  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.
