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

Space-bounded Church-Turing thesis and computational tractability of closed systems