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Horn Clause Solvers for Program Verification

Part of the Lecture Notes in Computer Science book series (LNPSE,volume 9300)

Abstract

Automatic program verification and symbolic model checking tools interface with theorem proving technologies that check satisfiability of formulas. A theme pursued in the past years by the authors of this paper has been to encode symbolic model problems directly as Horn clauses and develop dedicated solvers for Horn clauses. Our solvers are called Duality, HSF, SeaHorn, and \(\mu {Z}\) and we have devoted considerable attention in recent papers to algorithms for solving Horn clauses. This paper complements these strides as we summarize main useful properties of Horn clauses, illustrate encodings of procedural program verification into Horn clauses and then highlight a number of useful simplification strategies at the level of Horn clauses. Solving Horn clauses amounts to establishing Existential positive Fixed-point Logic formulas, a perspective that was promoted by Blass and Gurevich.

Keywords

  • Symbolic Model
  • Horn Clause
  • Satisfiability Modulo Theory
  • Proof Rule
  • Constraint Logic Programming

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    Note that we don’t need the clause \(s(x,y) \rightarrow q(y) \vee r(z)\) to preserve satisfiability because the sub-formula that s(xy) summarizes is only used in negative scope.

References

  1. Alberti, F., Ghilardi, S., Sharygina, N.: Booster: an acceleration-based verification framework for array programs. In: Cassez, F., Raskin, J.-F. (eds.) ATVA 2014. LNCS, vol. 8837, pp. 18–23. Springer, Heidelberg (2014)

    Google Scholar 

  2. Apt, K.R.: Logic programming. In: Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics (B), pp. 493–574. Elsevier (1990)

    Google Scholar 

  3. Ball, T., Rajamani, S.K.: Bebop: a path-sensitive interprocedural dataflow engine. In: Proceedings of the 2001 ACM SIGPLAN-SIGSOFT Workshop on Program Analysis for Software Tools and Engineering, PASTE 2001, Snowbird, Utah, USA, 18–19 June 2001, pp. 97–103 (2001)

    Google Scholar 

  4. Barnett, M., Chang, B.-Y.E., DeLine, R., Jacobs, B., M. Leino, K.R.: Boogie: a modular reusable verifier for object-oriented programs. In: de Boer, F.S., Bonsangue, M.M., Graf, S., de Roever, W.-P. (eds.) FMCO 2005. LNCS, vol. 4111, pp. 364–387. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  5. Barnett, M., Leino, K.R.M.: Weakest-precondition of unstructured programs. In: PASTE, pp. 82–87 (2005)

    Google Scholar 

  6. Barrett, C., Stump, A., Tinelli, C.: The Satisfiability Modulo Theories Library (SMT-LIB) (2010). www.SMT-LIB.org

  7. Barvinok, A.I.: A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. In: 34th Annual Symposium on Foundations of Computer Science, Palo Alto, California, USA, 3–5 November 1993, pp. 566–572 (1993)

    Google Scholar 

  8. Berdine, J., Bjørner, N., Ishtiaq, S., Kriener, J.E., Wintersteiger, C.M.: Resourceful reachability as HORN-LA. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19 2013. LNCS, vol. 8312, pp. 137–146. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  9. Beyene, T.A., Chaudhuri, S., Popeea, C., Rybalchenko, A.: A constraint-based approach to solving games on infinite graphs. In: POPL, pp. 221–234 (2014)

    Google Scholar 

  10. Beyer, D., Cimatti, A., Griggio, A., Erkan Keremoglu, M., Sebastiani, R.: Software model checking via large-block encoding. In: FMCAD, pp. 25–32 (2009)

    Google Scholar 

  11. Bjørner, N., Gurfinkel, A.: Property directed polyhedral abstraction. In: D’Souza, D., Lal, A., Larsen, K.G. (eds.) VMCAI 2015. LNCS, vol. 8931, pp. 263–281. Springer, Heidelberg (2015)

