Abstract
We consider quantifier free formulae of a first order theory without functions and with predicates x rewrites to y in one step with given rewrite systems. Variables are interpreted in the set of finite trees. The full theory is undecidable [Tre96] and recent results [STT97], [Mar97], [Vor97] have strengthened the undecidability result to formulae with small prefixes (\( \exists ^ * \forall ^ * \) ) and very restricted classes of rewriting systems (e.g. linear, shallow and convergent in [STTT98]). Decidability of the positive existential fragment has been shown in [NPR97]. We give a decision procedure for positive and negative existential formulae in the case when the rewrite systems are quasi-shallow, that is all variables in the rewrite rules occur at depth one. Our result extends to formulae with equalities and memberships relations of the form x ∈ L where L is a recognizable set of terms.
Partially supported by The Esprit working group CCL II (22457), and “GDR AMI” Groupement De Recherche 1116 du CNRS.
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Caron, AC., Seynhaeve, F., Tison, S., Tommasi, M. (1999). Deciding the Satisfiability of Quantifier Free Formulae on One-Step Rewriting. In: Narendran, P., Rusinowitch, M. (eds) Rewriting Techniques and Applications. RTA 1999. Lecture Notes in Computer Science, vol 1631. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48685-2_9
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