Abstract
We present a method for determining the satisfiability of quantifier-free first-order formulas modulo the theory of non-linear arithmetic over the reals augmented with transcendental functions. Our procedure is based on the fruitful combination of two main ingredients: unconstrained optimisation, to generate a set of candidate solutions, and a result from topology called the topological degree test to check whether a given bounded region contains at least a solution. We have implemented the procedure in a prototype tool called ugotNL, and integrated it within the MathSAT SMT solver. Our experimental evaluation over a wide range of benchmarks shows that it vastly improves the performance of the solver for satisfiable non-linear arithmetic formulas, significantly outperforming other available tools for problems with transcendental functions.
This work has been partly supported by project “AI@TN” funded by the Autonomous Province of Trento.
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
Similar content being viewed by others
Notes
- 1.
Note that, according to this definition, a problem could be unsat and \(\delta \)-sat at the same time.
- 2.
A formal definition of robustness can be found in Sect. 2 of [15].
- 3.
\(\partial B\) is the topological boundary of B, i.e. the set of points in the closure of B that are not in its interior.
- 4.
Available at https://www.cs.cas.cz/~ratschan/topdeg/topdeg.html.
- 5.
We remind that we are assuming to be in \(\mathcal {NRA}\) only here.
- 6.
This is not true for formulas containing strict inequalities, but we replaced strict inequalities in Algorithm 2 at line 1.
- 7.
References
Aberth, O.: Computation of topological degree using interval arithmetic, and applications. Math. Comput. 62(205), 171–178 (1994)
Barrett, C., Fontaine, P., Tinelli, C.: The Satisfiability Modulo Theories Library (SMT-LIB) (2016). www.SMT-LIB.org
Barrett, C., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability Modulo Theories, chapter 26 (2009)
Benhamou, F., Granvilliers, L.: Chapter 16 - continuous and interval constraints. In: Handbook of Constraint Programming (2006)
Brauße, F., Korovin, K., Korovina, M.V., Müller, N.T.: The ksmt calculus is a \(\delta \)-complete decision procedure for non-linear constraints. In: CADE (2021)
Barrett, C., et al.: CVC5 at the SMT Competition 2021 (2021)
Cimatti, A., Griggio, A., Irfan, A., Roveri, M., Sebastiani, R.: Incremental linearization for satisfiability and verification modulo nonlinear arithmetic and transcendental functions. ACM Trans. Comput. Logic 19(3), 1–52 (2018)
Cimatti, A., Griggio, A., Schaafsma, B.J., Sebastiani, R.: The MathSAT5 SMT solver. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 93–107. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36742-7_7
Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975). https://doi.org/10.1007/3-540-07407-4_17
Corzilius, F., Kremer, G., Junges, S., Schupp, S., Ábrahám, E.: SMT-RAT: an open source C++ toolbox for strategic and parallel SMT solving. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 360–368. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24318-4_26
de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24
de Moura, L., Jovanović, D.: A model-constructing satisfiability calculus. In: Giacobazzi, R., Berdine, J., Mastroeni, I. (eds.) VMCAI 2013. LNCS, vol. 7737, pp. 1–12. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35873-9_1
Dutertre, B.: Yices 2.2. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 737–744. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08867-9_49
Franek, P., Ratschan, S.: Effective topological degree computation based on interval arithmetic. Math. Comput. 84(293), 1265–1290 (2015)
Franek, P., Ratschan, S., Zgliczynski, P.: Quasi-decidability of a fragment of the first-order theory of real numbers. J. Autom. Reasoning 57(2), 157–185 (2015). https://doi.org/10.1007/s10817-015-9351-3
Fränzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. JSAT (2007)
Fu, Z., Su, Z.: XSat: a fast floating-point satisfiability solver. In: Chaudhuri, S., Farzan, A. (eds.) CAV 2016. LNCS, vol. 9780, pp. 187–209. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-41540-6_11
Gao, S., Avigad, J., Clarke, E.M.: \(\delta \)-complete decision procedures for satisfiability over the reals. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 286–300. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31365-3_23
Gao, S., Kong, S., Clarke, E.M.: dReal: an SMT solver for nonlinear theories over the reals. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 208–214. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38574-2_14
Jovanović, D., de Moura, L.: Solving non-linear arithmetic. ACM Commun. Comput. Algebra 46(3/4), 104–105 (2013)
Minh, D.D.L., Minh, D.L.P.: Understanding the hastings algorithm. Commun. Stat. Simul. Comput. 44(2), 332–349 (2015)
Moore, R., Kearfott, R., Cloud, M.: Introduction to Interval Analysis (2009)
Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press (1991)
O’Regan, D., Je, C.Y., Chen, Y.: Topological Degree Theory and Applications. Taylor and Francis (2006)
Ratschan, S.: Safety verification of non-linear hybrid systems is quasi-decidable. Formal Meth. Syst. Des. 44(1), 71–90 (2013). https://doi.org/10.1007/s10703-013-0196-2
Richardson, D.: Some undecidable problems involving elementary functions of a real variable. J. Symb. Log. 33(4), 514–520 (1968)
Xuan, T.V., Khanh, T., Ogawa, M.: rasat: an smt solver for polynomial constraints. Formal Meth. Syst. Des. 51, 12 (2017)
Wales, D.J., Doye, J.P.K.: Global optimization by basin-hopping and the lowest energy structures of lennard-jones clusters containing up to 110 atoms. J. Phys. Chem. A, (28) (1997)
Zhang, L., Madigan, C.F., Moskewicz, M.H., Malik, S.: Efficient conflict driven learning in a boolean satisfiability solver. In: ICCAD (2001)
Ábrahám, E., Davenport, J., England, M., Kremer, G.: Deciding the consistency of non-linear real arithmetic constraints with a conflict driven search using cylindrical algebraic coverings. J. Log. Algebraic Meth. Program. 119, 100633 (2020)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Cimatti, A., Griggio, A., Lipparini, E., Sebastiani, R. (2022). Handling Polynomial and Transcendental Functions in SMT via Unconstrained Optimisation and Topological Degree Test. In: Bouajjani, A., Holík, L., Wu, Z. (eds) Automated Technology for Verification and Analysis. ATVA 2022. Lecture Notes in Computer Science, vol 13505. Springer, Cham. https://doi.org/10.1007/978-3-031-19992-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-031-19992-9_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-19991-2
Online ISBN: 978-3-031-19992-9
eBook Packages: Computer ScienceComputer Science (R0)