  We report a new limitation on the ability of physical systems to perform
computation -- one that is based on generalizing the notion of memory, or
storage space, available to the system to perform the computation. Roughly, we
define memory as the maximal amount of information that the evolving system can
carry from one instant to the next. We show that memory is a limiting factor in
computation even in lieu of any time limitations on the evolving system - such
as when considering its equilibrium regime. We call this limitation the
Space-Bounded Church Turing Thesis (SBCT). The SBCT is supported by a
Simulation Assertion (SA), which states that predicting the long-term behavior
of bounded-memory systems is computationally tractable. In particular, one
corollary of SA is an explicit bound on the computational hardness of the
long-term behavior of a discrete-time finite-dimensional dynamical system that
is affected by noise. We prove such a bound explicitly.
