Polynomial Local Search in the Polynomial Hierarchy and Witnessing in Fragments of Bounded Arithmetic

The complexity class of Πpk-polynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the Σi -definable functions of T k+1 2 are characterized in terms of Π p k-PLS problems. These Π p kPLS problems can be defined in a weak base theory such as S 2 , and proved to be total in T k+1 2 . Furthermore, the Π p k-PLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new ∀Σ1(α)-principle that is conjectured to separate T k 2 (α) and T k+1 2 (α).