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Characterising definable search problems in bounded arithmetic via proof notations

Arnold Beckmann , Samuel R Buss

Ways of Proof Theory, Volume: 2, Pages: 65 - 134

Swansea University Author:

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DOI (Published version): 10.1515/9783110324907.65

Abstract

The complexity class of $\Pi^p_k$-Polynomial Local Search (PLS) problems with $\Pi^p_\ell$-goal is introduced, and is used to give new characterisations of definable search problems in fragments of Bounded Arithmetic. The characterisations are established via notations for propositional proofs obtai...

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Published in: Ways of Proof Theory Berlin De Gruyter 2010 http://www.degruyter.com/view/books/9783110324907/9783110324907.65/9783110324907.65.xml https://cronfa.swan.ac.uk/Record/cronfa8149 No Tags, Be the first to tag this record!
Abstract: The complexity class of $\Pi^p_k$-Polynomial Local Search (PLS) problems with $\Pi^p_\ell$-goal is introduced, and is used to give new characterisations of definable search problems in fragments of Bounded Arithmetic. The characterisations are established via notations for propositional proofs obtained by translating Bounded Arithmetic proofs using the Paris-Wilkie-translation. For $\ell\le k$, the $\Sigma^b_{\ell+1}$-definable search problems of $T^{k+1}_2$ are exactly characterised by $\Pi^p_k$-PLS problems with $\Pi^p_\ell$-goals. These $\Pi_p_k$-PLS problems can be defined in a weak base theory such as $S^1_2$, and proved to be total in $T^{k+1}_2$. Furthermore, the $\Pi^p_k$-PLS definitions can be Skolemised with simple polynomial time functions. The Skolemised $\Pi^p_k$-PLS definitions give rise to a new $\forall\Sigma^b_1(\alpha)$ principle conjectured to separate $\T^k_2(\alpha)$ from $T^{k+1}_2(\alpha)$. College of Science 65 134