Abstract
We introduce the notion of “δ-complete decision procedures” for solving SMT problems over the real numbers, with the aim of handling a wide range of nonlinear functions including transcendental functions and solutions of Lipschitz-continuous ODEs. Given an SMT problem ϕ and a positive rational number δ, a δ-complete decision procedure determines either that ϕ is unsatisfiable, or that the “δ-weakening” of ϕ is satisfiable. Here, the δ-weakening of ϕ is a variant of ϕ that allows δ-bounded numerical perturbations on ϕ. We establish the existence and complexity of δ-complete decision procedures for bounded SMT over reals with functions mentioned above. We propose to use δ-completeness as an ideal requirement for numerically-driven decision procedures. As a concrete example, we formally analyze the DPLL〈ICP〉 framework, which integrates Interval Constraint Propagation in DPLL(T), and establish necessary and sufficient conditions for its δ-completeness. We discuss practical applications of δ-complete decision procedures for correctness-critical applications including formal verification and theorem proving.
This research was sponsored by the National Science Foundation grants no. DMS1068829, no. CNS0926181, and no. CNS0931985, the GSRC under contract no. 1041377 (Princeton University), the Semiconductor Research Corporation under contract no. 2005TJ1366, General Motors under contract no. GMCMUCRLNV301, and the Office of Naval Research under award no. N000141010188.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Akbarpour, B., Paulson, L.C.: Metitarski: An automatic theorem prover for real-valued special functions. J. Autom. Reasoning 44(3), 175–205 (2010)
Avigad, J., Friedman, H.: Combining decision procedures for the reals. Logical Methods in Computer Science, 2(4) (2006)
Benhamou, F., Granvilliers, L.: Continuous and Interval Constraints. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, ch. 16. Elsevier (2006)
Borralleras, C., Lucas, S., Navarro-Marset, R., Rodríguez-Carbonell, E., Rubio, A.: Solving Non-linear Polynomial Arithmetic via SAT Modulo Linear Arithmetic. In: Schmidt, R.A. (ed.) CADE 2009. LNCS, vol. 5663, pp. 294–305. Springer, Heidelberg (2009)
Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms, pp. 425–491. Springer, New York (2008)
Brown, C.W., Davenport, J.H.: The complexity of quantifier elimination and cylindrical algebraic decomposition. In: ISSAC 2007 (2007)
Clarke, E.M., Biere, A., Raimi, R., Zhu, Y.: Bounded model checking using satisfiability solving. Formal Methods in System Design 19(1), 7–34 (2001)
Clarke, E.M., Grumberg, O., Peled, D.: Model checking. MIT Press (2001)
Collins, G.E.: Hauptvortrag: Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)
Dutertre, B., de Moura, L.: A Fast Linear-Arithmetic Solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006)
Eggers, A., Fränzle, M., Herde, C.: SAT Modulo ODE: A Direct SAT Approach to Hybrid Systems. In: Cha, S(S.), Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 171–185. Springer, Heidelberg (2008)
Eggers, A., Ramdani, N., Nedialkov, N., Fränzle, M.: Improving SAT Modulo ODE for Hybrid Systems Analysis by Combining Different Enclosure Methods. In: Barthe, G., Pardo, A., Schneider, G. (eds.) SEFM 2011. LNCS, vol. 7041, pp. 172–187. Springer, Heidelberg (2011)
Fränzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. JSAT 1(3-4), 209–236 (2007)
Ganai, M.K., Ivančić, F.: Efficient decision procedure for non-linear arithmetic constraints using cordic. In: Formal Methods in Computer Aided Design, FMCAD (2009)
Gao, S., Avigad, J., Clarke, E.: δ-Decision procedures for satisfiability over the reals. Extended version, http://arxiv.org/abs/1204.3513
Gao, S., Avigad, J., Clarke, E.: δ-Decidability over the reals. In: Logic in Computer Science, LICS (2012)
Gao, S., Ganai, M., Ivancic, F., Gupta, A., Sankaranarayanan, S., Clarke, E.: Integrating ICP and LRA solvers for deciding nonlinear real arithmetic. In: FMCAD (2010)
Goldsztejn, A., Mullier, O., Eveillard, D., Hosobe, H.: Including Ordinary Differential Equations Based Constraints in the Standard CP Framework. In: Cohen, D. (ed.) CP 2010. LNCS, vol. 6308, pp. 221–235. Springer, Heidelberg (2010)
Hales, T.C.: Introduction to the flyspeck project. In: Mathematics, Algorithms, Proofs (2005)
Hentenryck, P.V., McAllester, D., Kapur, D.: Solving polynomial systems using a branch and prune approach. SIAM Journal on Numerical Analysis 34(2), 797–827 (1997)
Ishii, D., Ueda, K., Hosobe, H.: An interval-based sat modulo ode solver for model checking nonlinear hybrid systems. STTT 13(5), 449–461 (2011)
Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. In: IEEE Conference on Computational Complexity, pp. 149–160. IEEE Computer Society (2009)
Ko, K.-I.: Complexity Theory of Real Functions. Birkhäuser (1991)
Ko, K.-I.: On the computational complexity of integral equations. Ann. Pure Appl. Logic 58(3), 201–228 (1992)
Kroening, D., Strichman, O.: Decision Procedures: An Algorithmic Point of View. Springer (2008)
Munoz, C., Narkawicz, A.: Formalization of an efficient representation of Bernstein polynomials and applications to global optimization, http://shemesh.larc.nasa.gov/people/cam/Bernstein/
Nedialkov, N.S., Jackson, K.R., Corliss, G.F.: Validated solutions of initial value problems for ordinary differential equations. Applied Mathematics and Computation 105(1), 21–68 (1999)
Nuzzo, P., Puggelli, A., Seshia, S.A., Sangiovanni-Vincentelli, A.L.: Calcs: Smt solving for non-linear convex constraints. In: Bloem, R., Sharygina, N. (eds.) FMCAD, pp. 71–79. IEEE (2010)
Platzer, A., Clarke, E.M.: The Image Computation Problem in Hybrid Systems Model Checking. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 473–486. Springer, Heidelberg (2007)
Ratschan, S.: Quantified constraints under perturbation. J. Symb. Comput. 33(4), 493–505 (2002)
Weihrauch, K.: Computable Analysis: An Introduction (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gao, S., Avigad, J., Clarke, E.M. (2012). δ-Complete Decision Procedures for Satisfiability over the Reals. In: Gramlich, B., Miller, D., Sattler, U. (eds) Automated Reasoning. IJCAR 2012. Lecture Notes in Computer Science(), vol 7364. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31365-3_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-31365-3_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31364-6
Online ISBN: 978-3-642-31365-3
eBook Packages: Computer ScienceComputer Science (R0)