Decision problems for systems of language equations and inequations

Abstract

Systems of language equations $\phi (X_1,\dots ,X_n)=\psi (X_1,\dots ,X_n)$ and inequations $\phi (X_1,\dots ,X_n)\ne \psi (X_1,\dots ,X_n)$ are studied, where φ and ψ may contain Boolean operations and concatenation. It is proved that the problem whether such a system has a solution is ${\Sigma }_2^0$-complete in the arithmetical hierarchy (cf. the earlier studied case of equations only, where it is co-r.e.-complete), the problem whether it has a unique solution is in ${\Sigma }_3^0\cap {\Pi }_3^0$, and is both ${\Sigma }_2^0$-hard and ${\Pi }_2^0$-hard, existence of a finite or regular solution is an r.e.-complete problem, while testing whether a system has finitely many solutions is ${\Sigma }_3^0$-complete. Furthermore, it is shown that the class of languages representable by unique solutions of such systems is exactly the class of recursive sets, but decision procedures for the set cannot be algorithmically constructed out of a system. All results hold already for equations over a unary alphabet.