Satisfiability Checking: Theory and Applications

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9763)

Abstract

Satisfiability checking aims to develop algorithms and tools for checking the satisfiability of existentially quantified logical formulas. Besides powerful SAT solvers for solving propositional logic formulas, sophisticated SAT-modulo-theories (SMT) solvers are available for a wide range of theories, and are applied as black-box engines for many techniques in different areas. In this paper we give a short introduction to the theoretical foundations of satisfiability checking, mention some of the most popular tools, and discuss the successful embedding of SMT solvers in different technologies.

References

  1. 1.
    Ábrahám, E.: Building bridges between symbolic computation and satisfiability checking. In: Proceedings of ISSAC 2015, pp. 1–6. ACM (2015)Google Scholar
  2. 2.
    Ansótegui, C., Bofill, M., Palahı, M., Suy, J., Villaret, M.: Satisfiability modulo theories: An efficient approach for the resource-constrained project scheduling problem. In: Proceedings of SARA 2011, pp. 2–9. AAAI (2011)Google Scholar
  3. 3.
  4. 4.
    Bae, K., Ölveczky, P.C., Kong, S., Gao, S., Clarke, E.M.: SMT-based analysis of virtually synchronous distributed hybrid systems. In: Proceedings of HSCC 2016 (2016). (to appear)Google Scholar
  5. 5.
    Ball, T., Bounimova, E., Levin, V., De Moura, L.: Efficient evaluation of pointer predicates with Z3 SMT solver in SLAM2. Technical report, Microsoft Research (2010)Google Scholar
  6. 6.
    Barrett, C., Conway, C.L., Deters, M., Hadarean, L., Jovanović, D., King, T., Reynolds, A., Tinelli, C.: CVC4. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 171–177. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Barrett, C.W., de Moura, L., Stump, A.: SMT-COMP: satisfiability modulo theories competition. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 20–23. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Barrett, C., Fontaine, P., Tinelli, C.: The Satisfiability Modulo Theories Library (SMT-LIB) (2016). www.SMT-LIB.org
  9. 9.
    Barrett, C., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, Chap. 26. Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 825–885. IOS Press, Amsterdam (2009)Google Scholar
  10. 10.
    Biere, A., Biere, A., Heule, M., van Maaren, H., Walsh, T.: Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press, Amsterdam (2009)MATHGoogle Scholar
  11. 11.
    Biere, A., Cimatti, A., Clarke, E., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, pp. 193–207. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. 12.
    Bjørner, N., Jayaraman, K.: Checking cloud contracts in microsoft azure. In: Natarajan, R., Barua, G., Patra, M.R. (eds.) ICDCIT 2015. LNCS, vol. 8956, pp. 21–32. Springer, Heidelberg (2015)Google Scholar
  13. 13.
    Bjørner, N., Phan, A.-D., Fleckenstein, L.: \(\nu \) Z - an optimizing SMT solver. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 194–199. Springer, Heidelberg (2015)Google Scholar
  14. 14.
    Bofill, M., Coll, J., Suy, J., Villaret, M.: A system for generation and visualization of resource-constrained projects. In: Proceedings of CCIA 2014. Frontiers in Artificial Intelligence and Applications, vol. 269, pp. 237–246. IOS Press (2014)Google Scholar
  15. 15.
    Boogie.: An intermediate verification language. http://research.microsoft.com/en-us/projects/boogie/
  16. 16.
    Bouton, Thomas, de Oliveira, D.C.B., Déharbe, D., Fontaine, P.: \({\sf { veriT}}\): an open, trustable and efficient SMT-solver. In: Schmidt, Renate A. (ed.) CADE-22. LNCS, vol. 5663, pp. 151–156. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Bradley, A.R.: SAT-based model checking without unrolling. In: Schmidt, D., Jhala, R. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 70–87. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Bruttomesso, R., Pek, E., Sharygina, N., Tsitovich, A.: The OpenSMT solver. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 150–153. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Cadar, C., Dunbar, D., Engler, D.: KLEE: Unassisted and automatic generation of high-coverage tests for complex systems programs. In: Proceedings of OSDI 2008, pp. 209–224. USENIX Association (2008)Google Scholar
  20. 20.
    Catan, M., et al.: Aeolus: mastering the complexity of cloud application deployment. In: Lau, K.-K., Lamersdorf, W., Pimentel, E. (eds.) ESOCC 2013. LNCS, vol. 8135, pp. 1–3. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  21. 21.
    Christ, J., Hoenicke, J., Nutz, A.: SMTInterpol: an interpolating SMT solver. In: Donaldson, A., Parker, D. (eds.) SPIN 2012. LNCS, vol. 7385, pp. 248–254. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  22. 22.
