Undecidability of theories of semirings with fixed points

TL;DR

This paper proves that many theories of semirings with fixed points are undecidable, using recursion theoretic methods, and applies to various theories like Conway and Park μ-semirings, impacting their computational understanding.

Contribution

It establishes the undecidability and $ ext{Σ}_1^0$-completeness of several theories of semirings with fixed points using effective inseparability techniques.

Findings

Many theories of semirings with fixed points are undecidable.

The results apply to Conway μ-semirings, Park μ-semirings, and Chomsky algebras.

The methods used are based on recursion theory and effective inseparability.

Abstract

In this work we prove the undecidability (and $\Sigma^0_1$-completeness) of several theories of semirings with fixed points. The generality of our results stems from recursion theoretic methods, namely the technique of effective inseperability. Our result applies to many theories proposed in the literature, including Conway $\mu$-semirings, Park $\mu$-semirings, and Chomsky algebras.