Abstract
In this paper we study a class of CQ Horn functions introduced in Boros et al. (Ann Math Artif Intell 57(3–4):249–291, 2010). We prove that given a CQ Horn function f, the maximal number of pairwise disjoint essential sets of implicates of f equals the minimum number of clauses in a CNF representing f. In other words, we prove that the maximum number of pairwise disjoint essential sets of implicates of f constitutes a tight lower bound on the size (the number of clauses) of any CNF representation of f.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ausiello, G., D’Atri, A., Sacca, D.: Minimal representation of directed hypergraphs. SIAM J. Comput. 15(2), 418–431 (1986)
Boros, E., Čepek, O.: On the complexity of Horn minimization. Tech. Rep. RUTCOR Research Report RRR 1-1994, Rutgers University, New Brunswick, NJ (1994)
Boros, E., Čepek, O., Kogan, A.: Horn minimization by iterative decomposition. Ann. Math. Artif. Intell. 23, 321–343 (1998)
Boros, E., Čepek, O., Kogan, A., Kučera, P.: Exclusive and essential sets of implicates of Boolean functions. Discrete Appl. Math. 158(2), 81–96 (2010)
Boros, E., Čepek, O., Kogan, A., Kučera, P.: A subclass of Horn cnfs optimally compressible in polynomial time. Ann. Math. Artif. Intell. 57(3–4), 249–291 (2010)
Büning, H.K., Letterman, T.: Propositional Logic: Deduction and Algorithms. Cambridge University Press, New York, NY (1999)
Čepek, O.: Structural properties and minimization of Horn Boolean functions. Ph.D. thesis, Rutgers University, New Brunswick, NJ (1995)
Čepek, O., Kučera, P., Savický, P.: Boolean functions with a simple certificate for CNF complexity. Discrete Applied Mathematics (2011, in print, available online)
Dechter, R., Pearl, J.: Structure identification in relational data. Artif. Intell. 58, 237–270 (1992)
Delobel, C., Casey, R.: Decomposition of a data base and the theory of Boolean switching functions. IBM J. Res. Develop. 17, 374–386 (1973)
Genesereth, M., Nilsson, N.: Logical Foundations of Artificial Intelligence. Morgan Kaufmann, Los Altos, CA (1987)
Hammer, P., Kogan, A.: Horn functions and their DNFs. IBM J. Res. Develop. 44, 23–29 (1992)
Hammer, P., Kogan, A.: Optimal compression of propositional Horn knowledge bases: complexity and approximation. Artif. Intell. 64, 131–145 (1993)
Hammer, P., Kogan, A.: Knowledge compression—logic minimization for expert systems. In: Computers as our Better Partners. Proceedings of the IISF/ACM Japan International Symposium, pp. 306–312. World Scientific, Singapore (1994)
Hammer, P., Kogan, A.: Quasi-acyclic propositional Horn knowledge bases: optimal compression. IEEE Trans. Knowl. Data Eng. 7(5), 751–762 (1995)
Maier, D.: Minimal covers in the relational database model. JACM 27, 664–674 (1980)
Quine, W.: A way to simplify truth functions. Am. Math. Mon. 62, 627–631 (1955)
Russel, S., Norvig, P.: Artificial Intelligence: A Modern Approach, 2nd edn. Pearson Education (2003)
Tarjan, R.: Depth first search and linear graph algorithms. SIAM J. Comput. 2, 146–160 (1972)
Umans, C.: The minimum equivalent DNF problem and shortest implicants. J. Comput. Syst. Sci. 63(4), 597–611 (2001)
Umans, C., Villa, T., Sangiovanni-Vincentelli, A.L.: Complexity of two-level logic minimization. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 25(7), 1230–1246 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Čepek, O., Kučera, P. Disjoint essential sets of implicates of a CQ Horn function. Ann Math Artif Intell 61, 231–244 (2011). https://doi.org/10.1007/s10472-011-9263-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10472-011-9263-9