A simple undecidable problem: Existential agreement of inverses of two morphisms on a regular language

Abstract

We show that it is undecidable whether or not for a given finite language F and for morphisms h and g the relation $\text{h}{}^{\mathrm{-1}}\text{(x) ⋓ (g}{}^{\mathrm{-1}}\text{(x) ≠ θ}$ holds for all x in $\text{F∗ ⋒ (}\text{dom}\text{(h}{}^{\mathrm{-1}}\text{) u}\text{dom}\text{(g}{}^{\mathrm{-1}}\text{)}$, i.e., whether or not h⁻¹ and g⁻¹ existentially agree on $\text{F∗}$.