Equilibrium and Termination

TL;DR

This paper demonstrates that determining whether a continuous-time Markov chain (CTMC) has an equilibrium is undecidable by reducing the problem from the Post correspondence problem, linking computability and stochastic processes.

Contribution

It introduces a reduction from the termination problem of Turing machines to the equilibrium problem in CTMCs, establishing undecidability in this context.

Findings

Decidability of equilibrium in CTMCs is impossible in general.

The reduction uses the Post correspondence problem as a basis.

Dissipativity of computable CTMCs is undecidable.

Abstract

We present a reduction of the termination problem for a Turing machine (in the simplified form of the Post correspondence problem) to the problem of determining whether a continuous-time Markov chain presented as a set of Kappa graph-rewriting rules has an equilibrium. It follows that the problem of whether a computable CTMC is dissipative (ie does not have an equilibrium) is undecidable.