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Solving Non-linear Arithmetic

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Automated Reasoning (IJCAR 2012)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7364))

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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.

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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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Chapter  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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)

    Google Scholar 

  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)

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  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)

    Chapter  Google Scholar 

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

    Google Scholar 

  16. Jovanović, 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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

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

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  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)

    Chapter  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  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)

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. New York University, USA

    Dejan Jovanović

  2. Microsoft Research, USA

    Leonardo de Moura

Authors
  1. Dejan Jovanović
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  2. Leonardo de Moura
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Editor information

Editors and Affiliations

  1. Fakultät für Informatik, Technische Universität Wien, Favoritenstr. 9, E185/2, 1040, Wien, Austria

    Bernhard Gramlich

  2. École Polytechnique, INRIA Saclay and Laboratoire d’Informatique, Route de Saclay, 91128, Palaiseau Cedex, France

    Dale Miller

  3. School of Computer Science, University of Manchester, Oxford Road, M13 9PL, Manchester, UK

    Uli Sattler

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Jovanović, D., de Moura, L. (2012). Solving Non-linear Arithmetic. 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_27

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

  • 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)

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