Algebrization: A new barrier in complexity theory

Scott Aaronson*, Avi Wigderson

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

28 Scopus citations

Abstract

Any proof of P≠NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a low-degree extension of A over a, finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known non-relativizing results based on arithmetization-both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP ⊄P/poly -do indeed algebrize. Second, we show that almost all of the major open problems- including P versus NP, P versus RP, and NEXP versus P/poly- will require non-algebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MA-protocol for the Inner Product function with O (√n log n) communication (essentially matching a lower bound of Klauck),

Original languageEnglish
Title of host publicationSTOC'08
Subtitle of host publicationProceedings of the 2008 ACM Symposium on Theory of Computing
Pages731-740
Number of pages10
StatePublished - 2008
Externally publishedYes
Event40th Annual ACM Symposium on Theory of Computing, STOC 2008 - Victoria, BC, Canada
Duration: 17 May 200820 May 2008

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference40th Annual ACM Symposium on Theory of Computing, STOC 2008
Country/TerritoryCanada
CityVictoria, BC
Period17/05/0820/05/08

Keywords

  • Theory

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