Uniform proofs of ACC representations

Uniform proofs of ACC representations We give a uniform proof of the theorems of Yao and Beigel–Tarui representing ACC predicates as constant depth circuits with $$\hbox {MOD}_{m}$$ 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 $$\hbox {AC}^{0}[p]$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Uniform proofs of ACC representations

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
D.O.I.
10.1007/s00153-017-0560-9
Publisher site
See Article on Publisher Site

Abstract

We give a uniform proof of the theorems of Yao and Beigel–Tarui representing ACC predicates as constant depth circuits with $$\hbox {MOD}_{m}$$ 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 $$\hbox {AC}^{0}[p]$$ 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.

Journal

Archive for Mathematical LogicSpringer Journals

Published: May 26, 2017

References

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