Characterising Complexity Classes by Inductive Definitions in Bounded Arithmetic

TL;DR

This paper characterizes complexity classes P and PSPACE using inductive definitions within second order bounded arithmetic, providing a logical framework that captures their computational power.

Contribution

It introduces axioms of inductive definitions in second order bounded arithmetic to distinguish P and PSPACE based on inflationary and non-inflationary inductive principles.

Findings

P is characterized by inflationary inductive definitions

PSPACE is characterized by non-inflationary inductive definitions

Provides a logical framework for complexity class characterization

Abstract

Famous descriptive characterisations of P and PSPACE are restated in terms of the Cook-Nguyen style second order bounded arithmetic. We introduce an axiom of inductive definitions over second order bounded arithmetic. We show that P can be captured by the axiom of inflationary inductive definitions whereas PSPACE can be captured by the axiom of non-inflationary inductive definitions.