Inductive equivalence in clausal logic and nonmonotonic logic programming

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.

[1]  Katsumi Inoue.  On Abductive Equivalence , .

[2]  Chiaki Sakama,et al.  Inductive Equivalence of Logic Programs , 2005, ILP.

[3]  Chiaki Sakama,et al.  Abductive Equivalence in First-order Logic , 2006, Log. J. IGPL.

[4]  J. McCarthy.  Circumscription|a Form of Nonmonotonic Reasoning , 1979 .

[5]  Pierre Siegel,et al.  Saturation, Nonmonotonic Reasoning and the Closed-World Assumption , 1985, Artif. Intell..

[6]  Nicolas Lachiche.  Abduction and Induction from a Non-Monotonic Reasoning Perspective , 2000 .

[7]  Luc De Raedt,et al.  A Theory of Clausal Discovery , 1993, IJCAI.

[8]  Michael J. Maher.  Equivalences of Logic Programs , 1988, Foundations of Deductive Databases and Logic Programming..

[9]  Luc De Raedt,et al.  The Many Faces of Inductive Logic Programming , 1993, ISMIS.

[10]  Robert A. Kowalski,et al.  The Semantics of Predicate Logic as a Programming Language , 1976, JACM.

[11]  Paolo Mancarella,et al.  Abductive Logic Programming , 1992, LPNMR.

[12]  Chiaki Sakama,et al.  Equivalence of Logic Programs Under Updates , 2004, JELIA.

[13]  Luc De Raedt,et al.  Learning from satisfiability , 1997 .

[14]  David Pearce,et al.  Strongly equivalent logic programs , 2001, ACM Trans. Comput. Log..

[15]  Peter A. Flach,et al.  Abduction and induction: essays on their relation and integration , 2000 .

[16]  Keith L. Clark,et al.  Negation as Failure , 1987, Logic and Data Bases.

[17]  Yehoshua Sagiv.  Optimizing Datalog Programs , 1988, Foundations of Deductive Databases and Logic Programming..

[18]  Stephen Muggleton,et al.  Machine Invention of First Order Predicates by Inverting Resolution , 1988, ML.

[19]  Tomi Janhunen,et al.  LPEQ and DLPEQ - Translators for Automated Equivalence Testing of Logic Programs , 2004, LPNMR.

[20]  Stephen Muggleton,et al.  Inverse entailment and progol , 1995, New Generation Computing.

[21]  Antonis C. Kakas,et al.  Abduction in Logic Programming , 2002, Computational Logic: Logic Programming and Beyond.

[22]  Stephen Muggleton,et al.  Inductive Logic Programming , 2011, Lecture Notes in Computer Science.

[23]  Thomas Eiter,et al.  Uniform Equivalence of Logic Programs under the Stable Model Semantics , 2003, ICLP.

[24]  Michael Gelfond,et al.  Classical negation in logic programs and disjunctive databases , 1991, New Generation Computing.

[25]  Shan-Hwei Nienhuys-Cheng,et al.  Foundations of Inductive Logic Programming , 1997, Lecture Notes in Computer Science.

[26]  Ramón P. Otero,et al.  Induction of Stable Models , 2001, ILP.

[27]  Chitta Baral,et al.  Logic Programming and Knowledge Representation , 1994, J. Log. Program..

[28]  Chiaki Sakama,et al.  Abductive Framework for Nonmonotonic Theory Change , 1995, IJCAI.

[29]  Peter A. Flach.  Rationality Postulates for Induction , 1996, TARK.

[30]  Peter A. Flach,et al.  Abductive and inductive reasoning: background and issues , 2000 .

[31]  Lorenzo Magnani.  Model based reasoning in science and engineering : cognitive science, epistemology, logic , 2006 .

[32]  Chiaki Sakama,et al.  Brave induction: a logical framework for learning from incomplete information , 2009, Machine Learning.

[33]  Gordon Plotkin,et al.  A Further Note on Inductive Generalization , 2008 .

[34]  Fangzhen Lin.  Reducing Strong Equivalence of Logic Programs to Entailment in Classical Propositional Logic , 2002, KR.

[35]  Alberto Pettorossi,et al.  Transformation of Logic Programs: Foundations and Techniques , 1994, J. Log. Program..

[36]  Luc De Raedt,et al.  Clausal Discovery , 1997, Machine Learning.

[37]  Jack Minker,et al.  Logic and Data Bases , 1978, Springer US.

[38]  Jack Minker.  Foundations of deductive databases and logic programming , 1988 .

[39]  Chiaki Sakama,et al.  The Effect of Partial Deduction in Abductive Reasoning , 1995, ICLP.

[40]  Chiaki Sakama,et al.  Equivalence issues in abduction and induction , 2009, J. Appl. Log..

[41]  Katsumi Inoue,et al.  Induction as Consequence Finding , 2004, Machine Learning.

[42]  Antonis C. Kakas,et al.  Computational Logic: Logic Programming and Beyond: Essays in Honour of Robert A. Kowalski, Part I , 2002 .

[43]  John McCarthy,et al.  Circumscription - A Form of Non-Monotonic Reasoning , 1980, Artif. Intell..

[44]  Patrick Brézillon,et al.  Lecture Notes in Artificial Intelligence , 1999 .

[45]  Liviu Badea,et al.  Refinement Operators Can Be (Weakly) Perfect , 1999, ILP.

[46]  Hisao Tamaki,et al.  Unfold/Fold Transformation of Logic Programs , 1984, ICLP.

[47]  Charles S. Peirce,et al.  Studies in Logic , 2008 .

[48]  Henrik Bostr,et al.  Specialization of Logic Programs by Pruning SLD-Trees , 2007 .

[49]  Alex S. Taylor,et al.  Machine intelligence , 2009, CHI.

[50]  J. R. Quinlan.  Learning Logical Definitions from Relations , 1990 .

[51]  Luc De Raedt,et al.  Logical Settings for Concept-Learning , 1997, Artif. Intell..

[52]  Mauricio Osorio,et al.  Equivalence in Answer Set Programming , 2001, LOPSTR.

[53]  Jack Minker,et al.  On Indefinite Databases and the Closed World Assumption , 1987, CADE.

[54]  Chiaki Sakama,et al.  Equivalence in Abductive Logic , 2005, IJCAI.

[55]  Hudson Turner,et al.  Strong equivalence made easy: nested expressions and weight constraints , 2003, Theory and Practice of Logic Programming.