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.

This document is originally a working paper recording our results in progress. It is hoped that with some re-organization, addition of basic definitions, introduction and conclusion, and of course

Abstract. We describe a natural generalization of ordinary computation to a third-order setting and give a function calculus with nice properties and recursion-theoretic characterizations of several large complexity classes. We then present a number of third-order theories of bounded arithmetic whose definable functions are the classes of the EXP-time hierarchy in the third-order setting.

Abstract. This paper is concerned with the complexity of computing winning strategies for poset games. While it is reasonably clear that such strategies can be computed in PSPACE, we give a simple proof of this fact by a reduction to the game of geography. We also show how to formalize the reasoning about poset games in Skelley’s theory W 1 1 for PSPACE reasoning. We conclude that W 1 1 can use the “strategy stealing argument ” to prove that in poset games with a supremum the first player always has a winning strategy.