We define an extension of tree automata by adding some tests in the rules. The goal is to handle non linearity. We obtain a family which has good closure and decidability properties and we give some applications. IntroductionAutomata, on words or on terms, are an important tool, the decision results they helped to establish are numerous. The main idea of this paper is to add tests in the rules of automata of finite terms, in order to take account of the non-linearity of terms. Let us remind that a term is non-linear if and only if the same variable occurs at least twice in it. All instances of a non-linear term have identical subtrees, obtained by substitution of the same variable. Non linearlty is an important property, appearing in many domains, such as logic programming, rewriting, .... Phenomena generated by non-linearity are complex and there is a gap between the"linear case" and the "non-linear case": many properties satisfied in the first case, are lost in the second one (the homomorphic image of a recognizable set is recognizable when the homomorphism is linear, it is not otherwise; the problem of inductive reducibility is polynomial when the left-hand terms are linear [Jouannaud,Kounalis], exponential otherwise [Plaisted, Kaput] ). The tools already defined to explicitly manipulate comparisons between terms are rare, axiomatization for algebras of trees [Comon, Maher] is the main one.In 1981, M. Dauchet and J. Mongy [MonSt] have defined a class of automata, called Rateg. In Rateg automata, left-hand sides of transitions rules may be non-linear terms with variables at depth more than one. States appear at depth one. For instance a (ql (b(x, x)), q2 (c(y, a(x), y)), q2 (b(x, y)), q3 (Y)) is a correct left-hand side of a Rateg rule. This rule, in fact, imposes equalities of several subterins. This class, called Rateg, is closed under finite intersection and union, but the emptiness is undecidable, as it was proved by J. Mongy. From the order-sorted algebras point of view, bottom up tree automata correspond to a usual signature -states are equivalents to sorts, and transition rules to declarations-.As for Rateg automata, they are close to signatures with "term declarations"; this notion was introduced.by Schmidt-Schauss [Sch88]: the declarations are of the form t:S, which means that the term t is of sort S; the terms in the declarations may be non linear and of any depth. The corresponding tree language class is strictly included in Rateg. Schmid~-Schanss has proved the unification of terms to be undecidable in the general term declaration case and this result can also be obtained as a corollary of Mongy results.Our goal is to extend the classical definition of automata, while keeping both (good) closure properties (especially under boolean operations) and decision properties (emptiness). Intuitively, the power of Rateg come from the ability to overlay equality constraints, and, so, to generate non local tests between subterms (the Post problem can easily be coded with Rateg). In order to avoid this phenom...
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