Skip to main content Accessibility help
×
×
Home

Causal Graph Justifications of Logic Programs*

  • PEDRO CABALAR (a1), JORGE FANDINNO (a1) and MICHAEL FINK (a2)
Abstract

In this work we propose a multi-valued extension of logic programs under the stable models semantics where each true atom in a model is associated with a set of justifications. These justifications are expressed in terms of causal graphs formed by rule labels and edges that represent their application ordering. For positive programs, we show that the causal justifications obtained for a given atom have a direct correspondence to (relevant) syntactic proofs of that atom using the program rules involved in the graphs. The most interesting contribution is that this causal information is obtained in a purely semantic way, by algebraic operations (product, sum and application) on a lattice of causal values whose ordering relation expresses when a justification is stronger than another. Finally, for programs with negation, we define the concept of causal stable model by introducing an analogous transformation to Gelfond and Lifschitz's program reduct. As a result, default negation behaves as “absence of proof” and no justification is derived from negative literals, something that turns out convenient for elaboration tolerance, as we explain with a running example.

Copyright
Footnotes
Hide All
*

This research was partially supported by Spanish MEC project TIN2009-14562-C05-04, Xunta projects GPC2013/070 and INCITE 2011, Inditex-University of Corunna 2013 grants and the Austrian Science Fund (FWF) project P24090.

Footnotes
References
Hide All
Artëmov, S. N. 2001. Explicit provability and constructive semantics. Bulletin of Symbolic Logic 7, 1, 136.
Brewka, G., Eiter, T., and Truszczynski, M. 2011. Answer set programming at a glance. Commun. ACM 54, 12, 92103.
Broda, K., Gabbay, D., Lamb, L. and Russo, A. 2004. Compiled Labelled Deductive Systems: A Uniform Presentation of Non-Classical Logics. Research Studies Press.
Cabalar, P. 2011. Logic programs and causal proofs. In AAAI Spring Symposium: Logical Formalizations of Commonsense Reasoning. AAAI.
Cabalar, P. and Fandinno, J. 2013. An algebra of causal chains. In Proc. of the 6th Workshop on Answer Set Programming and Other Computing Paradigms (ASPOCP'13).
Damásio, C. V., Analyti, A., and Antoniou, G. 2013. Justifications for logic programming. In Proc. of the 12th Intl. Conf. on Logic Programming and Nonmonotonic Reasoning, (LPNMR'13). Lecture Notes in Computer Science, vol. 8148. Springer, 530542.
Denecker, M. and De Schreye, D. 1993. Justification semantics: A unifiying framework for the semantics of logic programs. In Proc. of the Logic Programming and Nonmonotonic Reasoning Workshop. 365–379.
Fages, F. 1991. A new fixpoint semantics for general logic programs compared with the well-founded and the stable model semantics. New Generation Computing 9, 3–4, 425443.
Gebser, M., Pührer, J., Schaub, T., and Tompits, H. 2008. Meta-programming technique for debugging answer-set programs. In Proc. of the 23rd Conf. on Artificial Inteligence (AAAI'08). 448–453.
Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Logic Programming: Proc. of the Fifth International Conference and Symposium (Volume 2), Kowalski, R. A. and Bowen, K. A., Eds. MIT Press, Cambridge, MA, 10701080.
Hall, N. 2004. Two concepts of causality. 181–276.
Halpern, J. Y. 2008. Defaults and normality in causal structures. In Proc. of the Eleventh International Conference on Principles of Knowledge Representation and Reasoning (KR 2008). 198–208.
Halpern, J. Y. and Pearl, J. 2005. Causes and explanations: A structural-model approach. part I: Causes. British Journal for Philosophy of Science 56, 4, 843887.
Hitchcock, C. and Knobe, J. 2009. Cause and norm. Journal of Philosophy 11, 587612.
Hume, D. 1748. An enquiry concerning human understanding. Reprinted by Open Court Press, LaSalle, IL, 1958.
Kifer, M. and Subrahmanian, V. S. 1992. Theory of generalized annotated logic programming and its applications. Journal of Logic Programming 12.
Liang, S. and Kifer, M. 2013. A practical analysis of non-termination in large logic programs. TPLP 13, 4–5, 705719.
Lin, F. 1995. Embracing causality in specifying the indirect effects of actions. In Proc. of the Intl. Joint Conf. on Artificial Intelligence (IJCAI), Mellish, C. S., Ed. Morgan Kaufmann, Montreal, Canada.
McCain, N. and Turner, H. 1997. Causal theories of action and change. In Proc. of the AAAI-97. 460–465.
McCarthy, J. 1977. Epistemological problems of Artificial Intelligence. In Proc. of the Intl. Joint Conf. on Artificial Intelligence (IJCAI). MIT Press, Cambridge, MA, 10381044.
McCarthy, J. 1998. Elaboration tolerance. In Proc. of the 4th Symposium on Logical Formalizations of Commonsense Reasoning (Common Sense 98). London, UK, 198217. Updated version at http://www-formal.stanford.edu/jmc/elaboration.ps.
Pearce, D. 2006. Equilibrium logic. Ann. Math. Artif. Intell. 47, 1-2, 341.
Pearl, J. 2000. Causality: models, reasoning, and inference. Cambridge University Press, New York, NY, USA.
Pereira, L. M., Aparício, J. N., and Alferes, J. J. 1991. Derivation procedures for extended stable models. In Proceedings of the 12th International Joint Conference on Artificial Intelligence, Mylopoulos, J. and Reiter, R., Eds. Morgan Kaufmann, 863869.
Pontelli, E., Son, T. C., and El-Khatib, O. 2009. Justifications for logic programs under answer set semantics. Theory and Practice of Logic Programming 9, 1, 156.
Schulz, C., Sergot, M., and Toni, F. 2013. Argumentation-based answer set justification. In Proc. of the 11th Intl. Symposium on Logical Formalizations of Commonsense Reasoning (Commonsense'13).
Stumme, G. 1997. Free distributive completions of partial complete lattices. Order 14, 179189.
Thielscher, M. 1997. Ramification and causality. Artificial Intelligence Journal 89, 1–2, 317364.
van Emden, M. H. and Kowalski, R. A. 1976. The semantics of predicate logic as a programming language. J. ACM 23, 4, 733742.
Vennekens, J. 2011. Actual causation in cp-logic. TPLP 11, 4–5, 647662.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Theory and Practice of Logic Programming
  • ISSN: 1471-0684
  • EISSN: 1475-3081
  • URL: /core/journals/theory-and-practice-of-logic-programming
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Type Description Title
PDF
Supplementary materials

CABALAR et al.
Causal Graph Justifications of Logic Programs

 PDF (159 KB)
159 KB

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed