Journal of Automated Reasoning

, Volume 38, Issue 4, pp 353–373 | Cite as

A Decision Procedure for Linear “Big O” Equations

  • Jeremy Avigad
  • Kevin Donnelly


Let F be the set of functions from an infinite set, S, to an ordered ring, R. For f, g, and h in F, the assertion f = g + O(h) means that for some constant C, |f(x) − g(x)| ≤C |h(x)| for every x in S. Let L be the first-order language with variables ranging over such functions, symbols for 0, +, −, min , max , and absolute value, and a ternary relation f = g + O(h). We show that the set of quantifier-free formulas in this language that are valid in the intended class of interpretations is decidable and does not depend on the underlying set, S, or the ordered ring, R. If R is a subfield of the real numbers, we can add a constant 1 function, as well as multiplication by constants from any computable subfield. We obtain further decidability results for certain situations in which one adds symbols denoting the elements of a fixed sequence of functions of strictly increasing rates of growth.


Decision procedures Asymptotic equations Big O 


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  1. 1.
    Apt, K.: Principles of Constraint Programming. Cambridge University Press, Cambridge (2003)Google Scholar
  2. 2.
    Avigad, J., Donnelly, K.: Formalizing O notation in Isabelle/HOL. In: Basin, D., Rusinowitch, M. (eds.) Automated Reasoning: Second International Joint Conference, IJCAR 2004, pp. 357–371. Springer, Berlin Heidelberg New York (2004)Google Scholar
  3. 3.
    Avigad, J., Donnelly, K., Gray, D., Raff, P.: A formally verified proof of the prime number theorem (to appear in ACM Transactions on Computational Logic)Google Scholar
  4. 4.
    Avigad, J., Friedman, H.: Combining decision procedures for the reals. Logical Methods in Computer Science 2(4:4), 1–42 (2006)MathSciNetGoogle Scholar
  5. 5.
    Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development: Coq’art: The Calculus of Inductive Constructions. Springer, Berlin Heidelberg New York (2004)MATHGoogle Scholar
  6. 6.
    Gordon, M.J.C., Melham, T.F.: Introduction to HOL: A Theorem Proving Environment for Higher-order Logic. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  7. 7.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Addison-Wesley, Reading, MA (1994)MATHGoogle Scholar
  8. 8.
    Hall Jr., M.: Combinatorial Theory, 2nd ed. Wiley, New York (1986)MATHGoogle Scholar
  9. 9.
    Karmarkar, N.: A new polynomial time algorithm for linear programming. Combinatorica 4, 373–395 (1984)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kaufmann, M., Manolios, P., Moore, J.S.: Computer-aided Reasoning: An Approach. Kluwer, Boston (2000)Google Scholar
  11. 11.
    Loos, R., Weispfenning, V.: Applying linear quantifier elimination. The Computer Journal 36, 450–461 (1993)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-order Logic. Springer, Berlin Heidelberg New York (2002)MATHGoogle Scholar
  13. 13.
    Owre, S., Rushby, J.M., Shankar, N.: PVS: A prototype verification system. In: Proceedings of the 11th International Conference on Automated Ceduction (CADE), pp. 748–752. Springer, Berlin Heidelberg New York (1992)Google Scholar
  14. 14.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover, Mineola, NY (1998) (Corrected reprint of the 1982 original, Prentice-Hall, New Jersey)MATHGoogle Scholar
  15. 15.
    Weispfenning, V.: The complexity of linear problems in fields. J. Symb. Comput. 5, 3–27 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Weispfenning, V.: Parametric linear and quadratic optimization by elimination. Technical report MIP-9404, Universität Passau (1994)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of PhilosophyCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Computer ScienceBoston UniversityBostonUSA

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