Abstract Partial Cylindrical Algebraic Decomposition I: The Lifting Phase

  • Grant Olney Passmore
  • Paul B. Jackson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)


Though decidable, the theory of real closed fields (RCF) is fundamentally infeasible. This is unfortunate, as automatic proof methods for nonlinear real arithmetic are crucially needed in both formalised mathematics and the verification of real-world cyber-physical systems. Consequently, many researchers have proposed fast, sound but incomplete RCF proof procedures which are useful in various practical applications. We show how such practically useful, sound but incomplete RCF proof methods may be systematically utilised in the context of a complete RCF proof method without sacrificing its completeness. In particular, we present an extension of the RCF quantifier elimination method Partial CAD (P-CAD) which uses incomplete ∃ RCF proof procedures to “short-circuit” expensive computations during the lifting phase of P-CAD. We present the theoretical framework and preliminary experiments arising from an implementation in our RCF proof tool RAHD.


Sample Point Satisfying Assignment Expensive Computation Proof Procedure Real Closed Field 
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.
    Avigad, J., Friedman, H.: Combining Decision Procedures for the Reals. In: Logical Methods in Computer Science (2006)Google Scholar
  2. 2.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, USA (2006)zbMATHGoogle Scholar
  3. 3.
    Collins, G.E., Hong, H.: Partial Cylindrical Algebraic Decomposition for Quantifier Elimination. J. Sym. Comp. 12(3), 299–328 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Daumas, M., Lester, D., Muñoz, C.: Verified Real Number Calculations: A Library for Interval Arithmetic. IEEE Trans. Comp. 58(2), 226–237 (2009)CrossRefGoogle Scholar
  5. 5.
    Davenport, J.H., Heintz, J.: Real Quantifier Elimination is Doubly Exponential. J. Symb. Comput. 5, 29–35 (1988), MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gao, S., Ganai, M., Ivancic, F., Gupta, A., Sankaranarayanan, S., Clarke, E.: Integrating ICP and LRA solvers for deciding nonlinear real arithmetic problems. In: FMCAD 2010, pp. 81–89 (2010)Google Scholar
  7. 7.
    Harrison, J.: Verifying Nonlinear Real Formulas via Sums of Squares. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 102–118. Springer, Heidelberg (2007), CrossRefGoogle Scholar
  8. 8.
    Hong, H.: Comparison of Several Decision Algorithms for the Existential Theory of the Reals. Tech. rep., RISC (1991)Google Scholar
  9. 9.
    Passmore, G.O.: Combined Decision Procedures for Nonlinear Arithmetics, Real and Complex. Ph.D. thesis, University of Edinburgh (2011)Google Scholar
  10. 10.
    Passmore, G.O., Jackson, P.B.: Combined Decision Techniques for the Existential Theory of the Reals. In: Carette, J., Dixon, L., Coen, C.S., Watt, S.M. (eds.) MKM 2009, Held as Part of CICM 2009. LNCS (LNAI), vol. 5625, pp. 122–137. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Pfender, F., Ziegler, G.M.: Kissing Numbers, Sphere Packings, and Some Unexpected Proofs. Notices of the A.M.S. 51, 873–883 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Platzer, A., Quesel, J.-D., Rümmer, P.: Real World Verification. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 485–501. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry. RAND Corporation (1948)Google Scholar
  14. 14.
    Tiwari, A.: An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 248–262. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Grant Olney Passmore
    • 1
    • 2
  • Paul B. Jackson
    • 2
  1. 1.Clare Hall, University of CambridgeCambridgeUnited Kingdom
  2. 2.Laboratory for Foundations of Computer Science, School of InformaticsUniversity of EdinburghEdinburghScotland

Personalised recommendations