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Graph Isomorphism is Not AC0-Reducible to Group Isomorphism

Published:01 November 2013Publication History
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Abstract

We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with O(log log n) depth and O(log2 n) nondeterministic bits, where n is the number of group elements. This improves the existing upper bound for the problems. In the previous bound the circuits have bounded fan-in but depth O(log2 n) and also O(log2 n) nondeterministic bits. We then prove that the kind of circuits from our upper bound cannot compute the Parity function. Since Parity is AC0-reducible to Graph Isomorphism, this implies that Graph Isomorphism is strictly harder than Group or Quasigroup Isomorphism under the ordering defined by AC0 reductions. We extend this result to the stronger ACC0[p] reduction and its randomized version.

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        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 5, Issue 4
        November 2013
        103 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/2539126
        Issue’s Table of Contents

        Copyright © 2013 ACM

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 1 November 2013
        • Revised: 1 May 2013
        • Accepted: 1 May 2013
        • Received: 1 April 2012
        Published in toct Volume 5, Issue 4

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