Verifying Mixed Real-Integer Quantifier Elimination

  • Amine Chaieb
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4130)


We present a formally verified quantifier elimination procedure for the first order theory over linear mixed real-integer arithmetics in higher-order logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for linear real arithmetics.


Decision Procedure Theorem Prover Automate Reasoning High Order Logic Output Size 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Appel, A.W., Felty, A.P.: Dependent types ensure partial correctness of theorem provers. J. Funct. Program 14(1), 3–19 (2004)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barendregt, H.: Reflection and its use: from science to meditation (2002)Google Scholar
  3. 3.
    Barendregt, H., Barendsen, E.: Autarkic computations in formal proofs. J. Autom. Reasoning 28(3), 321–336 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barras, B.: Programming and computing in HOL. In: Proceedings of the 13th International Conference on Theorem Proving in Higher Order Logics, pp. 17–37. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  5. 5.
    Berezin, S., Ganesh, V., Dill, D.L.: An online proof-producing decision procedure for mixed-integer linear arithmetic. In: Garavel, H., Hatcliff, J. (eds.) ETAPS 2003 and TACAS 2003. LNCS, vol. 2619, pp. 521–536. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Berghofer, S.: Towards generating proof producing code from HOL definitions. Private communicationGoogle Scholar
  7. 7.
    Berghofer, S., Nipkow, T.: Executing higher order logic. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 24–40. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Bertot, Y., Castéran, P.: Coq’Art: The Calculus of Inductive Constructions. Text in theor. comp. science: an EATCS series, vol. XXV. Springer, Heidelberg (2004)MATHGoogle Scholar
  9. 9.
    Boigelot, B., Jodogne, S., Wolper, P.: An effective decision procedure for linear arithmetic over the integers and reals. ACM Trans. Comput. Log. 6(3), 614–633 (2005)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Chaieb, A., Nipkow, T.: Generic proof synthesis for presburger arithmetic. Technical report, Technische Universität München (2003)Google Scholar
  11. 11.
    Chaieb, A., Nipkow, T.: Verifying and reflecting quantifier elimination for Presburger arithmetic. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Cooper, D.C.: Theorem proving in arithmetic without multiplication. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence, vol. 7, pp. 91–100. Edinburgh University Press (1972)Google Scholar
  13. 13.
    Crégut, P.: Une procédure de décision réflexive pour un fragment de l’arithmétique de Presburger. In: Informal proceedings of the 15th journées francophones des langages applicatifs (2004) (In French)Google Scholar
  14. 14.
    Davis, M.: A computer program for presburger’s algorithm. In: Summaries of talks presented at the Summer Inst. for Symbolic Logic, Cornell University, Inst. for Defense Analyses, Princeton, NJ, pp. 215–233 (1957)Google Scholar
  15. 15.
    Ferrante, J., Rackoff, C.: A decision procedure for the first order theory of real addition with order. SIAM J. Comput. 4(1), 69–76 (1975)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ferrante, J., Rackoff, C.: The Computational Complexity of Logical Theories. Lecture Notes in Mathematics, vol. 718. Springer, Heidelberg (1979)MATHGoogle Scholar
  17. 17.
    Fischer, R.: Super-exponential complexity of presburger arithmetic. In: SIAMAMS: Complexity of Computation: Proc. of a Symp. in Appl. Math. of the AMS and the Society for Industrial and Applied Mathematics (1974)Google Scholar
  18. 18.
    Fourier, J.: Solution d’une question particulière du calcul des inegalités. Nouveau Bulletin des Sciences par la Scociété Philomatique de Paris, pp. 99–100 (1823)Google Scholar
  19. 19.
    Harrison, J.: Metatheory and reflection in theorem proving: A survey and critique. Technical Report CRC-053, SRI Cambridge, Millers Yard, Cambridge, UK (1995),
  20. 20.
    Harrison, J.R.: Introduction to logic and theorem proving (to appear)Google Scholar
  21. 21.
    Howe, D.J.: Computational Metatheory in Nuprl. In: Lusk, E.L., Overbeek, R.A. (eds.) CADE 1988. LNCS, vol. 310, pp. 238–257. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  22. 22.
    Klaedtke, F.: On the automata size for Presburger arithmetic. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS 2004), pp. 110–119. IEEE Computer Society Press, Los Alamitos (2004)CrossRefGoogle Scholar
  23. 23.
    Klapper, R., Stump, A.: Validated Proof-Producing Decision Procedures. In: Tinelli, C., Ranise, S. (eds.) 2nd Int. Workshop Pragmatics of Decision Procedures in Automated Reasoning (2004)Google Scholar
  24. 24.
    Loos, R., Weispfenning, V.: Applying linear quantifier elimination. Comput. J. 36(5), 450–462 (1993)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    McLaughlin, S., Harrison, J.: A proof-producing decision procedure for real arithmetic. . In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 295–314. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  26. 26.
    McLauglin, S.: An Interpretation of Isabelle/HOL in HOL Light. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, Springer, Heidelberg (2006)Google Scholar
  27. 27.
    Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002), MATHCrossRefGoogle Scholar
  28. 28.
    Norrish, M.: Complete integer decision procedures as derived rules in HOL. In: Basin, D., Wolff, B. (eds.) TPHOLs 2003. LNCS, vol. 2758, pp. 71–86. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  29. 29.
    Obua, S., Skalberg, S.: Importing HOL into Isabelle/HOL. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  30. 30.
    Oppen, D.C.: Elementary bounds for presburger arithmetic. In: STOC 1973: Proceedings of the fifth annual ACM symposium on Theory of computing, pp. 34–37. ACM Press, New York (1973)CrossRefGoogle Scholar
  31. 31.
    Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes Rendus du I Congrès des Math. des Pays Slaves, pp. 92–101 (1929)Google Scholar
  32. 32.
    Pugh, W.: The Omega test: a fast and practical integer programming algorithm for dependence analysis. In: Proceedings of the 1991 ACM/IEEE conference on Supercomputing, pp. 4–13. ACM Press, New York (1991)CrossRefGoogle Scholar
  33. 33.
    Reddy, C.R., Loveland, D.W.: Presburger arithmetic with bounded quantifier alternation. In: STOC 1978: Proceedings of the tenth annual ACM symposium on Theory of computing, pp. 320–325. ACM Press, New York (1978)CrossRefGoogle Scholar
  34. 34.
    Skolem, T.: Über einige Satzfunktionen in der Arithmetik. In: Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo, I. Matematisk naturvidenskapelig klasse, volume 7, pp. 1–28. Oslo (1931)Google Scholar
  35. 35.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press (1951)Google Scholar
  36. 36.
    Weispfenning, V.: The complexity of linear problems in fields. J. Symb. Comput. 5(1/2), 3–27 (1988)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Weispfenning, V.: The complexity of almost linear diophantine problems. J. Symb. Comput. 10(5), 395–404 (1990)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Weispfenning, V.: Mixed real-integer linear quantifier elimination. In: ISSAC ’99: Proceedings of the 1999 international symposium on Symbolic and algebraic computation, pp. 129–136. ACM Press, New York (1999)CrossRefGoogle Scholar
  39. 39.
    Wolper, P., Boigelot, B.: An automata-theoretic approach to presburger arithmetic constraints (extended abstract). In: Mycroft, A. (ed.) SAS 1995. LNCS, vol. 983, pp. 21–32. Springer, Heidelberg (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Amine Chaieb
    • 1
  1. 1.Institut für InformatikTechnische Universität München 

Personalised recommendations