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Toda's theorem in bounded arithmetic with parity quantifiers and bounded depth proof systems with parity gates

Presented by: 
L Kolodziejczyk Uniwersytet Warszawski
Date: 
Thursday 29th March 2012 - 11:00 to 11:30
Venue: 
INI Seminar Room 1
Abstract: 
The "first part" of Toda's theorem states that every language in the polynomial hierarchy is probabilistically reducible to a language in $\oplus P$. The result also holds for the closure of the polynomial hierarchy under a parity quantifier. We use Jerabek's framework for approximate counting to show that this part of Toda's theorem is provable in a relatively weak fragment of bounded arithmetic with a parity quantifier. We discuss the significance of the relativized version of this result for bounded depth propositional proof systems with parity gates. Joint work with Sam Buss and Konrad Zdanowski.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons