Extension into trees of first order theories

Abstract : We present in this paper an automatic way to combine any first-order theory T with the theory of finite or infinite trees. First of all, we present a new class of theories that we call zero-infinite-decomposable and show that every decomposable theory T accepts a decision procedure in the form of six rewriting rules which for every first order proposition give either true or false in T. We present then the axiomatization T of the extension of T into trees and show that if T is flexible then its extension into trees T is zero-infinite-decomposable and thus complete. The flexible theories are theories having elegant properties which enable us to eliminate quantifiers in particular cases.
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Khalil Djelloul, Thi-Bich-Hanh Dao. Extension into trees of first order theories. The 8th International conference on artificial intelligence and symbolic computation, 2006, China. pp. LNAI, Vol 4120. P 53-67. ⟨hal-00202317⟩

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