Computational Flows in Arithmetic

Abstract

A computational flow is a pair consisting of a sequence of computational problems of a certain sort and a sequence of computational reductions among them. In this paper we will develop a theory for these computational flows and we will use it to make a sound and complete interpretation for bounded theories of arithmetic. This property helps us to decompose a first order arithmetical proof to a sequence of computational reductions by which we can extract the computational content of low complexity statements in some bounded theories of arithmetic such as $I\Delta_0$, $T^k_n$, $I\Delta_0+EXP$ and $PRA$. In the last section, by generalizing term-length flows to ordinal-length flows, we will extend our investigation from bounded theories to strong unbounded ones such as $I\Sigma_n$ and $PA+TI(\alpha)$ and we will capture their total $NP$ search problems as a consequence.