Characterising Definable Search Problems in Bounded Arithmetic via Proof Notations

The complexity class of Πpk-Polynomial Local Search (PLS) problems with Πp` -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 ` ≤ k, the Σ`+1-definable search problems of T 2 are exactly characterised by Π p k-PLS problems with Π p ` goals. These Πpk-PLS problems can be defined in a weak base theory such as S2, and proved to be total in T k+1 2 . Furthermore, the Π p k-PLS definitions can be Skolemised with simple polynomial time functions. The Skolemised Πpk-PLS definitions give rise to a new ∀Σ1(α) principle conjectured to separate T2(α) from T k+1 2 (α).