Decidability of Regularity and Related Properties of Ground Normal Form Languages

Abstract

We study language-theoretical properties of the set of reducible ground terms and its complement - the set of ground normal forms induced by a given rewriting system. As a tool for our analysis we introduce the property of finite irreducibility of a term with respect to a variable and prove it to be decidable. It turns out that this property generalizes numerous interesting properties of the language of ground normal forms. In particular, we show that testing regularity of this language can be reduced to verifying this property. In this way we prove the decidability of the regularity of the set of ground normal forms, the problem mentioned in the list of open problems in rewriting by Dershowitz, Jouannaud, and Klop ( in "Proceedings, 4th Conference on Rewriting Techniques and Applications" ( R. V. Book, Ed.), Springer-Verlag, Berlin/New York, 1991). Also, the decidability of the existence of an equivalent ground term rewriting system and some other results are proved.