A1 Refereed original research article in a scientific journal
Language Equations with Symmetric Difference
Authors: Okhotin A
Publisher: IOS PRESS
Publication year: 2012
Journal: Fundamenta Informaticae
Journal name in source: FUNDAMENTA INFORMATICAE
Journal acronym: FUND INFORM
Number in series: 41730
Volume: 116
Issue: 41730
First page : 205
Last page: 222
Number of pages: 18
ISSN: 0169-2968
DOI: https://doi.org/10.3233/FI-2012-679(external)
Abstract
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.
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.
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