Some Undecidability Results Concerning the Property of Preserving Regularity

A finite string-rewriting system R preserves regularity if and only if it preserves Σ-regularity, where Σ is the alphabet containing exactly those letters that have occurrences in the rules of R. This proves a conjecture of Gyenizse and Vagvolgyi (1997). In addition, some undecidability results are presented that generalize results of Gilleron and Tison (1995) from term-rewriting systems to string-rewriting systems. It follows that the property of being regularity preserving is undecidable for term-rewriting systems, thus answering another question of Gyenizse and Vagvolgyi (1997). Finally, it is shown that it is undecidable in general whether a finite, lengthreducing, and confluent string-rewriting system yields a regular set of normal forms for each regular language.

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