Boolean Functions with a Simple Certificate for CNF Complexity

0351382 - ÚI 2013 RIV NL eng J - Článek v odborném periodiku
Čepek, O. - Kučera, P. - Savický, Petr
Boolean Functions with a Simple Certificate for CNF Complexity.
Discrete Applied Mathematics. Roč. 160, 4-5 (2012), s. 365-382. ISSN 0166-218X. E-ISSN 1872-6771
Grant CEP: GA MŠMT(CZ) 1M0545
Grant ostatní: GA ČR(CZ) GP201/07/P168; GA ČR(CZ) GAP202/10/1188
Výzkumný záměr: CEZ:AV0Z10300504
Klíčová slova: Boolean functions * CNF representations
Kód oboru RIV: BA - Obecná matematika
Impakt faktor: 0.718, rok: 2012

In this paper we study relationships between CNF representations of a given Boolean function and its essential sets of implicates. It is known that every CNF representation and every essential set must intersect. Therefore the maximum number of pairwise disjoint essential sets provides a lower bound on the size of any CNF representation. We are interested in functions, for which this lower bound is tight, and call such functions coverable. We prove that for every coverable function there exists a polynomially verifiable certificate for its minimum CNF size. On the other hand, we show that not all functions are coverable and construct examples of non-coverable functions. Moreover, we prove that computing the lower bound, i.e. the maximum number of pairwise disjoint essential sets, is NP-hard under various restrictions on the function and on its input representation.
Trvalý link: http://hdl.handle.net/11104/0191148