Block rigidity: Strong multiplayer parallel repetition implies super-linear lower bounds for turing machines

Kunal Mittal, Ran Raz

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

We prove that a sufficiently strong parallel repetition theorem for a special case of multiplayer (multiprover) games implies super-linear lower bounds for multi-tape Turing machines with advice. To the best of our knowledge, this is the first connection between parallel repetition and lower bounds for time complexity and the first major potential implication of a parallel repetition theorem with more than two players. Along the way to proving this result, we define and initiate a study of block rigidity, a weakening of Valiant’s notion of rigidity [36]. While rigidity was originally defined for matrices, or, equivalently, for (multi-output) linear functions, we extend and study both rigidity and block rigidity for general (multi-output) functions. Using techniques of Paul, Pippenger, Szemerédi and Trotter [28], we show that a block-rigid function cannot be computed by multi-tape Turing machines that run in linear (or slightly super-linear) time, even in the non-uniform setting, where the machine gets an arbitrary advice tape. We then describe a class of multiplayer games, such that, a sufficiently strong parallel repetition theorem for that class of games implies an explicit block-rigid function. The games in that class have the following property that may be of independent interest: for every random string for the verifier (which, in particular, determines the vector of queries to the players), there is a unique correct answer for each of the players, and the verifier accepts if and only if all answers are correct. We refer to such games as independent games. The theorem that we need is that parallel repetition reduces the value of games in this class from v to vΩ(n), where n is the number of repetitions. As another application of block rigidity, we show conditional size-depth tradeoffs for boolean circuits, where the gates compute arbitrary functions over large sets.

Original languageEnglish (US)
Title of host publication12th Innovations in Theoretical Computer Science Conference, ITCS 2021
EditorsJames R. Lee
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771771
DOIs
StatePublished - Feb 1 2021
Event12th Innovations in Theoretical Computer Science Conference, ITCS 2021 - Virtual, Online
Duration: Jan 6 2021Jan 8 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume185
ISSN (Print)1868-8969

Conference

Conference12th Innovations in Theoretical Computer Science Conference, ITCS 2021
CityVirtual, Online
Period1/6/211/8/21

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Block-rigidity
  • Matrix rigidity
  • Parallel repetition
  • Size-depth tradeoffs
  • Turing machines

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