Abstract
This paper provides a logical framework for comparing inductive capabilities among agents having different background theories. A background theory is called inductively equivalent to another background theory if the two theories induce the same hypotheses for any observation. Conditions of inductive equivalence change depending on the logic of representation languages and the logic of induction or inductive logic programming (ILP). In this paper, we consider clausal logic and nonmonotonic logic programs as representation languages for background theories. Then we investigate conditions of inductive equivalence in four different frameworks of induction, cautious induction , brave induction , learning from satisfiability , and descriptive induction . We observe that several induction algorithms in Horn ILP systems require weaker conditions of equivalence under restricted problem settings. We address that inductive equivalence can be used for verification and evaluation of induction algorithms, and argue problems for optimizing background theories in ILP.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Badea, L., & Stanciu, M. (1999). Refinement operators can be (weakly) perfect. In Lecture notes in artificial intelligence : Vol. 1634. Proceedings of the 9th international workshop on inductive logic programming (pp. 21–32). Berlin: Springer.
Baral, C., & Gelfond, M. (1994). Logic programming and knowledge representation. Journal of Logic Programming, 19/20, 73–148.
Bossu, G., & Siegel, P. (1985). Saturation, nonmonotonic reasoning and the closed-world assumption. Artificial Intelligence, 25, 13–63.
Boström, H., & Idestam-Almquist, P. (1994) Specialization of logic programs by pruning SLD-trees. In Proceedings of the 4th international workshop on inductive logic programming (pp. 31–48).
Clark, K. L. (1978). Negation as failure. In H. Gallaire, & J. Minker (Eds.), Logic and data bases (pp. 293–322). New York: Plenum.
De Raedt, L. (1997). Logical settings for concept-learning. Artificial Intelligence, 95, 187–201.
De Raedt, L., & Bruynooghe, M. (1993). A theory of clausal discovery. In Proceedings of the 13th international joint conference on artificial intelligence (pp. 1058–1063). San Mateo: Morgan Kaufmann.
De Raedt, L., & Dehaspe, L. (1997a) Learning from satisfiability. In Proceedings of the 9th Dutch conference on artificial intelligence (pp. 303–312).
De Raedt, L., & Dehaspe, L. (1997b). Clausal discovery. Machine Learning, 26(2–3), 99–146.
De Raedt, L., & Lavrač, N. (1993). The many faces of inductive logic programming. In Lecture notes in computer science : Vol. 689. Methodologies for intelligent systems, 7th international symposium (pp. 435–449). Berlin: Springer.
Denecker, M., & Kakas, A. C. (2002). Abductive logic programming. In A. C. Kakas, & F. Sadri (Eds.), Lecture notes in artificial intelligence : Vol. 2407. Computational logic: logic programming and beyond—essays in honour of Robert A. Kowalski, Part I (pp. 402–436). Berlin: Springer.
Eiter, T., & Fink, M. (2003). Uniform equivalence of logic programs under the stable model semantics. In Lecture notes in computer sciences : Vol. 2916. Proceedings of the 19th international conference on logic programming (pp. 224–238). Berlin: Springer.
Flach, P. A. (1996). Rationality postulates for induction. In Proceedings of the 6th international conference on theoretical aspects of rationality and knowledge (pp. 267–281). San Mateo: Morgan Kaufmann.
Flach, P. A., & Kakas, A. C. (2000). Abductive and inductive reasoning: background and issues. In P. A. Flach, & A. C. Kakas (Eds.), Abduction and induction—essays on their relation and integration (pp. 1–27). Norwell: Kluwer Academic.
Gelfond, M., & Lifschitz, V. (1988). The stable model semantics for logic programming. In Proceedings of the 5th international conference and symposium on logic programming (pp. 1070–1080). Cambridge: MIT Press.
Gelfond, M., & Lifschitz, V. (1991). Classical negation in logic programs and disjunctive databases. New Generation Computing, 9, 365–385.
Inoue, K. (2004). Induction as consequent finding. Machine Learning, 55, 109–135.
Inoue, K., & Sakama, C. (1995). Abductive framework for nonmonotonic theory change. In Proceedings of the 14th international joint conference on artificial intelligence (pp. 204–210). San Mateo: Morgan Kaufmann.
Inoue, K., & Sakama, C. (2004). Equivalence of logic programs under updates. In Lecture notes in artificial intelligence : Vol. 3229. Proceedings of the 9th European conference on logics in artificial intelligence (pp. 174–186). Berlin: Springer.
Inoue, K., & Sakama, C. (2005) Equivalence in abductive logic. In Proceedings of the 19th international joint conference on artificial intelligence (pp. 472–477).
Inoue, K., & Sakama, C. (2006a). On abductive equivalence. In L. Magnani (Ed.), Model-based reasoning in science and engineering: cognitive science, epistemology, logic. Studies in logic (pp. 333–352). London: College Publications.
Inoue, K., & Sakama, C. (2006b). Abductive equivalence in first-order logic. Logic Journal of the IGPL. Special Issue: Abduction, Practical Reasoning, and Creative Inferences in Science, 14(2), 333–346.
