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Uniform proofs of ACC representations

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Abstract

We give a uniform proof of the theorems of Yao and Beigel–Tarui representing ACC predicates as constant depth circuits with \(\hbox {MOD}_{m}\) gates and a symmetric gate. The proof is based on a relativized, generalized form of Toda’s theorem expressed in terms of closure properties of formulas under bounded universal, existential and modular counting quantifiers. This allows the main proofs to be expressed in terms of formula classes instead of Boolean circuits. The uniform version of the Beigel–Tarui theorem is then obtained automatically via the Furst–Saxe–Sipser and Paris–Wilkie translations. As a special case, we obtain a uniform version of Razborov and Smolensky’s representation of \(\hbox {AC}^{0}[p]\) circuits. The paper is partly expository, but is also motivated by the desire to recast Toda’s theorem, the Beigel–Tarui theorem, and their proofs into the language of bounded arithmetic. However, no knowledge of bounded arithmetic is needed.

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  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093-0112, USA

    Sam Buss

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  1. Sam Buss
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Correspondence to Sam Buss.

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Supported in part by NSF Grants DMS-1101228 and CCR-1213151, Simons Foundation Grant 306202 and Skolkovo Institute of Science and Technology. Part of this work was performed during the Special Semester program in Complexity in St. Petersburg, Russia.

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Buss, S. Uniform proofs of ACC representations. Arch. Math. Logic 56, 639–669 (2017). https://doi.org/10.1007/s00153-017-0560-9

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