google Scholar
Billington, D. 2008. Propositional clausal defeasible logic. In Proc. of 11th European conference on Logics in Artificial Intelligence (JELIA '08). Lecture Notes in Computer Science, vol. 5293. Springer, Berlin, 34–47.Google Scholar
Billington, D., Coster, K. D. and Nute, D. 1990. A modular translation from defeasible nets to defeasible logics. Journal of Experimental and Theoretical Artificial Intelligence 2 (2), 151–177.CrossRefGoogle Scholar
Billington, D. and Rock, A. 2001. Propositional plausible logic: Introduction and implementation. Studia Logica 67 (2), 243–269.CrossRefGoogle Scholar
Brewka, G. 1996. Well-founded semantics for extended logic programs with dynamic preferences. Journal of Artificial Intelligence Research 4, 19–36.CrossRefGoogle Scholar
Brewka, G. 2001. On the relationship between defeasible logic and well-founded semantics. In Proc. of 6th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR '01). Lecture Notes in Computer Science, vol. 2173. Springer, Berlin, 121–132.Google Scholar
Donnelly, S. 1999. Semantics, Soundness, and Incompleteness for a Defeasible Logic, Master's thesis. The University of Georgia, Athens, Georgia.Google Scholar
Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Proc. of 5th International Conference on Logic Programming (ICLP '88). MIT Press, Cambridge, MA, 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
Governatori, G., Maher, M. J., Antoniou, G. and Billington, D. 2000. Argumentation semantics for defeasible logics. In Proc. of 6th Pacific Rim international conference on Artificial intelligence (PRICAI '00). Lecture Notes in Computer Science, vol. 1886. Springer, Berlin, 27–37.Google Scholar
Governatori, G., Maher, M. J., Antoniou, G. and Billington, D. 2004. Argumentation semantics for defeasible logic. Journal of Logic and Computation 14 (5), 675–702.CrossRefGoogle Scholar
Horty, J. F., Thomason, R. H. and Touretzky, D. S. 1990. A skeptical theory of inheritance in nonmonotonic semantic networks. Artificial Intelligence 42 (2–3), 311–348.Google Scholar
Kunen, K. 1987. Negation in logic programming. Journal of Logic Programming 4 (4), 289–308.CrossRefGoogle Scholar
Maher, M. J. and Governatori, G. 1999. A semantic decomposition of defeasible logics. In Proc. of 16th National Conference on Artificial Intelligence (AAAI '99). AAAI/MIT Press, Menlo Park, CA/Cambridge, MA, 299–305.Google Scholar
Maher, M. J., Rock, A., Antoniou, G., Billington, D. and Miller, T. 2001. Efficient defeasible reasoning systems. International Journal on Artificial Intelligence Tools 10 (4), 483–501.CrossRefGoogle Scholar
Maier, F. 2010. Well-founded semantics for defeasible logic. Synthese 176 (2), 243–274.Google Scholar
Maier, F. and Nute, D. 2006. Ambiguity propagating defeasible logic and the well-founded semantics. In Proc. of 10th European Conference on Logics in Artificial Intelligence (JELIA '06). Lecture Notes in Artificial Intelligence, vol. 4160, Springer, Berlin, 306–318.CrossRefGoogle Scholar
Makinson, D. and Schlechta, K. 1991. Floating conclusions and zombie paths: Two deep difficulties in the ‘directly skeptical’ approach to inheritance nets. Artificial Intelligence 48 (2), 199–209.Google Scholar
Marek, W. and Truszczynski, M. 1989. Stable semantics for logic programs and default theories. In Proc. of 1989 North American Conference on Logic Programming (NACLP '89). MIT Press, Cambridge, MA, 243–256.Google Scholar
Nute, D. 1986. LDR: A Logic for Defeasible Resasoning. ACMC Research Report 01-0013. The University of Georgia, Athens, Georgia.Google Scholar
Nute, D. 1994. Defeasible logic. In Handbook of Logic for Artificial Intelligence and Logic Programming, Vol. III, Gabbay, D. and Hogger, C., Eds. Oxford University Press, New York, 353–395.Google Scholar
Nute, D. 1997. Apparent obligation. In Defeasible Deontic Logic: Essays in Nonmonotonic Normative Reasoning, Nute, D., Ed. Kluwer Academic Publishers, Dordrecht, Netherlands, 287–316.CrossRefGoogle Scholar
Nute, D. 1999. Norms, priorities, and defeasibility. In Norms, Logics and Information Systems, McNamara, P. and Prakken, H., Eds. IOS Press, Amsterdam, 201–218.Google Scholar
Nute, D. 2003. Defeasible logic: Theory, implementation, and applications. In Proc. of 14th International Conference on Applications of Prolog (INAP '01). Lecture Notes in Computer Science, vol. 2543. Springer, Berlin, 151–169.Google Scholar
Nute, D., Billington, D. and Coster, K. D. 1989. Defeasible logic and inheritance hierarchies with exceptions. In Proc. of Tübingen Workshop on Semantic Networks and Nonmonotonic Reasoning, vol. I. SNS Bericht, 89-48, University of Tübingen, 69–82.Google Scholar
Reiter, R. 1980. A logic for default reasoning. Artificial Intelligence 13 (1–2), 81–132.Google Scholar
Sagonas, K. F., Swift, T. and Warren, D. S. 1994. XSB as an efficient deductive database engine. ACM SIGMOD Record 23 (2), 442–453.CrossRefGoogle Scholar
Schaub, T. and Wang, K. 2002. Preferred well-founded semantics for logic programming by alternating fixpoints. Preliminary report. In Proc. of 9th International Workshop on Non-Monotonic Reasoning (NMR '02), 238–246.Google Scholar
Syrjänen, T. and Niemelä, I. 2001. The Smodels system. In Proc. of 6th International Conference Logic Programming and Nonmonotonic Reasoning (LPNMR '01). Lecture Notes in Computer Science, vol. 2173. Springer, Berlin, 434–438.Google Scholar
Tarski, A. 1955. A lattice theoretic fixpoint theorem and its application. Pacific Journal of Mathematics 5 (2), 285–309.Google Scholar
van Emden, M. and Kowalski, R. A. 1976. The semantics of predicate logic as a programming language. Journal of the ACM 23 (4), 733–742.Google Scholar
Van Gelder, A., Ross, K. A. and Schlipf, J. 1991. The well-founded semantics for general logic programs. Journal of the ACM 38 (3), 619–649.Google Scholar