Bottom-up rewriting for words and terms
Résumé
For the whole class of linear term rewriting systems, we define \emph{bottom-up rewriting} which is a restriction of the usual notion of rewriting. We show that bottom-up rewriting effectively inverse-preserves recognizability.\\ The \emph{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 $\SBU(k), k \in \N$ of \emph{Strongly Bottom-Up$(k)$} systems for which we show that membership is decidable. We define the class of \emph{Strongly Bottom-Up} systems by $\SBU = \bigcup_{k \in \N} \SBU(k)$. We give a polynomial 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.