E-unification by means of tree tuple synchronized grammars

Abstract

The goat of this paper is both to give a E-unification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructor-based rewrite system, and that four additional restrictions are satisfied. We give a procedure that represents the (possibly infinite) set of solutions thanks to a new kind of grammar, called tree tuple synchronized gi.~mmar, and that can decide unifiability thanks to an emptiness test. Moreover we show that if only three of the four additional restrictions are satisfied then unifiability is undecidable.

Tree Tuple Languages from the Logic Programming Point of View

“…Formalisms extending tree grammars were introduced in [18,23]. There, tree tuple languages are generated from a tuple axiom by applying sets of productions simultaneously.…”

Unification in extensions of shallow equational theories

“…The decidability results are also obtained by termination analyses of saturation under basic superposition. Limet & R ety (1997) show the decidability of E-uni cation in theories represented by a particular class of con uent, constructor-based TRS. The set of possibly in nite solutions is represented by T ree Tuple Synchronized Grammars.…”

“…Tree automata and grammars have b e e n u s e d b y Kaji et al (1997) and Limet & R ety (1997) to solve w ord and uni ability problems. In the rst paper as well as in the papers by Comon (1995) and Jacquemard (1996) the recognizability of the closure of some recognizable set L with respect to term rewriting systems of restricted classes is investigated.…”

“…Limet & R ety (1997) use Tree Tuple Synchronized Grammars to generate solutions to uni cation problems by a s i m ulation of narrowing. In (Kaji, Toru & Kasami 1997) it is shown that the closure with respect to some kind of term rewriting system of the (recognizable set) of ground instances of a linear term is recognizable.…”

Synchronized Tree Languages for Reachability in Non-right-linear Term Rewrite Systems

“…This technique can handle the reachability problem of [5]. These synchronized tree languages [10,8] are recognized using CS-programs [11], i.e. a particular class of Horn clauses.…”