Abstract

This dissertation concerns theories of bounded arithmetic and propositional proof systems associated with PSPACE and classes from the exponential-time hierarchy.
The second-order viewpoint of Zambella and Cook associates second-order theories of bounded arithmetic with various complexity classes by studying the definable functions of strings, rather than numbers. This approach simplifies presentation of the theories and their propositional translations, and furthermore is applicable to complexity classes that previously had no corresponding theories. We adapt this viewpoint to large complexity classes from the exponential-time hierarchy by adding a third sort, intended to represent exponentially long strings (``superstrings''), and capable of coding, for example, the computation of an exponential-time Turing machine. Specifically, our main theories $W^i_1$ and $TW^i_1$ are associated with PSPACE$^{\Sigma^p_{i-1}}$ and EXP$^{\Sigma^p_{i-1}}$, respectively. We also develop a model for computation in this third-order setting including a function calculus, and define third-order analogues of ordinary complexity classes. We then obtain recursion-theoretic characterizations of our function classes for FP, FPSPACE and FEXP. We use our characterization of FPSPACE as the basis for an open theory for PSPACE that is a conservative extension of a weak PSPACE theory $HW0_1$.
Next we present strong propositional proof systems \QBP[i], which are based on the Boolean program proof system \BPLK but additionally with quantifiers on function symbols. We exhibit a translation of theorems of $W^i_1$ into polynomial-sized proofs in \QBP[i].

Table of Contents

1.    Introduction
2.    Preliminaries and Related Work
3.    The Third-Order Viewpoint
4.    Third-Order Theories
5.    Definability in the Theories
6.    A Universal Conservative Extension of $HW0_1$
7.    Witnessing Theorems
8.    Propositional Translations
9.    Future Work