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

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Abstract

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 AC0, and thus cannot be hard for AC0(2) or any superclass like NC1, 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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  1. Theoretische Informatik, Universität Hannover, Appelstr. 4, 30167, Hannover, Germany

    Heribert Vollmer

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  1. Heribert Vollmer
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Correspondence to Heribert Vollmer.

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Supported in part by DFG Grant Vo 630/5-2 and EPSRC Grant 531174.

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Vollmer, H. The Complexity of Deciding if a Boolean Function Can Be Computed by Circuits over a Restricted Basis. Theory Comput Syst 44, 82–90 (2009). https://doi.org/10.1007/s00224-007-9030-9

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  • DOI: https://doi.org/10.1007/s00224-007-9030-9

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