  A sliding-window algorithm of window size $t$ is an algorithm whose current
operation depends solely on the last $t$ symbols read. We construct
pseudorandom generators (PRGs) for low-space randomized sliding-window
algorithms that have access to a binary randomness source. More specifically,
we lift these algorithms to the non-uniform setting of branching programs and
study them as a subclass thereof that we call sliding-window branching programs
(SWBPs), accordingly. For general SWBPs, given a base PRG $G_\mathrm{base}$
with seed length $d_\mathrm{base}$ that $\varepsilon_\mathrm{base}$-fools
width-$w$, length-$t$ (general) branching programs, we give two PRG
constructions for fooling any same-width SWBP of length $n$ and window size $t$
(where we assume $w \ge n$). The first uses an additional $d_\mathrm{base} +
O(\log(n/t) \log(1/\varepsilon_\mathrm{base}))$ random bits, whereas the second
has a seed length of $O((d_\mathrm{base} + \log\log(n/t) +
\log(1/\varepsilon_\mathrm{base})) \log(d_\mathrm{base} +
\log(1/\varepsilon_\mathrm{base})))$. Both PRGs incur only a $(n/2t)^{O(1)}$
multiplicative loss in the error parameter.
  As an application, we show how to decide the language of a sublinear-time
probabilistic cellular automaton using small space. More specifically, these
results target the model of PACAs, which are probabilistic cellular automata
that accept if and only if all cells are simultaneously accepting. For
(sublinear) $T(n) = \Omega(\log n)^{1.01}$, we prove that every language
accepted by a $T$-time one-sided error PACA (the PACA equivalent of
$\mathsf{RP}$) can be decided using only $O(T)$ space. Meanwhile, forgoing the
previous requirement on $T$, we show the same holds for $T$-time two-sided
error PACA (the PACA equivalent of $\mathsf{BPP}$) if we use $\tilde{O}(T) +
O(\log n)$ space instead (where the $\tilde{O}$ notation hides only
$\mathsf{polylog}(T)$ factors).
