# First- and Second-Order Models of Recursive Arithmetics

## TL;DR

This paper investigates interrelated subsystems of arithmetic, exploring their structure, relationships, and computational complexity, and introduces new recursive arithmetic theories to characterize complexity classes.

## Contribution

It introduces and analyzes first- and second-order recursive arithmetic theories capable of characterizing various computational complexity classes.

## Key findings

- Identifies relationships among multiple subsystems of arithmetic.
- Introduces new recursive arithmetic theories ARA_1 and ARA_2.
- Characterizes computational complexity classes using these theories.

## Abstract

We study a quadruple of interrelated subexponential subsystems of arithmetic WKL$_0^-$, RCA$^-_0$, I$\Delta_0$, and $\Delta$RA$_1$, which complement the similarly related quadruple WKL$_0$, RCA$_0$, I$\Sigma_1$, and PRA studied by Simpson, and the quadruple WKL$_0^\ast$, RCA$_0^\ast$, I$\Delta_0$(exp), and EFA studied by Simpson and Smith. We then explore the space of subexponential arithmetic theories between I$\Delta_0$ and I$\Delta_0$(exp). We introduce and study first- and second-order theories of recursive arithmetic $A$RA$_1$ and $A$RA$_2$ capable of characterizing various computational complexity classes and based on function algebras $A$, studied by Clote and others.