  Motivated by a concrete problem and with the goal of understanding the sense
in which the complexity of streaming algorithms is related to the complexity of
formal languages, we investigate the problem Dyck(s) of checking matching
parentheses, with $s$ different types of parenthesis.
  We present a one-pass randomized streaming algorithm for Dyck(2) with space
$\Order(\sqrt{n}\log n)$, time per letter $\polylog (n)$, and one-sided error.
We prove that this one-pass algorithm is optimal, up to a $\polylog n$ factor,
even when two-sided error is allowed. For the lower bound, we prove a direct
sum result on hard instances by following the "information cost" approach, but
with a few twists. Indeed, we play a subtle game between public and private
coins. This mixture between public and private coins results from a balancing
act between the direct sum result and a combinatorial lower bound for the base
case.
  Surprisingly, the space requirement shrinks drastically if we have access to
the input stream in reverse. We present a two-pass randomized streaming
algorithm for Dyck(2) with space $\Order((\log n)^2)$, time $\polylog (n)$ and
one-sided error, where the second pass is in the reverse direction. Both
algorithms can be extended to Dyck(s) since this problem is reducible to
Dyck(2) for a suitable notion of reduction in the streaming model.
