Abstract
A set constraint is of the form exp 1 \(\subseteq\) exp 2 where exp 1 and exp 2 are set expressions constructed using variables, function symbols, and the set union, intersection and complement symbols. An algorithm for solving such systems of set constraints was proposed by Aiken and Wimmers [1]. We present a new algorithm for solving this problem. Indeed, we define a new class of tree automata called Tree Set Automata. We prove that, given a system of set constraints, we can associate a tree set automaton such that the set of tuples of tree languages recognized by this automaton is the set of tuples of solutions of the system. We also prove the converse property. Furthermore, if the system has a solution, we prove, in a constructive way, that there is a regular solution (i.e. a tuple of regular tree languages) and a minimal solution and a maximal solution which are actually regular.
This research was partially supported by “GDR Mathématiques et Informatique” and ESPRIT Basic Research Action 6317 ASMICS2.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
A. Aiken and E.L. Wimmers. Solving System of Set Constraints. In 7th Symposium on LICS, pages 329–340, 1992.
L. Bachmair, H. Ganzinger, and U. Waldmann. Solving Set Constraints by Ordered Resolution with Simplification. Draft manuscript, September 92.
T. Früwirth, E.Shapiro, M.Y. Vardi, and E.Yardeni. Logic Programs as Types for Logic Programs. In 6th Symposium on LICS, 1991.
F. Gecseg and M. Steinby. Tree Automata. Akademiai Kiado, 1984.
R. Gilleron, S. Tison, and M. Tommasi. Solving System of Set Constraints using Tree Automata. Technical Report IT-92-235, Laboratoire d'Informatique Fondamentale de Lille, Université des Sciences et Technologies de Lille, Villeneuve d'Ascq, France, July 1992.
N. Heintze and J. Jaffar. A Decision Procedure for a Class of Set Constraints. In 5th Symposium on LICS, 1990.
N. Heintze and J. Jaffar. A Finite Presentation Theorem for Approximating Logic Programs. IBM technical report RC 16089 (#71415), IBM, August 1990.
N.D. Jones and S.S. Muchnick. Flow Analysis and Optimization of LISP-like Structures. In Proceedings 6 th ACM Symposium on Principles of Programming Languages, pages 244–246, 1979.
P. Mishra. Towards a Theory of Types in PROLOG. In Proceedings 1st IEEE Symposium on Logic Programming, pages 456–461, Atlantic City, 1984. 10.
M.O. Rabin. Decidability of Second-Order Theories and Automata on Infinite Trees. Trans. Amer. Math. Soc., 141:1–35, 1969.
J.C. Reynolds. Automatic Computation of Data Set Definition. Information Processing, 68:456–461, 1969.
W. Thomas. Handbook of Theoretical Computer Science, volume B, chapter Automata on Infinite Objects, pages 134–191. Elsevier, 1990.
T. E. Uribe. Sorted Unification Using Set Constraints. In D. Kapur, editor, Lecture Notes in Computer Science, New York, 1992. 11th International Conference on Automated Deduction.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gilleron, R., Tison, S., Tommasi, M. (1993). Solving systems of set constraints using tree automata. In: Enjalbert, P., Finkel, A., Wagner, K.W. (eds) STACS 93. STACS 1993. Lecture Notes in Computer Science, vol 665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56503-5_50
Download citation
DOI: https://doi.org/10.1007/3-540-56503-5_50
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56503-1
Online ISBN: 978-3-540-47574-3
eBook Packages: Springer Book Archive