    Cimatti, A., Griggio, A., Schaafsma, B.J., Sebastiani, R.: The MathSAT5 SMT solver. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013 (ETAPS 2013). LNCS, vol. 7795, pp. 93–107. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  23. 23.
    Cimatti, A., Mover, S., Tonetta, S.: A quantifier-free SMT encoding of non-linear hybrid automata. In: Proceedings of FMCAD 2012, pp. 187–195. IEEE (2012)Google Scholar
  24. 24.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) Automata Theory and Formal Languages. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  25. 25.
    Conchon, S., Iguernelala, M., Mebsout, A.: A collaborative framework for non-linear integer arithmetic reasoning in Alt-Ergo. In: Proceedings of SYNASC 2013, pp. 161–168. IEEE (2013)Google Scholar
  26. 26.
    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., et al. (eds.) SAT 2015. LNCS, vol. 9340, pp. 360–368. Springer, Heidelberg (2015). doi:10.1007/978-3-319-24318-4_26 CrossRefGoogle Scholar
  27. 27.
    Craciunas, S.S., Oliver, R.S.: SMT-based task- and network-level static schedule generation for time-triggered networked systems. In: Proceedings of RTNS 2014, p. 45. ACM (2014)Google Scholar
  28. 28.
    Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)MATHGoogle Scholar
  29. 29.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Commun. ACM 5(7), 394–397 (1962)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. J. ACM 7(3), 201–215 (1960)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Déharbe, D., Fontaine, P., Guyot, Y., Voisin, L.: Integrating SMT solvers in Rodin. Sci. Comput. Program. 94(P2), 130–143 (2014)CrossRefGoogle Scholar
  32. 32.
    Dutertre, B.: Yices 2.2. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 737–744. Springer, Heidelberg (2014)Google Scholar
  33. 33.
    Eggers, A., Ramdani, N., Nedialkov, N.S., Fränzle, M.: Improving the SAT modulo ODE approach to hybrid systems analysis by combining different enclosure methods. Softw. Syst. Model. 14(1), 121–148 (2012)CrossRefMATHGoogle Scholar
  34. 34.
    Fränzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient solving of large non-linear arithmetic constraint systems with complex Boolean structure. J. Satisf. Boolean Model. Comput. 1(3–4), 209–236 (2007)MATHGoogle Scholar
  35. 35.
    Ganesh, V., Dill, D.L.: A decision procedure for bit-vectors and arrays. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 519–531. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  36. 36.
    Gao, S., Ganai, M., Ivančić, F., Gupta, A., Sankaranarayanan, S., Clarke, E.M.: Integrating ICP and LRA solvers for deciding nonlinear real arithmetic problems. In: Proceedings of FMCAD 2010, pp. 81–90. IEEE (2010)Google Scholar
  37. 37.
    Giesl, J., et al.: Proving termination of programs automatically with \({\sf { AProVE}}\). In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 184–191. Springer, Heidelberg (2014)Google Scholar
  38. 38.
    Hallin, M.: SMT-Based Reasoning and Planning in TAL. Master’s thesis, Linköping University (2010)Google Scholar
  39. 39.
    Herbort, S., Ratz, D.: Improving the efficiency of a nonlinear-system-solver using a componentwise Newton method. Technical report 2/1997, Inst. für Angewandte Mathematik, University of Karlsruhe (1997)Google Scholar
  40. 40.
    Jayaraman, K., Bjrner, N., Outhred, G., Kaufman, C.: Automated analysis and debugging of network connectivity policies. Technical report MSR-TR-2014-102, Microsoft Research (2014). http://research.microsoft.com/apps/pubs/default.aspx?id=225826
  41. 41.
    Jovanović, D., de Moura, L.: Solving non-linear arithmetic. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 339–354. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  42. 42.
    Kahsai, T., Tinelli, C.: PKIND: A parallel \(k\)-induction based model checker. arXiv preprint (2011). arXiv:1111.0372
  43. 43.
    Khanh, T.V., Vu, X., Ogawa, M.: raSAT: SMT for polynomial inequality. In: Proceedings of SMT 2014, p. 67 (2014)Google Scholar
  44. 44.
    Kong, S., Gao, S., Chen, W., Clarke, E.: \({\sf dReach:} \delta \)-reachability analysis for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 200–205. Springer, Heidelberg (2015)Google Scholar
  45. 45.
    Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean termination tool 2. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 295–304. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  46. 46.
    Kroening, D., Strichman, O.: Decision Procedures: An Algorithmic Point of View. Springer, New York (2008)MATHGoogle Scholar
  47. 47.
    Kroening, D., Tautschnig, M.: CBMC – C bounded model checker. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014 (ETAPS). LNCS, vol. 8413, pp. 389–391. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  48. 48.
