A space-time tradeoff for implementing a function with master equation dynamics

TL;DR

This paper demonstrates how any function over visible states can be implemented using master equation dynamics by utilizing hidden states and timesteps, revealing a tradeoff between complexity and implementation cost.

Contribution

It introduces a space-time tradeoff framework for implementing functions with master equations using hidden states and steps, expanding the understanding of physical system dynamics.

Findings

Any function can be implemented with master equations using hidden states.

Decomposition of master equations into hidden timesteps is possible.

A tradeoff exists between the number of hidden states and timesteps.

Abstract

Master equations are commonly used to model the dynamics of physical systems, including systems that implement single-valued functions like a computer's update step. However, many such functions cannot be implemented by any master equation, even approximately, which raises the question of how they can occur in the real world. Here we show how any function over some "visible" states can be implemented with master equation dynamics--if the dynamics exploits additional, "hidden" states at intermediate times. We also show that any master equation implementing a function can be decomposed into a sequence of "hidden" timesteps, demarcated by changes in what state-to-state transitions have nonzero probability. In many real-world situations there is a cost both for more hidden states and for more hidden timesteps. Accordingly, we derive a "space-time" tradeoff between the number of hidden states and the number of hidden timesteps needed to implement any given function.