S. Tulipani. Decidability of the existential theory of infinite terms with subterm relation. Journal on Information and Computation, 103(2), 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The existential fragment of the first order theory of subtree constraints ss over finite trees is decidable and NP complete. fragment of this theory is undecidable. A proof has been given by Venkataraman in [28] Note that Theorem 2 carries over to the subtree relation on infinite trees [27, 25]. In this case, explicit equality constraints s=s have to be provided, since the equivalence XX X X X=X does not hold over infinite trees in contrast to finite trees. Definition 3. An equality up to constraint is uniform if all its conjuncts are of the form s 1 =s=s 2 =s. Lemma 4. ....

S. Tulipani. Decidability of the existential theory of infinite terms with subterm relation. Journal on Information and Computation, 103(2), 1993.

....formula is a first order formula over the predicate symbols = and 2 s. Note that there is in general infinitely many such predicate symbols) 1.7. 3 Introduction of Inequalities Also interesting is the introduction of the symbol (inequations) Formulas involving such symbols are studied in [Ven87, LM89, Com90c, JO91, Tul91] and many others. differs from = in that, given an F algebra, it has several distinct possible interpretations, as we shall discuss in the next section. 1.7.4 Arbitrary first order formulas The case of formulas involving arbitrary predicate symbols is investigated in [Com88b, Lug89] However, ....

....real closed fields, and so on require specialized techniques and are out of the scope of this paper. Even when the domain under consideration is the algebra of finite trees, many interpretations of inequality can be considered. For example, inequality is interpreted as the subterm relation in [Ven87, Tul91]. It is interpreted as the matching quasi ordering in [KK89] and as a recursive path ordering 4 in [Com90a, JO91] Each interpretation is motivated by some kind of application. 4 The recursive path ordering is defined in e.g. Der87] To each total ordering on F is associated a total ....

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Sauro Tulipani. Decidability of the existential theory of infinite terms with subterm relation. Unpublished Draft, October 1991.

....[26] that the Sigma 2 fragment of the theory of subterm ordering is undecidable, which sets up a quite precise boundary between decidability and undecidability in this case. Subterm ordering is also studied in the case of infinite trees: again the existential fragment of the theory is decidable [25] and the Sigma 2 fragment is undecidable [24] Let us finally consider yet another ordering on trees: the encompassment ordering. We say that s encompasses t (noted s t) if some instance of t is a subterm of s. For example, s = g(f(f(a; b) f(a; b) encompasses t = f(x; x) since instantiating ....

S. Tulipani. Decidability of the existential theory of infinite terms with subterm relation. To appear in Information and Computation, 1994.

....in [Mah88, CL89] it turns out that T (F ) the Herbrand Universe, is completely axiomatizable. Finite trees over an infinite alphabet are also completely axiomatizable, as well as rational trees [Mah88] Extensions to other structures have been considered: extension by adding an inequality symbol [Ven87, Com90b, Tre90, Tul91], extension with membership constraints [Com90a] What we consider here is a simple structure: the Herbrand Universe without any predicate symbol other than equality. However, we do not assume that the model is freely generated: equality is assumed to be generated by a finite set of equations E. ....

Sauro Tulipani. Decidability of the existential theory of infinite terms with subterm relation. To appear in Information and Computation, 1991.

....[26] that the Sigma 2 fragment of the theory of subterm ordering is undecidable, which sets up a quite precise boundary between decidability and undecidability in this case. Subterm ordering is also studied in the case of infinite trees: again the existential fragment of the theory is decidable [25] and the Sigma 2 fragment is undecidable [24] Let us finally consider yet another ordering on trees: the encompassment ordering. We say that s encompasses t (noted s t) if some instance of t is a subterm of s. For example, s = g(f(f(a; b) f(a; b) encompasses t = f(x; x) since instantiating ....

S. Tulipani. Decidability of the existential theory of infinite terms with subterm relation. To appear in Information and Computation, 1994.

.... satisfiability of ordering constraints has been studied for some orderings on terms: Venkataraman showed that the existential fragment of the theory of the subterm ordering is decidable, while the Sigma 3 fragment is undecidable [10] These results have been extended recently to infinite trees [9]. Comon showed that the existential fragment of the theory of any total lexicographic path ordering is decidable [1] This result has been extended to any recursive path ordering over a total precedence by Jouannaud and Okada [6] On the other hand, the Sigma 4 fragment of the theory of any ....

....not solve here the decidability problem in its full generality, but we hope to contribute to the general solution: we show that the positive existential fragment of the theory of tree embedding is decidable. The proof is carried out by elementary techniques which are quite different from those in [10, 9, 1, 6]. Indeed, for subterm problems, 10, 9] use some test sets showing that, if there is a solution, there is some solution which has a small size. They also use normal forms of inequations systems in which all inequations s t have a variable on the left. As we will see, it is not possible to ....

[Article contains additional citation context not shown here]

Sauro Tulipani. Decidability of the existential theory of infinite terms with subterm relation. To appear in Information and Computation, 1991.

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