Language Equations with Symmetric Difference

: Okhotin A

Publisher: IOS PRESS

: 2012

: Fundamenta Informaticae

: FUNDAMENTA INFORMATICAE

: FUND INFORM

: 41730

: 116

: 41730

: 205

: 222

: 18

: 0169-2968

DOI: https://doi.org/10.3233/FI-2012-679

The paper investigates the expressive power of language equations with the operations of concatenation and symmetric difference. For equations over every finite alphabet Sigma with vertical bar Sigma vertical bar >= 1, it is demonstrated that the sets representable by unique solutions of such equations are exactly the recursive sets over Sigma, and the sets representable by their least (greatest) solutions are exactly the recursively enumerable sets (their complements, respectively). If vertical bar Sigma vertical bar >= 2, the same characterization holds already for equations using symmetric difference and linear concatenation with regular constants. In both cases, the solution existence problem is Pi(0)(1)-complete, the existence of a unique, a least or a greatest solution is Pi(0)(2)-complete, while the existence of finitely many solutions is Sigma(0)(3)-complete.