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# Monotone Boolean dualization is in co-NP[log2n]

Department of Mathematics, University of Patras, Rhion, West Greece, Greece
(Impact Factor: 0.48). 01/2003; 85(1):1-6. DOI: 10.1016/S0020-0190(02)00346-0
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ABSTRACT In 1996, Fredman and Khachiyan [J. Algorithms 21 (1996) 618–628] presented a remarkable algorithm for the problem of checking the duality of a pair of monotone Boolean expressions in disjunctive normal form. Their algorithm runs in no(logn) time, thus giving evidence that the problem lies in an intermediate class between P and co-NP. In this paper we show that a modified version of their algorithm requires deterministic polynomial time plus O(log2n) nondeterministic guesses, thus placing the problem in the class co-NP[log2n]. Our nondeterministic version has also the advantage of having a simpler analysis than the deterministic one.

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##### Conference Paper: On Resolution Like Proofs of Monotone Self-Dual Functions
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ABSTRACT: We examine the time complexity of resolution like proofs for monotone self-dual functions. For a class of monotone boolean functions that strictly contain the class of self-dual functions we give a new characterization of self-duality. We show that an obvious implementation of a resolution like proof system has an exponential lower bound on the number of step in resolution proofs for monotone self-dual functions. We pose some questions, resolution of which would further the program on self-dual monotone boolean functions. Gen-eral lower bounds on the length of resolution like proofs for self-dual monotone boolean functions is an interesting open question.
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##### Conference Paper: Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic
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ABSTRACT: The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs G and H, decide whether H consists precisely of all minimal transversals of G (in which case we say that G is the dual of H). This problem is equivalent to decide whether two given non-redundant monotone DNFs are dual. It is known that co-DUAL, the complementary problem to DUAL, is in GC(log^2 n, PTIME), where GC(f(n), C) denotes the complexity class of all problems that after a nondeterministic guess of O(f(n)) bits can be decided (checked) within complexity class C. It was conjectured that co-DUAL is in GC(log^2 n, LOGSPACE). In this paper we prove this conjecture and actually place the co-DUAL problem into the complexity class GC(log^2 n, TC^0) which is a subclass of GC(log^2 n, LOGSPACE). We here refer to the logtime-uniform version of TC^0, which corresponds to FO(COUNT), i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for co-DUAL that requires to guess O(log^2 n) bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in FO(COUNT).
CSL-LICS '14 - Proceedings of the Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic (CSL) and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS); 07/2014
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