Linear generalized semi-monadic rewrite systems effectively preserve recognizability

Abstract

We introduce the notion of the generalized semi-monadic rewrite system, which is a generalization of well-known rewrite systems: the ground rewrite system, the monadic rewrite system, and the semi-monadic rewrite system. We show that linear generalized semi-monadic rewrite systems effectively preserve recognizability. We show that a tree language L is recognizable if and only if there exists a rewrite system R such that R∪ R −1 is a linear generalized semi-monadic rewrite system and that L is the union of finitely many ↔ R ★-classes. We show several decidability and undecidability results on rewrite systems effectively preserving recognizability and on generalized semi-monadic rewrite systems. For example, we show that for a rewrite system R effectively preserving recognizability, it is decidable if R is locally confluent. Moreover, we show that preserving recognizability and effectively preserving recognizability are modular properties of linear collapse-free rewrite systems. Finally, as a consequence of our results on trees we get that restricted right-left overlapping string rewrite systems effectively preserve recognizability.