We present in a uniform manner simple two-sorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.

Let H be a proof system for the quantified propositional calculus (QPC). We j -witnessing problem for H to be: given a prenex S j -formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S witnessing problems for the systems G 1 and G 1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce

In this thesis we are concerned with building logical foundations for Linear Algebra, from the perspective of proof complexity. As the cornerstone of our logical theories, we use Berkowitz's parallel algorithm for computing the coecients of the characteristic polynomial of a matrix. STandard

Abstract. There are many ways to define complexity in logic. In finite model theory, it is the complexity of describing properties, whereas in proof complexity it is the complexity of proving properties in a proof system. Here we consider several notions of complexity in logic, the connections among them, and their relationship with computational complexity. In particular, we show how the complexity of logics in the setting of finite model theory is used to obtain results in bounded arithmetic, stating which functions are provably total in certain weak systems of arithmetic. For example, the transitive closure function (testing reachability between two given points in a directed graph) is definable using only NL-concepts (where NL is non-deterministic log-space complexity class), and its totality is provable within NL-reasoning. 1