Preprint:

Arnold Beckmann and Samuel R. Buss.
"Characterizing Definable Search Problems in Bounded Arithmetic via Proof Notations"
In Ways of Proof Theory, Ontos Series in Mathematical Logic, pp. 65-134, 2010.

Abstract:  The complexity class of $\pip k$-Polynomial Local Search (PLS) problems with $\pip\ell$-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 $\sib{\ell+1}$-definable search problems of $\rT^{k+1}_2$ are exactly characterised by $\pip k$-PLS problems with $\pip\ell$-goals. These $\pip k$-PLS problems can be defined in a weak base theory such as $\rS^1_2$, and proved to be total in $\rT^{k+1}_2$. Furthermore, the $\pip k$-PLS definitions can be Skolemised with simple polynomial time functions. The Skolemised $\pip k$-PLS definitions give rise to a new $\forall\sib1(\al)$ principle conjectured to separate $\rT^{k}_2(\al)$ from $\rT^{k+1}_2(\al)$.