Interpolant Synthesis for Quadratic Polynomial Inequalities and Combination with EUF

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9706)


An algorithm for generating interpolants for formulas which are conjunctions of quadratic polynomial inequalities (both strict and nonstrict) is proposed. The algorithm is based on a key observation that quadratic polynomial inequalities can be linearized if they are concave. A generalization of Motzkin’s transposition theorem is proved, which is used to generate an interpolant between two mutually contradictory conjunctions of polynomial inequalities, using semi-definite programming in time complexity \(\mathcal {O}(n^3+nm)\), where n is the number of variables and m is the number of inequalities (This complexity analysis assumes that despite the numerical nature of approximate SDP algorithms, they are able to generate correct answers in a fixed number of calls.). Using the framework proposed in [22] for combining interpolants for a combination of quantifier-free theories which have their own interpolation algorithms, a combination algorithm is given for the combined theory of concave quadratic polynomial inequalities and the equality theory over uninterpreted functions (EUF).


Program verification Interpolant Concave quadratic polynomial Motzkin’s theorem SOS Semi-definite programming 



The first three authors are supported partly by NSFC under grants 11290141, 11271034 and 61532019; the fourth and sixth authors are supported partly by “973 Program” under grant No. 2014CB340701, by NSFC under grant 91418204, by CDZ project CAP (GZ 1023), and by the CAS/SAFEA International Partnership Program for Creative Research Teams; the fifth author is supported partly by NSF under grant DMS-1217054 and by the CAS/SAFEA International Partnership Program for Creative Research Teams.


  1. 1.
  2. 2.
    Beyer, D., Zufferey, D., Majumdar, R.: CSIsat: interpolation for LA+EUF. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 304–308. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Cimatti, A., Griggio, A., Sebastiani, R.: Efficient interpolant generation in satisfiability modulo theories. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 397–412. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Dai, L., Gan, T., Xia, B., Zhan, N.: Barrier certificate revisited. J. Symbolic Comput. (2016, to appear)Google Scholar
  5. 5.
    Dai, L., Xia, B., Zhan, N.: Generating non-linear interpolants by semidefinite programming. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 364–380. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    D’Silva, V., Kroening, D., Purandare, M., Weissenbacher, G.: Interpolant strength. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 129–145. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Fujie, T., Kojima, M.: Semidefinite programming relaxation for nonconvex quadratic programs. J. Global Optim. 10(4), 367–380 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gan, T., Dai, L., Xia, B., Zhan, N., Kapur, D., Chen, M.: Interpolation synthesis for quadratic polynomial inequalities and combination with EUF. CoRR, abs/1601.04802 (2016)Google Scholar
  9. 9.
    Graf, S., Saïdi, H.: Construction of abstract state graphs with PVS. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 72–83. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  10. 10.
    Henzinger, T., Jhala, R., Majumdar, R., McMillan, K.: Abstractions from proofs. In: POPL 2004, pp. 232–244 (2004)Google Scholar
  11. 11.
    Jung, Y., Lee, W., Wang, B.-Y., Yi, K.: Predicate generation for learning-based quantifier-free loop invariant inference. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 205–219. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Kapur, D., Majumdar, R., Zarba, C.: Interpolation for data structures. In: FSE 2006, pp. 105–116 (2006)Google Scholar
  13. 13.
    Kovács, L., Voronkov, A.: Interpolation and symbol elimination. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 199–213. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Krajíc̆cek, J.: Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. J. Symbolic Logic 62(2), 457–486 (1997)Google Scholar
  15. 15.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol. 149, pp. 157–270. Springer, New York (2009)CrossRefGoogle Scholar
  16. 16.
    McMillan, K.L.: Interpolation and SAT-based model checking. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    McMillan, K.: An interpolating theorem prover. Theor. Comput. Sci. 345(1), 101–121 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    McMillan, K.L.: Quantified invariant generation using an interpolating saturation prover. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 413–427. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Pudlák, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. J. Symbolic Logic 62(3), 981–998 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rybalchenko, A., Sofronie-Stokkermans, V.: Constraint solving for interpolation. J. Symb. Comput. 45(11), 1212–1233 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1998)zbMATHGoogle Scholar
  22. 22.
    Sofronie-Stokkermans, V.: Interpolation in local theory extensions. Logical Methods Comput. Sci. 4(4), 1–31 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Stengle, G.: A nullstellensatz and a positivstellensatz in semialgebraic geometry. Ann. Math. 207, 87–97 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. J. Math. Program. 95(2), 189–217 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yang, Z., Lin, W., Wu, M.: Exact safety verification of hybrid systems based on bilinear SOS representation. ACM Trans. Embed. Comput. Syst. 14(1), 16:1–16:19 (2015)CrossRefGoogle Scholar
  26. 26.
    Yorsh, G., Musuvathi, M.: A combination method for generating interpolants. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 353–368. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  27. 27.
    Zhao, H., Zhan, N., Kapur, D.: Synthesizing switching controllers for hybrid systems by generating invariants. In: Liu, Z., Woodcock, J., Zhu, H. (eds.) Theories of Programming and Formal Methods. LNCS, vol. 8051, pp. 354–373. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  28. 28.
    Zhao, H., Zhan, N., Kapur, D., Larsen, K.G.: A “Hybrid” approach for synthesizing optimal controllers of hybrid systems: a case study of the oil pump industrial example. In: Giannakopoulou, D., Méry, D. (eds.) FM 2012. LNCS, vol. 7436, pp. 471–485. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.LMAM & School of Mathematical SciencesPeking UniversityBeijingChina
  2. 2.State Key Laboratory of Computer ScienceInstitute of Software, Chinese Academy of SciencesBeijingChina
  3. 3.Department of Computer ScienceUniversity of New MexicoAlbuquerqueUSA
  4. 4.State Key Laboratory of Software EngineeringWuhan UniversityWuhanChina

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