    Google Scholar 

  12. Bjørner, N., McMillan, K.L., Rybalchenko, A.: Program verification as satisfiability modulo theories. In: SMT at IJCAR, pp. 3–11 (2012)

    Google Scholar 

  13. Bjørner, N., McMillan, K.L., Rybalchenko, A.: Higher-order program verification as satisfiability modulo theories with algebraic data-types. CoRR, abs/1306.5264 (2013)

    Google Scholar 

  14. Bjørner, N., McMillan, K., Rybalchenko, A.: On solving universally quantified horn clauses. In: Logozzo, F., Fähndrich, M. (eds.) SAS 2013. LNCS, vol. 7935, pp. 105–125. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  15. Blass, A., Gurevich, Y.: Existential fixed-point logic. In: Börger, E. (ed.) Computation Theory and Logic. LNCS, vol. 270, pp. 20–36. Springer, Heidelberg (1987)

    CrossRef  Google Scholar 

  16. Blass, A., Gurevich, Y.: Inadequacy of computable loop invariants. ACM Trans. Comput. Log. 2(1), 1–11 (2001)

    MathSciNet  CrossRef  Google Scholar 

  17. Bradley, A.R.: SAT-based model checking without unrolling. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 70–87. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  18. Burstall, R.M., Darlington, J.: A transformation system for developing recursive programs. JACM 24, 44–67 (1977)

    MathSciNet  CrossRef  MATH  Google Scholar 

  19. Ceri, S., Gottlob, G., Tanca, L.: Logic Programming and Databases. Springer, Heidelberg (1990)

    CrossRef  Google Scholar 

  20. Clarke, E.M.: Programming language constructs for which it is impossible to obtain good hoare axiom systems. J. ACM 26(1), 129–147 (1979)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Cook, S.A.: Soundness and completeness of an axiom system for program verif. SIAM J. Comput. 7(1), 70–90 (1978)

    MathSciNet  CrossRef  MATH  Google Scholar 

  22. Craig, W.: Three uses of the herbrand-gentzen theorem in relating model theory and proof theory. J. Symb. Log. 22(3), 269–285 (1957)

    MathSciNet  CrossRef  MATH  Google Scholar 

  23. De Angelis, E., Fioravanti, F., Pettorossi, A., Proietti, M.: Program verification via iterated specialization. Sci. Comput. Program. 95, 149–175 (2014)

    CrossRef  Google Scholar 

  24. De Angelis, E., Fioravanti, F., Pettorossi, A., Proietti, M.: VeriMAP: a tool for verifying programs through transformations. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014 (ETAPS). LNCS, vol. 8413, pp. 568–574. Springer, Heidelberg (2014)

    CrossRef  Google Scholar 

  25. Dellunde, P., Jansana, R.: Some characterization theorems for infinitary universal horn logic without equality. J. Symb. Log. 61(4), 1242–1260 (1996)

    MathSciNet  CrossRef  MATH  Google Scholar 

  26. Dijkstra, E.W.: A Discipline of Programming. Prentice-Hall, New Jersey (1976)

    MATH  Google Scholar 

  27. Fietzke, A., Weidenbach, C.: Superposition as a decision procedure for timed automata. Math. Comput. Sci. 6(4), 409–425 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  28. Flanagan, C., Leino, K.R.M., Lillibridge, M., Nelson, G., Saxe, J.B., Stata, R.: Extended static checking for java. In: PLDI, pp. 234–245 (2002)

    Google Scholar 

  29. Floyd, R.W.: Assigning meaning to programs. In: Proceedings of Symposium on Applied Mathematics, vol. 19, pp. 19–32. American Math. Soc. (1967)

    Google Scholar 

  30. Gallagher, J.P., Kafle, B.: Analysis and transformation tools for constrained horn clause verification. CoRR, abs/1405.3883 (2014)

    Google Scholar 

  31. German, S.M., Clarke, E.M., Halpern, J.Y.: Reasoning about procedures as parameters in the language L4. Inf. Comput. 83(3), 265–359 (1989)

