The Minimum Equivalent DNF Problem and Shortest Implicants

We prove that the Minimum Equivalent DNF problem is ?P2-complete, resolving a conjecture due to Stockmeyer. We also consider the complexity and approximability of a related optimization problem in the second level of the polynomial hierarchy, that of finding shortest implicants of a Boolean function. We show that when the input is given as a DNF, this problem is complete for a complexity class above coNP utilizing O(log2n)-limited nondeterminism. When the input is given as a formula or circuit, the problem is ?P2-complete, and ?P2-hard to approximate within factors of n1/2?? and n1??, respectively.

[1]  Desh Ranjan,et al.  The circuit subfunction relations are $sum^P_2$-complete , 1993 .

[2]  Olivier Coudert,et al.  Two-level logic minimization: an overview , 1994, Integr..

[3]  Desh Ranjan,et al.  On the Computational Complexity of Some Classical Equivalence Relations on Boolean Functions , 1998, Theory of Computing Systems.

[4]  Klaus W. Wagner.  More Complicated Questions About Maxima and Minima, and Some Closures of NP , 1987, Theor. Comput. Sci..

[5]  Kurt Keutzer,et al.  Logic Synthesis , 1994 .

[6]  Tadeusz Strzemecki,et al.  Polynomial-time algorithms for generation of prime implicants , 1992, J. Complex..

[7]  Patrick C. Fischer,et al.  Refining Nondeterminism in Relativized Polynomial-Time Bounded Computations , 1980, SIAM J. Comput..

[8]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[9]  Ingo Wegener,et al.  The complexity of Boolean functions , 1987 .

[10]  CoudertOlivier.  Two-level logic minimization: an overview , 1994 .

[11]  James F. Gimpel.  A Method of Producing a Boolean Function Having an Arbitrarily Prescribed Prime Implicant Table , 1965, IEEE Trans. Electron. Comput..

[12]  Manindra Agrawal,et al.  The Boolean isomorphism problem , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[13]  Liming Cai,et al.  On the Amount of Nondeterminism and the Power of Verifying , 1997, SIAM J. Comput..

[14]  Willard Van Orman Quine,et al.  The Problem of Simplifying Truth Functions , 1952 .

[15]  Christos H. Papadimitriou,et al.  Computational complexity , 1993 .

[16]  Mihalis Yannakakis,et al.  On limited nondeterminism and the complexity of the V-C dimension , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[17]  Edith Hemaspaandra,et al.  The Minimization Problem for Boolean Formulas , 2002, SIAM J. Comput..

[18]  Marcus Schaefer.  Graph Ramsey theory and the polynomial hierarchy , 1999, STOC '99.

[19]  E. McCluskey.  Minimization of Boolean functions , 1956 .