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Dual VP Classes

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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 \({i \geq 1}\).

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Authors and Affiliations

  1. Department of Computer Science, Rutgers University, Piscataway, NJ, USA

    Eric Allender & Ian Mertz

  2. Department of Computer Science, University of Texas, Austin, TX, USA

    Anna Gál

Authors
  1. Eric Allender
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  2. Anna Gál
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  3. Ian Mertz
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Correspondence to Eric Allender.

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Allender, E., Gál, A. & Mertz, I. Dual VP Classes. comput. complex. 26, 583–625 (2017). https://doi.org/10.1007/s00037-016-0146-7

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

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