article

Dual VP Classes

Authors:

Ian MertzAuthors Info & Claims

Computational Complexity, Volume 26, Issue 3

Pages 583 - 625

https://doi.org/10.1007/s00037-016-0146-7

Published: 01 September 2017 Publication History

Abstract

We consider the complexity class ACC 1 and related families of arithmetic circuits. We prove a variety of collapse results, showing several settings in which no loss of computational power results if fan-in of gates is severely restricted, as well as presenting a natural class of arithmetic circuits in which no expressive power is lost by severely restricting the algebraic degree of the circuits. We draw attention to the strong connections that exist between ACC 1 and VP, via connections to the classes CC 1[m] for various m. These results tend to support a conjecture regarding the computational power of the complexity class VP over finite algebras, and they also highlight the significance of a class of arithmetic circuits that is in some sense dual to VP. In particular, these dual-VP classes provide new characterizations of ACC 1 and TC 1 in terms of circuits of semiunbounded fan-in. As a corollary, we show that ACC i = CC i for all $${i \geq 1}$$iź1.

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Cited By

Allender E(2023)Guest ColumnACM SIGACT News10.1145/3586165.358617554:1(63-81)Online publication date: 28-Feb-2023
https://dl.acm.org/doi/10.1145/3586165.3586175
Cook JMertz I(2022)Trading time and space in catalytic branching programsProceedings of the 37th Computational Complexity Conference10.4230/LIPIcs.CCC.2022.8(1-21)Online publication date: 20-Jul-2022
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2022.8
Chen LTell RCharikar MCohen E(2019)Bootstrapping results for threshold circuits “just beyond” known lower boundsProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316333(34-41)Online publication date: 23-Jun-2019
https://dl.acm.org/doi/10.1145/3313276.3316333

Index Terms

Dual VP Classes
1. Theory of computation

Index terms have been assigned to the content through auto-classification.

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cover image Computational Complexity

Computational Complexity Volume 26, Issue 3

September 2017

230 pages

ISSN:1016-3328

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Copyright © Copyright © 2017 Springer International Publishing AG.

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Birkhauser Verlag

Switzerland

Publication History

Published: 01 September 2017

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Cited By

Allender E(2023)Guest ColumnACM SIGACT News10.1145/3586165.358617554:1(63-81)Online publication date: 28-Feb-2023
https://dl.acm.org/doi/10.1145/3586165.3586175
Cook JMertz I(2022)Trading time and space in catalytic branching programsProceedings of the 37th Computational Complexity Conference10.4230/LIPIcs.CCC.2022.8(1-21)Online publication date: 20-Jul-2022
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2022.8
Chen LTell RCharikar MCohen E(2019)Bootstrapping results for threshold circuits “just beyond” known lower boundsProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316333(34-41)Online publication date: 23-Jun-2019
https://dl.acm.org/doi/10.1145/3313276.3316333

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