Space Efficiency of Propositional Knowledge Representation Formalisms

We investigate the space efficiency of a Propositional Knowledge Representation (PKR) formalism. Intuitively, the space efficiency of a formalism F in representing a certain piece of knowledge α, is the size of the shortest formula of F that represents α. In this paper we assume that knowledge is either a set of propositional interpretations (models) or a set of propositional formulae (theorems). We provide a formal way of talking about the relative ability of PKR formalisms to compactly represent a set of models or a set of theorems. We introduce two new compactness measures, the corresponding classes, and show that the relative space efficiency of a PKR formalism in representing models/theorems is directly related to such classes. In particular, we consider formalisms for nonmonotonic reasoning, such as circumscription and default logic, as well as belief revision operators and the stable model semantics for logic programs with negation. One interesting result is that formalisms with the same time complexity do not necessarily belong to the same space efficiency class.

[1]  Georg Gottlob,et al.  Complexity Results for Nonmonotonic Logics , 1992, J. Log. Comput..

[2]  Bart Selman,et al.  Knowledge compilation and theory approximation , 1996, JACM.

[3]  Bernhard Nebel,et al.  How Hard is it to Revise a Belief Base , 1996 .

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

[5]  Raymond Reiter,et al.  A Logic for Default Reasoning , 1987, Artif. Intell..

[6]  Francesco M. Donini,et al.  Feasibility and Unfeasibility of Off-Line Processing , 1996, ISTCS.

[7]  Bart Selman,et al.  Horn Approximations of Empirical Data , 1995, Artif. Intell..

[8]  David W. Etherington.  Reasoning With Incomplete Information , 1988 .

[9]  Teodor C. Przymusinski,et al.  On the Relationship Between Circumscription and Negation as Failure , 1989, Artif. Intell..

[10]  Marco Schaerf,et al.  The Complexity of Model Checking for Belief Revision and Update , 1996, AAAI/IAAI, Vol. 1.

[11]  Francesco M. Donini,et al.  The size of a revised knowledge base , 1995, PODS '95.

[12]  Georg Gottlob,et al.  On the complexity of propositional knowledge base revision, updates, and counterfactuals , 1992, Artif. Intell..

[13]  Tomasz Imielinski,et al.  Results on Translating Defaults to Circumscription , 1985, IJCAI.

[14]  Ronald Fagin,et al.  On the semantics of updates in databases , 1983, PODS.

[15]  Bart Selman,et al.  Forming Concepts for Fast Inference , 1992, AAAI.

[16]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[17]  Chee-Keng Yap,et al.  Some Consequences of Non-Uniform Conditions on Uniform Classes , 1983, Theor. Comput. Sci..

[18]  Tomi Janhunen.  On the Intertranslatability of Autoepistemic, Default and Priority Logics, and Parallel Circumscription , 1998, JELIA.

[19]  G. Gottlob,et al.  Propositional circumscription and extended closed-world reasoning are ΠP2-complete , 1993 .

[20]  Marianne Winslett.  Sometimes Updates Are Circumscription , 1989, IJCAI.

[21]  Osamu Watanabe,et al.  New Collapse Consequences of NP Having Small Circuits , 1995, ICALP.

[22]  David S. Johnson,et al.  A Catalog of Complexity Classes , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[23]  Francesco M. Donini,et al.  Is Intractability of Non-Monotonic Reasoning a Real Drawback? , 1994, AAAI.

[24]  Ravi B. Boppana,et al.  The Complexity of Finite Functions , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[25]  Francesco M. Donini,et al.  Comparing Space Efficiency of Propositional Knowledge Representation Formalisms , 1996, KR.

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

[27]  Marco Cadoli,et al.  The Complexity of Model Checking for Circumscriptive Formulae , 1992, Inf. Process. Lett..

[28]  Richard J. Lipton,et al.  Some connections between nonuniform and uniform complexity classes , 1980, STOC '80.

[29]  Bart Selman,et al.  Hard Problems for Simple Default Logics , 1989, Artif. Intell..

[30]  Francesco M. Donini,et al.  On Compact Representations of Propositional Circumscription , 1995, STACS.

[31]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[32]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[33]  Marianne Winslett,et al.  Updating logical databases , 1990, Cambridge tracts in theoretical computer science.

[34]  Victor W. Marek,et al.  Autoepistemic logic , 1991, JACM.

[35]  Christos H. Papadimitriou,et al.  Some computational aspects of circumscription , 1988, JACM.

[36]  Dan Roth,et al.  Reasoning with Models , 1994, Artif. Intell..

[37]  John S. Schlipf,et al.  Decidability and definability with circumscription , 1987, Ann. Pure Appl. Log..

[38]  Francesco M. Donini,et al.  Preprocessing of Intractable Problems , 2002, Inf. Comput..

[39]  Rina Dechter,et al.  Default Logic, Propositional Logic, and Constraints , 1991, AAAI.

[40]  John S. Schlipf.  A Survey of Complexity and Undecidability Results in Logic Programming , 1992, Structural Complexity and Recursion-theoretic methods in Logic-Programming.

[41]  Georg Gottlob,et al.  Translating default logic into standard autoepistemic logic , 1995, JACM.

[42]  Dan Roth,et al.  Defaults and Relevance in Model-Based Reasoning , 1997, Artif. Intell..

[43]  Bart Selman,et al.  The Comparative Linguistics of Knowledge Representation , 1995, IJCAI.

[44]  Moshe Tennenholtz,et al.  Off-Line Reasoning for On-Line Efficiency: Knowledge Bases , 1996, Artif. Intell..