  The Askaryan Radio Array (ARA) is an ultra-high energy (UHE, $>10^{17}$ eV)
neutrino detector designed to observe neutrinos by searching for the radio
waves emitted by the relativistic products of neutrino-nucleon interactions in
Antarctic ice. In this paper, we present constraints on the diffuse flux of
ultra-high energy neutrinos between $10^{16}-10^{21}$ eV resulting from a
search for neutrinos in two complementary analyses, both analyzing four years
of data (2013-2016) from the two deep stations (A2, A3) operating at that time.
We place a 90 % CL upper limit on the diffuse all flavor neutrino flux at
$10^{18}$ eV of $EF(E)=5.6\times10^{-16}$
$\textrm{cm}^{-2}$$\textrm{s}^{-1}$$\textrm{sr}^{-1}$. This analysis includes
four times the exposure of the previous ARA result, and represents
approximately 1/5 the exposure expected from operating ARA until the end of
2022.
