Abstract

The complexity class of Π p k-Polynomial Local Search (PLS) problems with Π p ℓ-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 ℓ ≤ k, the Σb ℓ+1-definable search problems of T k+1 2 are exactly characterised by Π p k-PLS problems with Πp ℓ-goals. These Π p k-PLS problems can be defined in a weak base theory such as S1 2, 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 Π p k-PLS definitions give rise to a new ∀Σb1(α) principle conjectured to separate Tk 2(α) from T k+1 2 (α). 1

Citations