Lecture Notes in Artificial Intelligence

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1 Lecture Notes in Artificial Intelligence 941 Subseries of Lecture Notes in Computer Science Edited by J. G. Carbonell and J. Siekmann Lecture Notes in Computer Science Edited by G. Goos, J. Hartmanis and J. van Leeuwen

2 Marco Cadoli Tractable Reasoning in Artificial " " " Inte llig ence Springer

3 Series Editors Jaime G. Carbonell School of Computer Science Carnegie Mellon University Pittsburgh, PA , USA J6rg Siekmann University of Saarland German Research Center forartificial Intelligence (DFKI) Stuhlsatzenhausweg 3, D Saarbriicken, Germany Author Marco Cadoli Dipartimento di Informatica e Sistemistica Universith di Roma "La Sapienza" Via Salaria, 113, Roma, Italy Die Deutsche Bibliothek - CIP-Einheitsaufnahme Cadoll, Marco: Tractable reasoning in artificial intelligence / Marco Cadoli. - Berlin ; Heidelberg ; New York : Springer, 1995 (Lecture notes in computer science ; Vol. 941 : Lecture notes in artificial intelligence) ISBN (Berlin...) ISBN (New York...) NE: GT CR Subject Classification (1991): , F.4.1, El.3 ISBN Springer-Verlag Berlin Heidelberg New York CIP data applied for This work is subject to copyright.all rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera ready by author SPIN: / Printed on acid-free paper

4 I Preface Logic is one of the most popular approaches to Artificial Intelligence, as it provides a language within which it is possible to express a set of basic truths, from which other truths can be mechanically generated by applying some rules. A potential obstacle to use of logic is its high computational complexity. Logical inference is indeed an extraordinarily powerful computational device, and problems in logic are known among computer scientists for their high computational complexity. As efficiency of AI systems is important for their success, if we are to use logic as the major tool for AI we have to deal with its computational intractability. This book is concerned with computational aspects of the logical approach to Artificial Intelligence. The focus is on two strategies for achieving computational tractability in Knowledge Representation and Reasoning. The first strategy is language restriction: the language used for representing knowledge is restricted so that the formalization of interesting cases is still possible, but resulting tasks are computationally feasible. The second strategy is theory approximation: we approximate a form of logic by using anotherone that allows weaker inferential power but is computationally feasible even with a full expressiveness in the language. As far as the language restriction strategy is concerned, we consider a form of non-monotonic reasoning called minimal reasoning, which is a technique developed in the AI literature as a tool for the formalization of common-sense reasoning, now popular also in Database Theory and in Logic Programming. Closed world reasoning and circumscription are two forms of minimal reasoning addressed in our work. We analyze the tractability threshold of several problems, finding polynomially tractable cases --for which we give polynomialtime algorithms-- and intractable cases. We consider propositional as well as first-order formulae, decision as well as constructive problems, modeling several

reasoning patterns. Moreover we apply the complexity results to obtain efficient strategies for the solution of reasoning problems like non-monotonic inheritance and model-based diagnosis.

5 reasoning patterns. Moreover we apply the complexity results to obtain efficient strategies for the solution of reasoning problems like non-monotonic inheritance and model-based diagnosis. For what concerns theory approximation, we define a strategy for the approximation of decision reasoning problems based on multi-valued logic. The strategy is designed taking into account semantical as well as computational requirements and differs from existing ones in several aspects. The strategy is applied to a wide range of reasoning problems in propositional logic, fragments of first-order logic (concept description languages), propositional non-monotonic logics and modal logics. Moreover we provide a meta-level language for the representation of approximate knowledge owned by a non-omniscient agent. In the last part we show some results relating the two studies. In particular we use techniques and results developed in our analysis of the tractability threshold of minimal reasoning to perform an analysis of a famous form of logic-based approximation. As logic and computational analysis provide a ground which is common to Artificial Intelligence, Database Theory and Theoretical Computer Science, we believe that this book belongs to the intersection of the three disciplines. The work describes research that has been carried out when the author was a doctoral student at Universit~ di Roma "La Sapienza', in the period November February Acknowledgments First of all I want to thank God, who made all this possible, and wonderful. Many colleagues deserve my deepest thanks for helping me in various ways during the preparation of this book. I am grateful to my advisor Maurizio Lenzerini, who taught me a lot, helped me very much in my research and encouraged me continuously. Marco Schaerf was so kind to allow me to use results about the approximation techniques that we defined and studied together (now in Chapter 3 of this book.) Other friends gave me the same privilege of using portions of work done together: Thomas Eiter and Georg Gottlob (Section 2.6), Maurizio Lenzerini (Sections 2.3 and 2.7.1), Daniele Nardi and Fiora Pirri (Section ) I found in Dipartimento di Informatica e Sistemistica dell'universitd "La Sapienza" di Roma an ideal place to do my research. In particular I thank Luigia Carlucci Aiello for her continuous advice and encouragement. Finally, many other people read this book or portions of it that have been previously published, and gave me their feedback. Some others gave me clarifications on their work. I am grateful to all of them for their useful advice. Among them, I remember of Wolfgang Bibel, Piero Bonatti, Gerd Brewka, Marta Cialdea, Pierluigi Crescenzi, Francesco Donini, Bruno Errico, Henry Kautz, Hector Levesque, Alberto Marchetti Spaccamela, Eugenio Omodeo, Riccardo Rosati, Andrea Schaerf, Torsten Schaub, Bart Selman, Carolyn Talcott.

6 Financial support for the preparation of the present work has been partially provided by ESPRIT Basic Research Action N.6810-COMPULOG II and Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo del Consiglio Nazionale delle Ricerche.