    Lange, T., Neuhäußer, M.R., Noll, T.: IC3 software model checking on control flow automata. In: Proceedings of FMCAD 2015, pp. 97–104. IEEE (2015)Google Scholar
  49. 49.
    Li, Y., Albarghouthi, A., Kincaid, Z., Gurfinkel, A., Chechik, M.: Symbolic optimization with SMT solvers. In: Proceedings of POPL 2014, pp. 607–618. ACM (2014)Google Scholar
  50. 50.
    Marques-silva, J.P., Sakallah, K.A.: Grasp: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48, 506–521 (1999)MathSciNetCrossRefGoogle Scholar
  51. 51.
    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)CrossRefGoogle Scholar
  52. 52.
    de Moura, L., Passmore, G.O.: The strategy challenge in SMT solving. In: Bonacina, M.P., Stickel, M.E. (eds.) Automated Reasoning and Mathematics. LNCS, vol. 7788, pp. 15–44. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  53. 53.
    Nedunuri, S., Prabhu, S., Moll, M., Chaudhuri, S., Kavraki, L.E.: SMT-based synthesis of integrated task and motion plans from plan outlines. In: Proceedings of ICRA 2014, pp. 655–662. IEEE (2014)Google Scholar
  54. 54.
    Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. Program. Lang. Syst. 1(2), 245–257 (1979)CrossRefMATHGoogle Scholar
  55. 55.
    Niemetz, A., Preiner, M., Biere, A.: Boolector 2.0. J. Satisf. Boolean Model. Comput. 9, 53–58 (2015)Google Scholar
  56. 56.
    Peleska, J., Vorobev, E., Lapschies, F.: Automated test case generation with SMT-solving and abstract interpretation. In: Bobaru, M., Havelund, K., Holzmann, G.J., Joshi, R. (eds.) NFM 2011. LNCS, vol. 6617, pp. 298–312. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  57. 57.
    Phothilimthana, P.M., Thakur, A., Bodik, R., Dhurjati, D.: GreenThumb: Superoptimizer construction framework. In: Proceedings of CCC 2016, pp. 261–262. ACM (2016)Google Scholar
  58. 58.
    Pike, L.: Modeling time-triggered protocols and verifying their real-time schedules. In: Proceedings of FMCAD 2007, pp. 231–238. IEEE (2007)Google Scholar
  59. 59.
    Rintanen, J.: Discretization of temporal models with application to planning with SMT. In: Proceedings of AAAI 2015, pp. 3349–3355. AAAI (2015)Google Scholar
  60. 60.
    Symbolic analysis laboratory. http://sal.csl.sri.com/introduction.shtml
  61. 61.
    Scala, E., Ramirez, M., Haslum, P., Thiebaux, S.: Numeric planning with disjunctive global constraints via SMT. In: Proceedings of ICASP 2016 (2016, to appear)Google Scholar
  62. 62.
    Scheibler, K., Kupferschmid, S., Becker, B.: Recent improvements in the SMT solver iSAT. In: Proceedings of MBMV 2013, pp. 231–241. Institut für Angewandte Mikroelektronik und Datentechnik, Fakultät für Informatik und Elektrotechnik, Universität Rostock (2013)Google Scholar
  63. 63.
    Sebastiani, R., Trentin, P.: OptiMathSAT: a tool for optimization modulo theories. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 447–454. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  64. 64.
    SMT-COMP 2015 result summary (2015). http://smtcomp.sourceforge.net/2015/results-summary.shtml
  65. 65.
  66. 66.
    Tiwari, A., Gascón, A., Dutertre, B.: Program synthesis using dual interpretation. In: Felty, A., Middeldorp, A. (eds.) CADE-25. Lecture Notes in Computer Science, vol. 9195, pp. 482–497. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  67. 67.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Siekmann, J.H., Wrightson, G. (eds.) Automation of Reasoning, pp. 466–483. Springer, New York (1983)CrossRefGoogle Scholar
  68. 68.
    Weispfenning, V.: A new approach to quantifier elimination for real algebra. In: Caviness, B.F., Johnson, J.R. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation, pp. 376–392. Springer, NEw York (1998)CrossRefGoogle Scholar
  69. 69.
    Weispfenning, V.: Quantifier elimination for real algebra - the quadratic case and beyond. Appl. Algebra Eng. Commun. Comput. 8(2), 85–101 (1997)MathSciNetCrossRefMATHGoogle Scholar
  70. 70.
    Wintersteiger, C.M., Hamadi, Y., de Moura, L.: A concurrent portfolio approach to SMT solving. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 715–720. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  71. 71.
    Yamada, A., Kusakari, K., Sakabe, T.: Nagoya termination tool. In: Dowek, G. (ed.) RTA-TLCA 2014. LNCS, vol. 8560, pp. 466–475. Springer, Heidelberg (2014)Google Scholar
  72. 72.
    Yuan, M., He, X., Gu, Z.: Hardware/software partitioning and static task scheduling on runtime reconfigurable FPGAs using an SMT solver. In: Proceedings of RTAS 2008, pp. 295–304. IEEE (2008)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.RWTH Aachen UniversityAachenGermany

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