  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.
