Abstract:
We present a new framework for the study of search problems and their definability in bounded arithmetic. We identify two notions of complexity of search problems: verification complexity and computational complexity. Notions of exact solvability and exact reducibility are developed, and exact $\Sigma^b_i$-definability of search problems in bounded arithmetic is introduced. We specify a new machine model called the oblivious witness-oracle Turing machines.

Based on work of Buss and Krajicek, we present a type-2 search problem ITERATION (ITER) that characterizes the class PLS and the exactly $\Sigma^b_1$-definable search problems of the theory T$^1_2$. We show that the type-2 problems of Beame et. al. are not Turing reducible to ITER. The separations of the corresponding type-2 classes and the unprovability of certain combinatorial principles in a relativized version of T$^1_2$ are obtained as corollaries.

We also present the first characterization of the exactly $\Sigma^b_2$-definable search problems of S$^1_2$ and T$^1_2$.