Complexity Results for Propositional Closed World Reasoning and Circumscription from Tractable Knowledge Bases

This paper presents new complexity results for propositional closed world reasoning (CWR) from tractable knowledge bases (KBs). Both (basic) CWR, generalized CWR, extended generalized CWR, careful CWR and extended CWR (equivalent to circumscription) are considered. The focus is laid on tractable KBs belonging to target classes for exact compilation functions: Blake formulas, DNFs, disjunctions of Horn formulas, and disjunctions of renamable Horn formulas. The complexity of inference is identified for all the forms of CWR listed above. For each of them, new tractable fragments are exhibited. Our results suggest knowledge compilation as a valuable approach to deal with the complexity of CWR in some situations.

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

[2]  Archie Blake.  Canonical expressions in Boolean algebra , 1938 .

[3]  Christos H. Papadimitriou,et al.  Computational complexity , 1993 .

[4]  Maurizio Lenzerini,et al.  The Complexity of Propositional Closed World Reasoning and Circumscription , 1994, J. Comput. Syst. Sci..

[5]  Michael Gelfond,et al.  Negation as Failure: Careful Closure Procedure , 1986, Artif. Intell..

[6]  Jean H. Gallier,et al.  Linear-Time Algorithms for Testing the Satisfiability of Propositional Horn Formulae , 1984, J. Log. Program..

[7]  Anil Nerode,et al.  Computing Circumscriptive Databases: I. Theory and Algorithms , 1995, Inf. Comput..

[8]  Johan de Kleer,et al.  A Minimality Maintenance System , 1992, KR.

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

[10]  Raymond Reiter,et al.  Foundations of Assumption-based Truth Maintenance Systems: Preliminary Report , 1987, AAAI.

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

[12]  Lakhdar Sais,et al.  Tractable Cover Compilations , 1997, IJCAI.

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

[14]  Marco Cadoli,et al.  A Survey on Knowledge Compilation , 1997, AI Commun..

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

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

[17]  Robert Schrag,et al.  Compilation for Critically Constrained Knowledge Bases , 1996, AAAI/IAAI, Vol. 1.

[18]  John McCarthy,et al.  Applications of Circumscription to Formalizing Common Sense Knowledge , 1987, NMR.

[19]  Rina Dechter,et al.  Default Reasoning Using Classical Logic , 1996, Artif. Intell..

[20]  Claudette Cayrol,et al.  Using the Davis and Putnam Procedure for an Efficient Computation of Preferred Models , 1996, ECAI.

[21]  Harry R. Lewis,et al.  Renaming a Set of Clauses as a Horn Set , 1978, JACM.

[22]  Georg Gottlob,et al.  Propositional Circumscription and Extended Closed-World Reasoning are IIp2-Complete , 1993, Theor. Comput. Sci..

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

[24]  Yacine Boufkhad.  Algorithms for Propositional KB Approximation , 1998, AAAI/IAAI.

[25]  Paolo Liberatore,et al.  On the Compilability of Diagnosis, Planning, Reasoning about Actions, Belief Revision, etc , 1998, KR.