A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The st-connectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless the paths cross somewhere.

The complexity class of Π p k-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 Σ p i-definable functions of T k+1 2 are characterized in terms of Π p k-PLS problems. 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 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 ∀Σb 1(α)-principle that is conjectured to separate T k 2 (α) and T k+1 2 (α). 1

We discuss the Paris-Wilkie translation from bounded arithmeticproofs to bounded depth propositional proofs in both relativized and non-relativized forms. We describe normal forms for proofs in boundedarithmetic, and a definition of \Sigma 0-depth for PK-proofs that makes the translation from bounded arithmetic to propositional logic particularlytransparent. Using this, we give new proofs of the witnessing theorems for S12and T 12; namely, new proofs that the \Sigma b1-definable functions of S12are polynomial time computable and that the \Sigma b1-definable functions of T 12 are in Polynomial Local Search (PLS). Both proofs generalize to \Sigma