# Block Rigidity: Strong Multiplayer Parallel Repetition Implies Super-Linear Lower Bounds for Turing Machines

## Author(s): Mittal, Kunal; Raz, Ran

DownloadTo refer to this page use:

`http://arks.princeton.edu/ark:/88435/pr12v2c95k`

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 [Valiant, 1977]. 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 [Paul et al., 1983], 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 v^Ω(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. |

Publication Date: | 2021 |

Citation: | Mittal, Kunal, and Ran Raz. "Block Rigidity: Strong Multiplayer Parallel Repetition Implies Super-Linear Lower Bounds for Turing Machines." In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021) 185 (2021): pp. 71:1-71:15. doi:10.4230/LIPIcs.ITCS.2021.71 |

DOI: | 10.4230/LIPIcs.ITCS.2021.71 |

ISSN: | 1868-8969 |

Pages: | 71:1 - 71:15 |

Type of Material: | Conference Article |

Series/Report no.: | Leibniz International Proceedings in Informatics (LIPIcs); |

Journal/Proceeding Title: | 12th Innovations in Theoretical Computer Science Conference (ITCS 2021) |

Version: | Final published version. This is an open access article. |

Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.