Article

Solving non-linear arithmetic

Authors:

Dejan Jovanović,

Leonardo de MouraAuthors Info & Claims

IJCAR'12: Proceedings of the 6th international joint conference on Automated Reasoning

Pages 339 - 354

https://doi.org/10.1007/978-3-642-31365-3_27

Published: 26 June 2012 Publication History

Abstract

We present a new algorithm for deciding satisfiability of non-linear arithmetic constraints. The algorithm performs a Conflict-Driven Clause Learning (CDCL)-style search for a feasible assignment, while using projection operators adapted from cylindrical algebraic decomposition to guide the search away from the conflicting states.

References

[1]

Akbarpour, B., Paulson, L.C.: MetiTarski: An automatic theorem prover for real-valued special functions. Journal of Automated Reasoning 44(3), 175-205 (2010).

Digital Library

[2]

Barrett, C., Tinelli, C.: CVC3. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 298-302. Springer, Heidelberg (2007).

Digital Library

[3]

Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry. Springer (2006).

Digital Library

[4]

Brown, C.W.: Solution formula construction for truth invariant CAD's. PhD thesis, University of Delaware (1999).

Digital Library

[5]

Brown, C.W.: QEPCAD B: a program for computing with semi-algebraic sets using CADs. ACM SIGSAM Bulletin 37(4), 97-108 (2003).

Digital Library

[6]

Brown, W.S., Traub, J.F.: On Euclid's algorithm and the theory of subresultants. Journal of the ACM 18(4), 505-514 (1971).

Digital Library

[7]

Buchberger, B., Collins, G.E., Loos, R., Albrecht, R. (eds.): Computer algebra. Symbolic and algebraic computation. Springer (1982).

Digital Library

[8]

Caviness, B.F., Johnson, J.R. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation. Springer (2004).

[9]

Cohen, H.: A Course in Computational Algebraic Number Theory. Springer (1993).

Digital Library

[10]

Collins, G.E.: 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).

Digital Library

[11]

de Moura, L., Bjørner, N.S.: Z3: An Efficient SMT Solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337-340. Springer, Heidelberg (2008).

Digital Library

[12]

Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2-9 (1997).

Digital Library

[13]

Fränzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient solving of large non-linear arithmetic constraint systems with complex Boolean structure. Journal on Satisfiability, Boolean Modeling and Computation 1(3-4), 209-236 (2007).

[14]

Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: SAT Solving for Termination Analysis with Polynomial Interpretations. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 340-354. Springer, Heidelberg (2007).

Digital Library

[15]

Hong, H.: Comparison of several decision algorithms for the existential theory of the reals (1991).

[16]

Jovanovic, D., de Moura, L.: Cutting to the Chase Solving Linear Integer Arithmetic. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 338-353. Springer, Heidelberg (2011).

Digital Library

[17]

Korovin, K., Tsiskaridze, N., Voronkov, A.: Conflict Resolution. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 509-523. Springer, Heidelberg (2009).

Digital Library

[18]

Loos, R.: Generalized polynomial remainder sequences. Computer Algebra: Symbolic and Algebraic Computation, 115-137 (1982).

[19]

McLaughlin, S., Harrison, J.V.: A Proof-Producing Decision Procedure for Real Arithmetic. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 295-314. Springer, Heidelberg (2005).

Digital Library

[20]

McMillan, K.L., Kuehlmann, A., Sagiv, M.: Generalizing DPLL to Richer Logics. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 462-476. Springer, Heidelberg (2009).

Digital Library

[21]

Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: From an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). Journal of the ACM 53(6), 937-977 (2006).

Digital Library

[22]

Passmore, G.O.: Combined Decision Procedures for Nonlinear Arithmetics, Real and Complex. PhD thesis, University of Edinburgh (2011).

[23]

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).

Digital Library

[24]

Silva, J.P.M., Sakallah, K.A.: GRASP: A search algorithm for propositional satisfiability. IEEE Transactions on Computers 48(5), 506-521 (1999).

Digital Library

[25]

Strzebonski, A.W.: Cylindrical algebraic decomposition using validated numerics. Journal of Symbolic Computation 41(9), 1021-1038 (2006).

[26]

Tarski, A.: A decision method for elementary algebra and geometry. Technical Report R-109, Rand Corporation (1951).

[27]

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).

Digital Library

[28]

Weispfenning, V.: Quantifier elimination for real algebra - the quadratic case and beyond. AAECC 8, 85-101 (1993).

[29]

Zankl, H., Middeldorp, A.: Satisfiability of Non-linear (Ir)rational Arithmetic. In: Clarke, E.M., Voronkov, A. (eds.) LPAR-16 2010. LNCS, vol. 6355, pp. 481-500. Springer, Heidelberg (2010).

