Abstract
Two infinite structures (sets together with operations and relations) hold our attention here: the trees together with operations of construction and the rational numbers together with the operations of addition and substraction and a linear dense order relation without endpoints. The object of this paper is the study of the evaluated trees, a structure mixing the two preceding ones.
First of all, we establish a general theorem which gives a sufficient condition for the completeness of a first-order theory. This theorem uses a special quantifier, primarily asserting the existence of an infinity of individuals having a given first order property. The proof of the theorem is nothing other than the broad outline of a general algorithm which decides if a proposition or its negation is true in certain theories.
We introduce then the theory T E of the evaluated trees and show its completeness using our theorem. From our proof it is possible to extract a general algorithm for solving quantified constraints in T E .
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge University Press, Cambridge (1998) ISBN 0-521-45520-0
Benhamou, F., Bouvier, P., Colmerauer, A., Garetta, H., Giletta, B., Massat, J., Narboni, G., N’dong, S., Pasero, R., Pique, J., Van Touraivane, c.M.: Vetillard E. Le manuel de Prolog IV, PrologIA, Marseille, France (1996)
Chang, C., Keisler, H.: Model theory. Section 1.4 Theories and examples of theories. Elsevier, Amsterdam (1988), (fifth impression)
Colmerauer, A.: An introduction to Prolog III. Communication of the ACM 33(7), 68–90 (1990)
Colmerauer, A.: Equations and inequations on finite and infinite trees. In: Proceeding of the International conference on the fifth generation of computer systems, Tokyo, pp. 85–99 (1984)
Colmerauer, A., Dao, T.: Expressiveness of full first order constraints in the algebra of finite and infinite trees. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 172–186. Springer, Heidelberg (2000)
Courcelle, B.: Equivalences and Transformations of Regular Systems applications to Program Schemes and Grammars. Theretical Computer Science 42, 1–122 (1986)
Courcelle, B.: Fundamental Properties of Infinite Trees. Theoretical Computer Science 25(2), 95–169 (1983)
Dao, T.: Resolution de contraintes du premier ordre dans la theorie des arbres finis ou infinis. These d’informatique, Universite de la mediterranee (Decembre 2000)
Huet G.: Resolution d’equations dans les langages d’ordre 1, 2,...ω. These d’Etat, Universite Paris 7. France (1976)
Maher, M.: Complete axiomatization of the algebra of finite, rational and infinite trees. Technical report, IBM - T.J.Watson Research Center (1988)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Djelloul, K. (2005). About the Combination of Trees and Rational Numbers in a Complete First-Order Theory. In: Gramlich, B. (eds) Frontiers of Combining Systems. FroCoS 2005. Lecture Notes in Computer Science(), vol 3717. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11559306_6
Download citation
DOI: https://doi.org/10.1007/11559306_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29051-3
Online ISBN: 978-3-540-31730-2
eBook Packages: Computer ScienceComputer Science (R0)