Language Equations with Symmetric Difference

Issue title: Words, Graphs, Automata, and Languages; Special Issue Honoring the 60th Birthday of Professor Tero Harju

Article type: Research Article

Authors: Okhotin, Alexander

Affiliations: Department of Mathematics, University of Turku, Turku FI-20014, Finland, alexander.okhotin@utu.fi

Note: [] Address for correspondence: Department of Mathematics, University of Turku, Turku FI-20014, Finland

Abstract: The paper investigates the expressive power of language equations with the operations of concatenation and symmetric difference. For equations over every finite alphabet Σ with |Σ| ≥ 1, it is demonstrated that the sets representable by unique solutions of such equations are exactly the recursive sets over Σ, and the sets representable by their least (greatest) solutions are exactly the recursively enumerable sets (their complements, respectively). If |_| ≥ 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 Π01-complete, the existence of a unique, a least or a greatest solution is Π02-complete, while the existence of finitely many solutions is Π03-complete.

DOI: 10.3233/FI-2012-679

Journal: Fundamenta Informaticae, vol. 116, no. 1-4, pp. 205-222, 2012