Bottom-up rewriting for words and terms. (March 2015) - Durand, I.; Sénizergues, G.
Abstract: For the whole class of linear term rewriting systems, we define bottom-up rewriting which is a restriction of the usual notion of rewriting. We show that bottom-up rewriting effectively inverse-preserves recognizability. The Bottom-Up class ( BU ) is, by definition, the set of linear systems for which every derivation can be replaced by a bottom-up derivation. Since membership to BU turns out to be undecidable, we are led to define more restricted classes: the classes SBU ( k ), k ∈ N, of Strongly Bottom-Up ( k ) systems for which we show that membership is decidable. We define the class of Strongly Bottom-Up systems by SBU = ⋃ k ∈ N SBU ( k ) . We give a polynomial-time sufficient condition for a system to be in SBU . The class SBU contains (strictly) several classes of systems which were already known to inverse preserve recognizability: the inverse left-basic semi-Thue systems (viewed as unary term rewriting systems), the linear growing term rewriting systems, the inverse Linear-Finite-Path-Ordering systems.