Janhunen, T., & Oikarinen, E. (2004). LPEQ and DLPEQ—translators for automated equivalence testing of logic programs. In Lecture notes in artificial intelligence : Vol. 2923. Proceedings of the 7th international conference of logic programming and nonmonotonic reasoning (pp. 336–340). Berlin: Springer.
Lachiche, N. (2000). Abduction and induction from a non-monotonic reasoning perspective. In P. A. Flach, & A. C. Kakas (Eds.), Abduction and induction—essays on their relation and integration (pp. 107–116). Norwell: Kluwer Academic.
Lifschitz, V., Pearce, D., & Valverde, A. (2001). Strongly equivalent logic programs. ACM Transactions on Computational Logic, 2, 526–541.
Lin, F. (2002). Reducing strong equivalence of logic programs to entailment in classical propositional logic. In Proceedings of the 8th international conference on principles of knowledge representation and reasoning (pp. 170–176). San Mateo: Morgan Kaufmann.
Maher, M. J. (1988). Equivalence of logic programs. In J. Minker (Ed.), Foundations of deductive databases and logic programming (pp. 627–658). San Mateo: Morgan Kaufmann.
McCarthy, J. (1980). Circumscription—a form of nonmonotonic reasoning. Artificial Intelligence, 13, 27–39.
Minker, J. (1982). On indefinite data bases and the closed world assumption. In Lecture notes in computer science : Vol. 138. Proceedings of the 6th international conference on automated deduction (pp. 292–308). Berlin: Springer.
Muggleton, S. (Ed.) (1992). Inductive logic programming. San Diego: Academic Press.
Muggleton, S. (1995). Inverse entailment and progol. New Generation Computing, 13, 245–286.
Muggleton, S., & Buntine, W. (1992). Machine invention of first-order predicate by inverting resolution. In S. Muggleton (Ed.), Inductive logic programming (pp. 261–280). San Diego: Academic Press.
Muggleton, S., & Feng, C. (1990). Efficient induction algorithm. In S. Muggleton (Ed.), Inductive logic programming (pp. 281–298). San Diego: Academic Press.
Nienhuys-Cheng, S.-H., & De Wolf, R. (1997). Lecture notes in artificial intelligence : Vol. 228. Foundations of inductive logic programming. Berlin: Springer.
Osorio, M., Navarro, J. A., & Arrazola, J. (2001). Equivalence in answer set programming. In Lecture notes in computer science : Vol. 2372. Proceedings of the 11th international workshop on logic based program synthesis and transformation (pp. 57–75). Berlin: Springer.
Otero, R. P. (2001). Induction of stable models. In Lecture notes in artificial intelligence : Vol. 2157. Proceedings of the 11th international conference on inductive logic programming (pp. 193–205). Berlin: Springer.
Pettorossi, A., & Proietti, M. (1994). Transformation of logic programs: foundations and techniques. Journal of Logic Programming, 19/20, 261–320.
Plotkin, G. D. (1971). A further note on inductive generalization. In B. Meltzer, & D. Michie (Eds.), Machine intelligence (Vol. 6, pp. 101–124). Edinburgh: Edinburgh University Press.
Quinlan, R. (1990). Learning logical definitions from relations. Machine Learning, 5, 239–266.
Sagiv, Y. (1988). Optimizing datalog programs. In J. Minker (Ed.), Foundations of deductive databases and logic programming (pp. 659–668). San Mateo: Morgan Kaufmann.
Sakama, C., & Inoue, K. (1995). The effect of partial deduction in abductive reasoning. In Proceedings of the 12th international conference on logic programming (pp. 383–397). Cambridge: MIT Press.
Sakama, C., & Inoue, K. (2005). Inductive equivalence of logic programs. In Lecture notes in artificial intelligence : Vol. 3625. Proceedings of the 15th international conference on inductive logic programming (pp. 312–329). Berlin: Springer.
Sakama, C., & Inoue, K. (2009a). Equivalence issues in abduction and induction. Journal of Applied Logic, 7(3), 318–328.
Sakama, C., & Inoue, K. (2009b). Brave induction: a logical framework for learning from incomplete information. Machine Learning, 76(1), 3–35.
Tamaki, H., & Sato, T. (1984). Unfold/fold transformation of logic programs. In Proceedings of the 2nd international conference on logic programming (pp. 127–138).
Turner, H. (2003). Strong equivalence made easy: nested expressions and weight constraints. Theory and Practice of Logic Programming, 3(4–5), 609–622.
Van Emden, M. H., & Kowalski, R. A. (1976). The semantics of predicate logic as a programming language. Journal of the ACM, 23(4), 733–742.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor: David Page.
Rights and permissions
About this article
Cite this article
Sakama, C., Inoue, K. Inductive equivalence in clausal logic and nonmonotonic logic programming. Mach Learn 83, 1–29 (2011). https://doi.org/10.1007/s10994-010-5189-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10994-010-5189-4