7 Contents Introduction What is Artificial Intelligence? The role of Logic in Artificial Intelligence Logic and computational complexity The subject of this book... fl 1.5 Previous studies A closer look at the subject of this book Language restriction :2 Theory approximation Linking the two techniques Possible critics Who may be interested in reading this book? Prerequisites Structure of the book Terminology and preliminary notions Propositional logic First-order predicate logic Complexity classes Polynomial fragments of propositional logic... 25

8 Xll Contents Language restriction: Complexity of minimal reasoning Why non-monotonic reasoning? Minimal models, circumscription and closed world reasoning Minimal models and circumscription Minimal models and closed world reasoning Complexity of closed world reasoning : I Goal of the research and related results Results Complexity of minimal model checking Goal of the research and related results Results Complexity of finding a minimal model Goal of the research and related results Results Circumscription in first-order formulae Goal of the research and related results Results Applications of tractable minimal reasoning to practical reasoning Non-monotonic inheritance Syntax Semantics Computational analysis Model-based diagnosis Other applications Approximation of a logical theory Logic-based methods for approximation: existing techniques Levesque's limited inference Selman and Kautz's Horn knowledge compilation Comparison between the two approaches Guidelines of a new method for approximation Approximation of propositional logic Semantics Computational aspects Discussion and examples Approximate reasoning and non-omniscient agents Syntax and semantics Properties and applications Approximation of concept description languages Concept Languages First-order generalization of S-interpretations Approximation of.as Approximation of.ai:c Approximation of non-monotonic logics Approximation of default logic Preliminaries

9 Contents xiii Approximation method Approximation of circumscription Preliminaries Approximation method... i12 Approximation of propositional modal logics Semantics Properties Using complexity results for evaluating approximation techniques 4.1 Horn compilation: semantical and computational aspects Horn Horn Discussion GLBs From GLBs to minimal models From minimal models to GLBs Semantical consequences LUB Horn UBs with a limited number of clauses Dealing with the LUB implicitly Comments on the computational results A semantical remark Conclusions Comments on the results obtained Open problems Complexity of minimal reasoning Approximation of a logical theory A Appendix to Chapter 2 A.1 Appendix to Section 2.3 A.I.1 A.1.2 A.1.3 A.1.4 A.1.5 A.1.6 A Preliminaries Upper bounds and algorithms for CWR A Upper bounds of CWR A Algorithms for computing circumscription CWA GCWA A Intractable cases A Tractable cases EGCWA A Intractable cases A Tractable cases CCWA A Intractable cases A Tractable cases ECWA A Intractable cases

10 XlV Contents A Tractable cases A.1.8 Inference of new positive clauses A.2 Appendix to Section A.2.1 Intractable cases A.2.2 Tractable cases A.3 Appendix to Section A.3.1 Preliminaries A.3.2 Intractable cases A.3.3 Tractable cases A.4 Appendix to Section A.4.1 The reduction A.4.2 Example A.5 Appendix to Section A.5.1 Appendix to Section A Semantics of NINs A Intractable cases A Tractable cases A.5.2 Appendix to Section B Appendix to Chapter 3 B.1 Appendix to Section "[ B.I.I Proofs of theorems B.1.2 Reducing S-entailment to S-unsatisfiability B.1.3 Algorithms for testing S-satisfiability B Resolution and Enumeration B The algorithm B.2 Appendix to Section ' B.3 Appendix to Section B.3.1 Preliminaries B.3.2 Appendix to Section , B.3.3 Appendix to Section B.4 Appendix to Section B.4.1 Appendix to Section B.4.2 Appendix to Section B.5 Appendix to Section 3, C Appendix to Chapter 4 C.1 Appendix to Section 4.2 C.2 Appendix to Section Bibliography 227 Index 241

11 List of Figures 1.1 Using first-order logic for Knowledge Representation ~ Reasoning Representing somebody else's knowledge Using a plausible assmnption Testing if a formula of propositional logic is inconsistent is a polynomially intractable task A new fact makes a Knowledge Base inconsistent Two reasoners with limited expressiveness Sound and complete reasoner You can trust a sound reasoner on "yes" answers You can trust a complete reasoner on "no" answers Taxonomy of some propositional classes in which inference is polynomial The evolution of formalisms for CWR Different kinds of assertions and their representation Tweety the bird The Nixon diamond Approximation of a set of logical formulae... 73

12 List of Tables Clauses that each class of formulae can represent Complexity of clause inference for the various syntactic classes and CWR rules (problems INF- [C,CLAUSE] and INF-O~[C,CLAUSE,d]) 42 Complexity of literal inference for the various syntactic classes and CWR rules (problems INF-OI[C,LIT] and INF-O~[C,LIT,d]) Upper bound of the complexity Of clause inference for the various syntactic classes and CWR rules (problems INF-oI[c,CLAUSE] and INF-a[C,CLAUSE,d]) Can the CWR method infer new positive clauses (as opposed to only negative clauses)? Complexity of the problem (P; Q; Z)-MIN(S)... : Complexity of the problem (P; Q; Z)-FIND(S) Some properties of Horn G LBs and LUBs Various forms of interpretation of a propositional letter... 77

13 List of Algorithms A.I NonMinimal A.2 NotECWA A.3 CCWA A.4 CWA A.5 DefCCWA A.6 Krom CCWA A.7 HornECWA A.8 HK-ECWA t73 A.9 Lexicographically smallest model ]3.1 S-3-unsat/S-l-sat C.1 MinimalModel2LB

Lecture Notes in Artificial Intelligence. Lecture Notes in Computer Science

Lecture Notes in Artificial Intelligence 897 Subseries of Lecture Notes in Computer Science Edited by J. G. Carbonell and J. Siekmann Lecture Notes in Computer Science Edited by G. Goos, J. Hartmanis and