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

## TL;DR

This paper introduces the Space-Bounded Church-Turing Thesis (SBCT), proposing that limited memory constrains the computational capabilities of physical systems, even in equilibrium, and provides bounds on their long-term predictability.

## Contribution

It generalizes the notion of memory to include the maximal information carried over time, establishing a new fundamental limitation on physical computation and long-term system predictability.

## Key findings

- Memory limits affect computational power of physical systems.
- Explicit bounds on the complexity of long-term behavior of noisy dynamical systems.
- Validation of the Simulation Assertion (SA) for bounded-memory systems.

## Abstract

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.