A Quantifier-Free String Theory for ALOGTIME Reasoning

A Quantifier-Free String Theory for ALOGTIME Reasoning François Pitt Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2000 The main contribution of this work is the definition of a quantifier-free string theory T1 suitable for formalizing ALOGTIME reasoning. After describing L1—a new, simple, algebraic characterization of the complexity class ALOGTIME based on strings instead of numbers—the theory T1 is defined (based on L1), and a detailed formal development of T1 is given. Then, theorems of T1 are shown to translate into families of propositional tautologies that have uniform polysize Frege proofs, T1 is shown to prove the soundness of a particular Frege system F , and F is shown to provably p-simulate any proof system whose soundness can be proved in T1. Finally, T1 is compared with other theories for ALOGTIME reasoning in the literature. To our knowledge, this is the first formal theory for ALOGTIME reasoning whose basic objects are strings instead of numbers, and the first quantifier-free theory formalizing ALOGTIME reasoning in which a direct proof of the soundness of some Frege system has been given (in the case of first-order theories, such a proof was first given by Arai for his theory AID). Also, the polysize Frege proofs we give for the propositional translations of theorems of T1 are considerably simpler than those for other theories, and so is our proof of the soundness of a particular F-system in T1. Together with the simplicity of T1’s recursion schemes, axioms, and rules these facts suggest that T1 is one of the most natural theories available for ALOGTIME reasoning.