Abstract
The formalisation of proofs and algorithms in the theory of real closed fields yields a control over the structures handled by computations. At first we study the behaviour of the connectors and give a system which links proofs in linear logic to manipulations of polynomial structures. We extend the proof system with proper axioms which are results obtained from the completeness of the theory, and we show that the system is sound and that it controls when the computation may be done. Then we study the case of sign assignments upon roots of polynomials, and give a way to control cylindrical algebraic decompositions and thus to mix two cell decompositions.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
M. Ben-Or, D. Kozen and J. Reif.: The Complexity of Elementary Algebra and Geometry, J. of Comput. and Syst. Sci. 32 (1986) 251–264
G.E. Collins: “Quantifier elimination for real closed fields by cylindrical algebraic decomposition, proc.2nd GI Conference on Automata Theory and Formal languages”, Lect. Notes in Comput. Sci. 35 (1975) 134–183
J.Y. Girard: Linear Logic, theoretical Computer Sci. (1987) 1–102
D. Grigor'ev: Complexity of deciding Tarski algebra, J. Symbolic Computation 5 (1988) 65–108
J. Heintz, M.F. Roy, P. Solernó: Sur la complexiti du principe de Tarski-Seidenberg, Bull. Soc. Math. France. 118 (1990) 101–126
J. Renegar: On the computational complexity and geometry of the first-order theory of the reals, J. Symbolic Computation 13 (1992) 255–352
A. Tarski: The completeness of elementary algebra and geometry, schedulded to appear in Paris 1940 (1967)
C. Tollu: Planification en Logique Linéaire et Langage de RequÊtes Relationnelles avec Compteurs, thèse, Paris, (1993) 13–30
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jumpertz, P. (1995). Linear logic and real closed fields: A way to handle situations dynamically. In: Calmet, J., Campbell, J.A. (eds) Integrating Symbolic Mathematical Computation and Artificial Intelligence. AISMC 1994. Lecture Notes in Computer Science, vol 958. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60156-2_8
Download citation
DOI: https://doi.org/10.1007/3-540-60156-2_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60156-2
Online ISBN: 978-3-540-49533-8
eBook Packages: Springer Book Archive