Closure of Hedge-Automata Languages by Hedge Rewriting

Abstract

Abstract. We consider rewriting systems for unranked ordered terms, i.e. trees where the number of successors of a node is not determined by its label, and is not a priori bounded. The rewriting systems are defined such that variables in the rewrite rules can be substituted by hedges (sequences of terms) instead of just terms. Consequently, this notion of rewriting subsumes both standard term rewriting and word rewriting. We investigate some preservation properties for two classes of languages of unranked ordered terms under this generalization of term rewriting. The considered classes include languages of hedge automata (HA) and some extension (called CF-HA) with context-free languages in transitions, instead of regular languages. In particular, we show that the set of unranked terms reachable from a given HA language, using a so called inverse context-free rewrite system, is a HA language. The proof, based on a HA completion procedure, reuses and combines known techniques with non-trivial adaptations. Moreover, we prove, with different techniques, that the closure of CF-HA languages with respect to restricted context-free rewrite systems, the symmetric case of the above rewrite systems, is a CF-HA language. As a consequence, the problems of ground reachability and regular hedge model checking are decidable in both cases. We give several counter examples showing that we cannot relax the restrictions.

One-variable context-free hedge automata

“…The following result directly follows from a proof in [16] that the closure of a HA language under rewriting with a monadic HRS is a HA language.…”

“…The following theorem is a direct consequence of Theorem 1 in [16] on the rewrite closure of HA languages under monadic HRS.…”

“…This model, called CF 1 HA, is shown decidable and more expressive than previous ones. It permits us to capture significant classes of rewrite systems with computable closures, such as inverse-monadic rules, that are more general than the rules considered in [16]. Moreover, the representation of transitions as rewrite rules enables simpler constructions than in the case of HA [16,15].…”

Regular expression order-sorted unification and matching

“…In order to understand the correspondence, we recall the notion of finite bottom-up unranked tree automaton (a.k.a. hedge automaton, see, e.g., Comon et al (2007) , Jacquemard and Rusinowitch (2008) ). This is a tuple where Q is a finite set of states (nonterminals), F is a finite unranked alphabet (terminals), δ is a finite set of rules of the form or where , R is a regular expression over Q and , and q are from Q , and (final states) is a subset of Q .…”

Rewrite rules for search database systems

“…Essentially, a rule has the form E ⇒ F , where E and F are S-hedges that contain variables (in addition to terms and concepts). In terms of standard rewriting systems [25], our rules can be classified into the category of hedge rewriting [11], context sensitive [20] (actually, the full context is captured in each of the two sides of a rule), linear [25] (i.e., each variable has at most one occurrence in each of the two sides of a rule), and conditional [13] (i.e., the involved hedges need to conform to the schema). Examples are the rules Ei ⇒ Fi (for i = 1, 2, 3) in Figure 1(b).…”