Article

A formal quantifier elimination for algebraically closed fields

Authors:

Assia MahboubiAuthors Info & Claims

AISC'10/MKM'10/Calculemus'10: Proceedings of the 10th ASIC and 9th MKM international conference, and 17th Calculemus conference on Intelligent computer mathematics

Pages 189 - 203

Published: 05 July 2010 Publication History

Abstract

We prove formally that the first order theory of algebraically closed fields enjoys quantifier elimination, and hence is decidable. This proof is organized in two modular parts. We first reify the first order theory of rings and prove that quantifier elimination leads to decidability. Then we implement an algorithm which constructs a quantifier free formula from any first order formula in the theory of ring. If the underlying ring is in fact an algebraically closed field, we prove that the two formulas have the same semantic. The algorithm producing the quantifier free formula is programmed in continuation passing style, which leads to both a concise program and an elegant proof of semantic correctness.

References

[1]

Barras, B.: Sets in coq, coq in sets. In: Proceedings of the 1st Coq Workshop. Technical University of Munich Research Report (2009)

[2]

Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics. Springer, New York (2006)

Digital Library

[3]

Berardi, S., Valentini, S.: Krivine's intuitionistic proof of classical completeness for countable languages. Ann. Pure Appl. Logic 129(1-3), 93-106 (2004)

[4]

Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development, Coq'Art: the Calculus of Inductive Constructions. Springer, Heidelberg (2004)

Digital Library

[5]

Bertot, Y., Gonthier, G., Biha, S.O., Pasca, I.: Canonical big operators. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 86-101. Springer, Heidelberg (2008)

Digital Library

[6]

Buchberger, B.: Groebner Bases: Applications. In: Mikhalev, A.V., Pilz, G.F. (eds.) The Concise Handbook of Algebra, pp. 265-268. Kluwer Academic Publishers, Dordrecht (2002)

[7]

Chevalley, C., Cartan, H.: Schémas normaux; morphismes; ensembles constructibles. In: Séminaire Henri Cartan, Numdam, vol. 8, pp. 1-10 (1955-1956), http://www.numdam.org/item?id=SHC_1955-1956__8__A7_0

[8]

Garillot, F., Gonthier, G., Mahboubi, A., Rideau, L.: Packaging mathematical structures. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 327-342. Springer, Heidelberg (2009)

Digital Library

[9]

Geuvers, H., Pollack, R., Wiedijk, F., Zwanenburg, J.: A constructive algebraic hierarchy in Coq. Journal of Symbolic Computation 34(4), 271-286 (2002)

Digital Library

[10]

Harrison, J.: Complex quantifier elimination in HOL. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001: Supplemental Proceedings, Division of Informatics, University of Edinburgh, pp. 159-174 (2001); Published as Informatics Report Series EDI-INF-RR-0046, http://www.informatics.ed.ac.uk/publications/report/0046.html

[11]

Hodges, W.: A shorter model theory. Cambridge University Press, Cambridge (1997)

Digital Library

[12]

Gödel, K.: Über die Vollständigkeit des Logikkalküls. PhD thesis, University of Vienna, Austria (1929)

[13]

Nipkow, T.: Linear quantifier elimination. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 18-33. Springer, Heidelberg (2008)

Digital Library

[14]

O'Connor, R.: Incompleteness & Completeness, Formalizing Logic and Analysis in Type Theory. PhD thesis, Radboud University Nijmegen, Netherlands (2009)

[15]

Pottier, L.: Connecting gröbner bases programs with coq to do proofs in algebra, geometry and arithmetics. In: LPAR Workshops. CEUR Workshop Proceedings, vol. 418. CEUR-WS.org (2008)

[16]

Ridge, T., Margetson, J.: A mechanically verified, sound and complete theorem prover for first order logic. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 294-309. Springer, Heidelberg (2005)

Digital Library

[17]

Robinson, J.: Definability and decision problems in arithmetic. Journal of Symbolic Logic 14, 98-114 (1949)

[18]

Simpson, C.T.: Formalized proof, computation, and the construction problem in algebraic geometry (2004)

[19]

Tarski, A.: A decision method for elementary algebra and geometry. In: RAND Corp., Santa Monica, CA (1948) (manuscript); Republished as A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)

Cited By

Gonthier GAsperti AAvigad JBertot YCohen CGarillot FLe Roux SMahboubi AO'Connor RBiha SPasca IRideau LSolovyev ATassi EThéry L(2013)A machine-checked proof of the odd order theoremProceedings of the 4th international conference on Interactive Theorem Proving10.1007/978-3-642-39634-2_14(163-179)Online publication date: 22-Jul-2013
https://dl.acm.org/doi/10.1007/978-3-642-39634-2_14
Kaliszyk CIda T(2011)Proof assistant decision procedures for formalizing origamiProceedings of the 18th Calculemus and 10th international conference on Intelligent computer mathematics10.5555/2032713.2032718(45-57)Online publication date: 18-Jul-2011
https://dl.acm.org/doi/10.5555/2032713.2032718

A formal quantifier elimination for algebraically closed fields
1. Mathematics of computing
  1. Continuous mathematics
    1. Calculus
2. Theory of computation
  1. Models of computation

Recommendations

Quantifier Elimination over Algebraically Closed Fields in a Proof Assistant using a Computer Algebra System

We propose a decision procedure for algebraically closed fields based on a quantifier elimination method. The procedure is intended to build proofs for systems of polynomial equations and inequations. We describe how this procedure can be carried out in ...
Second Order Quantifier Elimination: Foundations, Computational Aspects and Applications
Improved Second-Order Quantifier Elimination in Modal Logic
JELIA '08: Proceedings of the 11th European conference on Logics in Artificial Intelligence

This paper introduces improvements for second-order quantifier elimination methods based on Ackermann's Lemma and investigates their application in modal correspondence theory. In particular, we define refined calculi and procedures for solving the ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

AISC'10/MKM'10/Calculemus'10: Proceedings of the 10th ASIC and 9th MKM international conference, and 17th Calculemus conference on Intelligent computer mathematics

July 2010

470 pages

ISBN:3642141277

Editors:
Serge Autexier
DFKI Bremen, Germany
,
Jacques Calmet
Karlsruhe Institute of Technology, Germany
,
David Delahaye
Conservatoire National des Arts et Métiers, Paris, France
,
Patrick D. F. Ion
AMS, Ann Arbor
,
Laurence Rideau
INRIA Sophia-Antipolis, France
,
Renaud Rioboo
ENSIIE, Evry, France
,
Alan P. Sexton
University of Birmingham, UK

Sponsors

ENSIEE: école Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise
CNAM: Conservatoire des Arts et Métiers
Texas Instruments
SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation
INRIA: Institut Natl de Recherche en Info et en Automatique

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 05 July 2010

Check for updates

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 01 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

Gonthier GAsperti AAvigad JBertot YCohen CGarillot FLe Roux SMahboubi AO'Connor RBiha SPasca IRideau LSolovyev ATassi EThéry L(2013)A machine-checked proof of the odd order theoremProceedings of the 4th international conference on Interactive Theorem Proving10.1007/978-3-642-39634-2_14(163-179)Online publication date: 22-Jul-2013
https://dl.acm.org/doi/10.1007/978-3-642-39634-2_14
Kaliszyk CIda T(2011)Proof assistant decision procedures for formalizing origamiProceedings of the 18th Calculemus and 10th international conference on Intelligent computer mathematics10.5555/2032713.2032718(45-57)Online publication date: 18-Jul-2011
https://dl.acm.org/doi/10.5555/2032713.2032718

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

Media

Figures

Other

Tables

View Table of Contents