Strict language inequalities and their decision problems

Abstract

Systems of language equations of the form {φ(X1, . . . , Xn) = ∅, ψ(X1, . . . , Xn) ≠ ∅} are studied, where φ, ψ may contain set-theoretic operations and concatenation; they can be equivalently represented as strict inequalities ξ(X1, . . . , Xn) ⊂ L0. It is proved that the problem whether such an inequality has a solution is Σ2-complete, the problem whether it has a unique solution is in (Σ3 ∩ ∏3)\ (Σ2 ∪ ∏2), the existence of a regular solution is a Σ1-complete problem, while testing whether there are finitely many solutions is Σ3-complete. The class of languages representable by their unique solutions is exactly the class of recursive sets, though a decision procedure cannot be algorithmically constructed out of an inequality, even if a proof of solution uniqueness is attached. © Springer-Verlag Berlin Heidelberg 2005.