Undecidability of theories of semirings with fixed points

In this work we prove the undecidability (and ${\mathrm{\Sigma }}_1^0$-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.