Abstract
We investigate the use of Büchi's techniques for Presburger arithmetic. More precisely, we show how to efficiently compute an automaton which accepts the set of solutions of a linear Diophantine equation (suitably encoded). Following Büchi, this gives a decision technique for the whole Presburger arithmetic. We show however how to compute more efficiently the automaton in the case of disequalities, inequalities and systems of linear Diophantine problems. We also show that such an “automaton algorithm” has a nearly optimal worst case complexity, both for the existential fragment and for the whole first-order theory.
This research was supported in part by the HCM Network SOL.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
H. Abdulrab and M. Maksimenko. General solutions of systems of linear diophantine equations and inequations. In J. Hsiang, editor, 6th International Conference on Rewriting Techniques and Applications, volume 914 of Lecture Notes in Computer Science, Kaiserslautern, Germany, Apr. 1995. Springer-Verlag.
J. Büchi. On a decision method in restricted second-order arithmetic. In E. e. a. Nagel, editor, Proc. Int. Congr. on logic, methodology and philosophy of science, Standford, 1960. Stanford Univ. Press.
E. Contejean and H. Devie. An efficient algorithm for solving systems of diophantine equations. Information and Computation, 113(1):143–172, Aug. 1994.
E. Domenjoud and A. P. Tomás. From elliotmac mahon to an algorithm for general linear constraints on naturals. In U. Montanari, editor, Principles and Practice of Constraint Programming, Cassis, France, Sept. 1995. To appear.
Elliot. On linear homogeneous diophantine equations. Quartely J. of Pure and Apllied Maths, 136, 1903.
J. Ferrante and C. W. Rackoff. The computational complexity of logical theories. Number 718 in Lecture Notes in Mathematics. Springer-Verlag, 1979.
A. Fortenbacher. An algebraic approach to unification under associativity and commutativity. In Proc. Rewriting Techniques and Applications 85, Dijon, LNCS 202. Springer-Verlag, May 1985.
P. MacMahon. Combinatory Analysis, volume 2, chapter II: A Syzygetic Theory, pages 111–114. Reprinted by Chelsea in 1960, 1919.
A. R. Meyer. Weak monadic second-order theory of successor is not elementary recursive. Manuscript, 1973.
M. Presburger. über die VollstÄndingen einer gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In Comptes Rendus du premier Congrès des Mathématiciens des Pays slaves, Warszawa, 1929.
L. Stockmeyer and A. Meyer. Word problems requiring exponential time. In Proc. 5th ACM Symp. on Theory of Computing, pages 1–9, 1973.
W. Thomas. Automata on infinite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, pages 134–191. Elsevier, 1990.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Boudet, A., Comon, H. (1996). Diophantine equations, Presburger arithmetic and finite automata. In: Kirchner, H. (eds) Trees in Algebra and Programming — CAAP '96. CAAP 1996. Lecture Notes in Computer Science, vol 1059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61064-2_27
Download citation
DOI: https://doi.org/10.1007/3-540-61064-2_27
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61064-9
Online ISBN: 978-3-540-49944-2
eBook Packages: Springer Book Archive