Boolean Functions with a Simple Certificate for CNF Complexity

0351382 - ÚI 2013 RIV NL eng J - Journal Article
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
R&D Projects: GA MŠMT(CZ) 1M0545
Grant - others:GA ČR(CZ) GP201/07/P168; GA ČR(CZ) GAP202/10/1188
Institutional research plan: CEZ:AV0Z10300504
Keywords : Boolean functions * CNF representations
Subject RIV: BA - General Mathematics
Impact factor: 0.718, year: 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.
Permanent Link: http://hdl.handle.net/11104/0191148