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Sound 3-Query PCPPs Are Long

Published:01 September 2009Publication History
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Abstract

We initiate the study of the trade-off between the length of a probabilistically checkable proof of proximity (PCPP) and the maximal soundness that can be guaranteed by a 3-query verifier with oracle access to the proof. Our main observation is that a verifier limited to querying a short proof cannot obtain the same soundness as that obtained by a verifier querying a long proof. Moreover, we quantify the soundness deficiency as a function of the proof-length and show that any verifier obtaining “best possible” soundness must query an exponentially long proof.

In terms of techniques, we focus on the special class of inspective verifiers that read at most 2 proof-bits per invocation. For such verifiers, we prove exponential length-soundness trade-offs that are later on used to imply our main results for the case of general (i.e., not necessarily inspective) verifiers. To prove the exponential trade-off for inspective verifiers, we show a connection between PCPP proof length and property-testing query complexity that may be of independent interest. The connection is that any linear property that can be verified with proofs of length ℓ by linear inspective verifiers must be testable with query complexity ≈ logℓ.

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    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 1, Issue 2
      September 2009
      89 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/1595391
      Issue’s Table of Contents

      Copyright © 2009 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 1 September 2009
      • Revised: 1 March 2009
      • Accepted: 1 March 2009
      • Received: 1 June 2008
      Published in toct Volume 1, Issue 2

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