Translation among CNFs, characteristic models and ordered binary decision diagrams

We consider translation among conjunctive normal forms (CNFs), characteristic models, and ordered binary decision diagrams (OBDDs) of Boolean functions. It is shown in this paper that Horn OBDDs can be translated into their Horn CNFs in polynomial time. As for the opposite direction, the problem can be solved in polynomial time if the ordering of variables in the resulting OBDD is specified as an input. In case that such ordering is not specified and the resulting OBDD must be of minimum size, its decision version becomes NP-complete. Similar results are also obtained for the translation in both directions between characteristic models and OBDDs. We emphasize here that the above results hold on any class of functions having a basis of polynomial size.

[1]  Shuzo Yajima,et al.  Optimizing OBDDs Is Still Intractable for Monotone Functions , 1998, MFCS.

[2]  Randal E. Bryant,et al.  Graph-Based Algorithms for Boolean Function Manipulation , 1986, IEEE Transactions on Computers.

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

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

[5]  Olivier Coudert,et al.  Doing two-level logic minimization 100 times faster , 1995, SODA '95.

[6]  R. P. Jacobi,et al.  Generating prime and irredundant covers for binary decision diagrams , 1992, [1992] Proceedings The European Conference on Design Automation.

[7]  Shin-ichi Minato,et al.  Fast Generation of Prime-Irredundant Covers from Binary Decision Diagrams , 1993 .

[8]  Alberto L. Sangiovanni-Vincentelli,et al.  Formal verification of combinational circuits , 1997, Proceedings Tenth International Conference on VLSI Design.

[9]  Toshihide Ibaraki,et al.  Complexity of Identification and Dualization of Positive Boolean Functions , 1995, Inf. Comput..

[10]  Georg Gottlob,et al.  Identifying the Minimal Transversals of a Hypergraph and Related Problems , 1995, SIAM J. Comput..

[11]  Toshihide Ibaraki,et al.  Ordered binary decision diagrams as knowledge-bases , 1999, Artif. Intell..

[12]  Randal E. Bryant,et al.  Efficient implementation of a BDD package , 1991, DAC '90.

[13]  Roni Khardon.  Translating between Horn Representations and their Characteristic Models , 1995, J. Artif. Intell. Res..

[14]  Nader H. Bshouty,et al.  Exact learning via the Monotone theory , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[15]  Shuzo Yajima,et al.  The Complexity of the Optimal Variable Ordering Problems of Shared Binary Decision Diagrams , 1993, ISAAC.

[16]  Michael Frazier,et al.  Learning conjunctions of Horn clauses , 2004, Machine Learning.

[17]  Bart Selman,et al.  Reasoning With Characteristic Models , 1993, AAAI.

[18]  Elias C. Stavropoulos,et al.  Evaluation of an Algorithm for the Transversal Hypergraph Problem , 1999, WAE.

[19]  Leonid Khachiyan,et al.  On the Complexity of Dualization of Monotone Disjunctive Normal Forms , 1996, J. Algorithms.