On the Tractable Counting of Theory Models and its Application to Truth Maintenance and Belief Revision

We address in this paper the problem of counting the models of a propositional theory under incremental changes to its literals. Specifcally, we show that if a propositional theory Δ is in a special form that we call smooth, deterministic, decomposable negation normal form (sd-DNNF), then for any consistent set of literals S, we can simultaneously count (in time linear in the size of Δ) the models of Δ ∪ S and the models of every theory Δ ∪ T where T results from adding, removing or flipping a literal in S. We present two results relating to the time and space complexity of compiling propositional theories into sd-DNNF. First, we show that if a conjunctive normal form (CNF) has a bounded treewidth, then it can be compiled into an sd-DNNF in time and space which are linear in its size. Second, we show that sd-DNNF is a strictly more space efficient representation than Free Binary Decision Diagrams (FBDDs). Finally, we discuss some applications of the counting results to truth maintenance systems, belief revision, and model-based diagnosis.

[1]  Ingo Wegener,et al.  Graph Driven BDDs - A New Data Structure for Boolean Functions , 1995, Theor. Comput. Sci..

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

[3]  Kenneth D. Forbus,et al.  Building Problem Solvers , 1993 .

[4]  Christoph Meinel,et al.  Efficient Boolean Manipulation With OBDD's can be Extended to FBDD's , 1994, IEEE Trans. Computers.

[5]  Adnan Darwiche,et al.  Model-Based Diagnosis using Structured System Descriptions , 1998, J. Artif. Intell. Res..

[6]  Raymond Reiter,et al.  Characterizing Diagnoses and Systems , 1992, Artif. Intell..

[7]  Randal E. Bryant,et al.  Symbolic Boolean manipulation with ordered binary-decision diagrams , 1992, CSUR.

[8]  Adnan Darwiche,et al.  Decomposable negation normal form , 2001, JACM.

[9]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[10]  H. Keisler,et al.  Handbook of mathematical logic , 1977 .

[11]  Adnan Darwiche.  Compiling Knowledge into Decomposable Negation Normal Form , 1999, IJCAI.

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

[13]  Beate Bollig,et al.  Complexity Theoretical Results on Partitioned (Nondeterministic) Binary Decision Diagrams , 1997, Theory of Computing Systems.

[14]  M. Iri.  Simultaneous computation of functions, partial derivatives and estimates of rounding errors —Complexity and practicality— , 1984 .

[15]  Peter Aczel etc.  HANDBOOK OF MATHEMATICAL LOGIC , 1999 .

[16]  DarwicheAdnan.  A differential approach to inference in Bayesian networks , 2003 .

[17]  G. Unter Rote.  Path Problems in Graphs , 1989 .

[18]  Rina Dechter,et al.  Directional Resolution: The Davis-Putnam Procedure, Revisited , 1994, KR.