Abstract Partial Cylindrical Algebraic Decomposition I: The Lifting Phase
Abstract
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.
Keywords
Sample Point Satisfying Assignment Expensive Computation Proof Procedure Real Closed FieldPreview
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References
- 1.Avigad, J., Friedman, H.: Combining Decision Procedures for the Reals. In: Logical Methods in Computer Science (2006)Google Scholar
- 2.Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, USA (2006)MATHGoogle Scholar
- 3.Collins, G.E., Hong, H.: Partial Cylindrical Algebraic Decomposition for Quantifier Elimination. J. Sym. Comp. 12(3), 299–328 (1991)MathSciNetMATHCrossRefGoogle Scholar
- 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.Davenport, J.H., Heintz, J.: Real Quantifier Elimination is Doubly Exponential. J. Symb. Comput. 5, 29–35 (1988), http://dx.doi.org/10.1016/S0747-71718880004-X MathSciNetMATHCrossRefGoogle Scholar
- 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.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), http://portal.acm.org/citation.cfm?id=1792233.1792242 CrossRefGoogle Scholar
- 8.Hong, H.: Comparison of Several Decision Algorithms for the Existential Theory of the Reals. Tech. rep., RISC (1991)Google Scholar
- 9.Passmore, G.O.: Combined Decision Procedures for Nonlinear Arithmetics, Real and Complex. Ph.D. thesis, University of Edinburgh (2011)Google Scholar
- 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.Pfender, F., Ziegler, G.M.: Kissing Numbers, Sphere Packings, and Some Unexpected Proofs. Notices of the A.M.S. 51, 873–883 (2004)MathSciNetMATHGoogle Scholar
- 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.Tarski, A.: A Decision Method for Elementary Algebra and Geometry. RAND Corporation (1948)Google Scholar
- 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