Abstract
Recent applications of decision procedures for nonlinear real arithmetic (the theory of real closed fields, or RCF) have presented a need for reasoning not only with polynomials but also with transcendental constants and infinitesimals. In full generality, the algebraic setting for this reasoning consists of real closed transcendental and infinitesimal extensions of the rational numbers. We present a library for computing over these extensions. This library contains many contributions, including a novel combination of Thom’s Lemma and interval arithmetic for representing roots, and provides all core machinery required for building RCF decision procedures. We describe the abstract algebraic setting for computing with such field extensions, present our concrete algorithms and optimizations, and illustrate the library on a collection of examples.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Akbarpour, B., Paulson, L.C.: Applications of MetiTarski in the Verification of Control and Hybrid Systems. In: Majumdar, R., Tabuada, P. (eds.) HSCC 2009. LNCS, vol. 5469, pp. 1–15. Springer, Heidelberg (2009)
Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical Algebraic Decomposition. I. The Basic Algorithm. SIAM J. Comp. 13(4), 865–877 (1984)
Basu, S., Pollack, R., Roy, M.: Algorithms in Real Algebraic Geometry. Springer, Secaucus (2006)
Ben-Or, M., Kozen, D., Reif, J.: The complexity of elementary algebra and geometry. In: STOC. ACM (1984)
Canny, J.: Some algebraic and geometric computations in PSPACE. In: Twentieth ACM Symposium on Theory of Computing, STOC. ACM (1988)
Coste, M., Roy, M.: Thom’s lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets. JSC 5(1-2) (1988)
de Moura, L., Passmore, G.O.: Exact nonlinear optimization on demand. In: Preparation (2013)
Denman, W., Akbarpour, B., Tahar, S., Zaki, M.H., Paulson, L.C.: Formal verification of analog designs using MetiTarski. In: FMCAD, pp. 93–100 (2009)
Fleuriot, J.D., Paulson, L.C.: A Combination of Nonstandard Analysis and Geometry Theorem Proving, with Application to Newton’s Principia. In: Kirchner, C., Kirchner, H. (eds.) CADE 1998. LNCS (LNAI), vol. 1421, pp. 3–16. Springer, Heidelberg (1998)
Fränzle, M., Herde, C., Ratschan, S., Schubert, T., Teige, T.: Efficient Solving of Large Non-linear Arithmetic Constraint Systems with Complex Boolean Structure. JSAT 1, 209–236 (2007)
Gamboa, R., Kaufmann, M.: Nonstandard Analysis in ACL2. JAR 27(4) (2001)
Gao, S., Avigad, J., Clarke, E.M.: δ-complete decision procedures for satisfiability over the reals. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 286–300. Springer, Heidelberg (2012)
Granvilliers, L., Benhamou, F.: RealPaver: An Interval Solver using Constraint Satisfaction Techniques. ACM Trans. on Maths. Software 32, 138–156 (2006)
Grigor’ev, D.Y., Vorobjov Jr., N.N.: Solving systems of polynomial inequalities in subexponential time. JSC 5(1-2), 37–64 (1988)
Hales, T.C., Harrison, J., McLaughlin, S., Nipkow, T., Obua, S., Zumkeller, R.: A revision of the proof of the Kepler conjecture. Discrete & Computational Geometry 44(1), 1–34 (2010)
Jovanović, D., de Moura, L.: Solving non-linear arithmetic. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 339–354. Springer, Heidelberg (2012)
Ligatsikas, Z., Rioboo, R., Roy, M.: Generic computation of the real closure of an ordered field. Maths. and Comp. in Sim. 42(4-6), 541–549 (1996)
Mishra, B., Pedersen, P.: Arithmetic with real algebraic numbers is in NC. In: ISSAC 1990, pp. 120–126. ACM, New York (1990)
Platzer, A., Clarke, E.M.: Formal verification of curved flight collision avoidance maneuvers: A case study. In: Cavalcanti, A., Dams, D.R. (eds.) FM 2009. LNCS, vol. 5850, pp. 547–562. Springer, Heidelberg (2009)
Ratschan, S.: Efficient Solving of Quantified Inequality Constraints over the Real Numbers. ACM Trans. on Comp. Logic 7(4), 723–748 (2006)
Rioboo, R.: Infinitesimals and real closure. Technical report, Laboratoire D’Informatique de Paris 6 (2001)
Strzebonski, A.: Computing in the field of complex algebraic numbers. JSC 24(6) (1997)
Strzeboński, A., Tsigaridas, E.P.: Univariate real root isolation in multiple extension fields. In: ISSAC 2012, pp. 343–350. ACM, New York (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
de Moura, L., Passmore, G.O. (2013). Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals. In: Bonacina, M.P. (eds) Automated Deduction – CADE-24. CADE 2013. Lecture Notes in Computer Science(), vol 7898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38574-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-38574-2_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38573-5
Online ISBN: 978-3-642-38574-2
eBook Packages: Computer ScienceComputer Science (R0)