The Complexity of Deciding if a Boolean Function Can Be Computed by Circuits over a Restricted Basis

We study the complexity of the following algorithmic problem: Given a Boolean function f and a finite set of Boolean functions B , decide if there is a circuit with basis B that computes  f . We show that if both f and all functions in B are given by their truth-table, the problem is in quasipolynom...

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Bibliographic Details
Published in:Theory of computing systems Vol. 44; no. 1; pp. 82 - 90
Main Author: Vollmer, Heribert
Format: Journal Article
Language:English
Published: New York Springer-Verlag 01.01.2009
Springer Nature B.V
Subjects:
Algebra
Algorithms
Boolean
Circuits
Codes
Computer Science
Mathematical analysis
Studies
Theory of Computation
Post’s lattice
Membership problem
Clones
Boolean circuit
Computational complexity
ISSN:1432-4350, 1433-0490
Online Access:Get full text
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Summary:We study the complexity of the following algorithmic problem: Given a Boolean function f and a finite set of Boolean functions B , decide if there is a circuit with basis B that computes  f . We show that if both f and all functions in B are given by their truth-table, the problem is in quasipolynomial-size AC 0 , and thus cannot be hard for AC 0 (2) or any superclass like NC 1 , L, or NL. This answers an open question by Bergman and Slutzki (SIAM J. Comput., 2000 ). Furthermore we show that, if the input functions are not given by their truth-table but in a succinct way, i.e., by circuits (over any complete basis), the above problem becomes complete for the class coNP.
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ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-007-9030-9