Digital Library

Cited By

Mikek BZhang Q(2024)SMT Theory Arbitrage: Approximating Unbounded Constraints using Bounded TheoriesProceedings of the ACM on Programming Languages10.1145/36563878:PLDI(246-271)Online publication date: 20-Jun-2024
https://dl.acm.org/doi/10.1145/3656387
Nalbach JÁbrahám E(2024)Merging Adjacent Cells During Single Cell ConstructionComputer Algebra in Scientific Computing10.1007/978-3-031-69070-9_15(252-272)Online publication date: 1-Sep-2024
https://dl.acm.org/doi/10.1007/978-3-031-69070-9_15
England M(2024)Recent Developments in Real Quantifier Elimination and Cylindrical Algebraic Decomposition (Extended Abstract of Invited Talk)Computer Algebra in Scientific Computing10.1007/978-3-031-69070-9_1(1-10)Online publication date: 1-Sep-2024
https://dl.acm.org/doi/10.1007/978-3-031-69070-9_1
Show More Cited By

Recommendations

Solving quantified linear arithmetic by counterexample-guided instantiation

This paper presents a framework to derive instantiation-based decision procedures for satisfiability of quantified formulas in first-order theories, including its correctness, implementation, and evaluation. Using this framework we derive decision ...
Solving Non-linear Polynomial Arithmetic via SAT Modulo Linear Arithmetic
CADE-22: Proceedings of the 22nd International Conference on Automated Deduction

Polynomial constraint-solving plays a prominent role in several areas of engineering and software verification. In particular, polynomial constraint solving has a long and successful history in the development of tools for proving termination of ...
Solving linear arithmetic with SAT-based model checking
FMCAD '17: Proceedings of the 17th Conference on Formal Methods in Computer-Aided Design

We present LIAMC, a novel decision procedure for (quantifier-free) linear arithmetic over both integers modulo 2^N (LIA_N) and integers (LIA).

There is no need to explain our motivation to design a new efficient decision procedure for the widely used LIA ...

Comments

Information & Contributors

Information

Published In

cover image Guide Proceedings

IJCAR'12: Proceedings of the 6th international joint conference on Automated Reasoning

June 2012

568 pages

ISBN:9783642313646

Editors:
Bernhard Gramlich
Fakultät für Informatik, Technische Universität Wien, Favoritenstr. 9, E185/2, Wien, Austria
,
Dale Miller
École Polytechnique, INRIA Saclay and Laboratoire d'Informatique, Route de Saclay, Palaiseau Cedex, France
,
Uli Sattler
School of Computer Science, University of Manchester, Oxford Road, Manchester, UK

Sponsors

Univ. of Manchester: University of Manchester
Microsoft Research: Microsoft Research

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 26 June 2012

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

80
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 14 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Mikek BZhang Q(2024)SMT Theory Arbitrage: Approximating Unbounded Constraints using Bounded TheoriesProceedings of the ACM on Programming Languages10.1145/36563878:PLDI(246-271)Online publication date: 20-Jun-2024
https://dl.acm.org/doi/10.1145/3656387
Nalbach JÁbrahám E(2024)Merging Adjacent Cells During Single Cell ConstructionComputer Algebra in Scientific Computing10.1007/978-3-031-69070-9_15(252-272)Online publication date: 1-Sep-2024
https://dl.acm.org/doi/10.1007/978-3-031-69070-9_15
England M(2024)Recent Developments in Real Quantifier Elimination and Cylindrical Algebraic Decomposition (Extended Abstract of Invited Talk)Computer Algebra in Scientific Computing10.1007/978-3-031-69070-9_1(1-10)Online publication date: 1-Sep-2024
https://dl.acm.org/doi/10.1007/978-3-031-69070-9_1
Tsiskaridze NBarrett CTinelli C(2024)Generalized Optimization Modulo TheoriesAutomated Reasoning10.1007/978-3-031-63498-7_27(458-479)Online publication date: 3-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-63498-7_27
Hader TKaufmann DIrfan AGraham-Lengrand SKovács L(2024)MCSat-Based Finite Field Reasoning in the Yices2 SMT Solver (Short Paper)Automated Reasoning10.1007/978-3-031-63498-7_23(386-395)Online publication date: 3-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-63498-7_23
Wang ZZhan BLi BCai S(2024)Efficient Local Search for Nonlinear Real ArithmeticVerification, Model Checking, and Abstract Interpretation10.1007/978-3-031-50524-9_15(326-349)Online publication date: 15-Jan-2024
https://dl.acm.org/doi/10.1007/978-3-031-50524-9_15
Bonacina MGraham-Lengrand SVauthier C(2023)QSMA: A New Algorithm for Quantified Satisfiability Modulo Theory and AssignmentAutomated Deduction – CADE 2910.1007/978-3-031-38499-8_5(78-95)Online publication date: 1-Jul-2023
https://dl.acm.org/doi/10.1007/978-3-031-38499-8_5
Li HXia BZhao T(2023)Local Search for Solving Satisfiability of Polynomial FormulasComputer Aided Verification10.1007/978-3-031-37703-7_5(87-109)Online publication date: 17-Jul-2023
https://dl.acm.org/doi/10.1007/978-3-031-37703-7_5
Nalbach JÁbrahám E(2023)Subtropical Satisfiability for SMT SolvingNASA Formal Methods10.1007/978-3-031-33170-1_26(430-446)Online publication date: 16-May-2023
https://dl.acm.org/doi/10.1007/978-3-031-33170-1_26
Sadeghimanesh AEngland M(2022)An SMT solver for non-linear real arithmetic inside mapleACM Communications in Computer Algebra10.1145/3572867.357288056:2(76-79)Online publication date: 23-Nov-2022
https://dl.acm.org/doi/10.1145/3572867.3572880
Show More Cited By

View Options

View options

Media

Figures

Other

Tables

View Table of Contents