    MathSciNet  CrossRef  MATH  Google Scholar 

  32. Grebenshchikov, S., Lopes, N.P., Popeea, C., Rybalchenko, A.: Synthesizing software verifiers from proof rules. In: PLDI (2012)

    Google Scholar 

  33. Gurfinkel, A., Chaki, S., Sapra, S.: Efficient Predicate Abstraction of Program Summaries. In: Bobaru, M., Havelund, K., Holzmann, G.J., Joshi, R. (eds.) NFM 2011. LNCS, vol. 6617, pp. 131–145. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  34. Gurfinkel, A., Kahsai, T., Komuravelli, A., Navas, J.A.: The seahorn verification framework. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 343–361. Springer, Heidelberg (2015)

    CrossRef  Google Scholar 

  35. Gurfinkel, A., Wei, O., Chechik, M.: Model checking recursive programs with exact predicate abstraction. In: Cha, S.S., Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 95–110. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  36. Hermenegildo, M.V., Bueno, F., Carro, M., Lopez-Garcia, P., Mera, E., Morales, J.F., Puebla, G.: An overview of ciao and its design philosophy. TPLP 12(1–2), 219–252 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Hoare, C.A.R.: An axiomatic basis for computer programming. Commun. ACM 12(10), 576–580 (1969)

    CrossRef  MATH  Google Scholar 

  38. Hoder, K., Bjørner, N.: Generalized property directed reachability. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 157–171. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  39. Hojjat, H., Iosif, R., Konečný, F., Kuncak, V., Rümmer, P.: Accelerating interpolants. In: Chakraborty, S., Mukund, M. (eds.) ATVA 2012. LNCS, vol. 7561, pp. 187–202. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  40. Horn, A.: On sentences which are true of direct unions of algebras. J. Symb. Log. 16(1), 14–21 (1951)

    MathSciNet  CrossRef  MATH  Google Scholar 

  41. Jaffar, J.: A CLP approach to modelling systems. In: Davies, J., Schulte, W., Barnett, M. (eds.) ICFEM 2004. LNCS, vol. 3308, p. 14. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  42. Jaffar, J., Maher, M.J.: Constraint logic programming: a survey. J. Log. Program. 19(20), 503–581 (1994)

    MathSciNet  CrossRef  MATH  Google Scholar 

  43. Jaffar, J., Santosa, A.E., Voicu, R.: An interpolation method for CLP traversal. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 454–469. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  44. Jhala, R., Majumdar, R., Rybalchenko, A.: HMC: verifying functional programs using abstract interpreters. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 470–485. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  45. Jones, N.D., Gomard, C.K., Sestoft, P.: Partial Evaluation and Automatic Program Generation. Prentice Hall international series in computer science. Prentice Hall, Englewood Cliff (1993)

    MATH  Google Scholar 

  46. Kafle, B., Gallagher, J.P.: Constraint specialisation in horn clause verification. In: PEPM, pp. 85–90 (2015)

    Google Scholar 

  47. Karbyshev, A., Bjørner, N., Itzhaky, S., Rinetzky, N., Shoham, S.: Property-directed inference of universal invariants or proving their absence (2015)

    Google Scholar 

  48. Komuravelli, A., Gurfinkel, A., Chaki, S.: SMT-based model checking for recursive programs. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 17–34. Springer, Heidelberg (2014)

    Google Scholar 

  49. Lal, A., Qadeer, S.: A program transformation for faster goal-directed search. In: Formal Methods in Computer-Aided Design, FMCAD 2014, Lausanne, Switzerland, 21–24 October 2014, pp. 147–154 (2014)

    Google Scholar 

  50. Rustan, K., Leino, M.: Efficient weakest preconditions. Inf. Process. Lett. 93(6), 281–288 (2005)

    MathSciNet  CrossRef  MATH  Google Scholar 

  51. Lopes, N.P., Bjørner, N., Godefroid, P., Jayaraman, K., Varghese, G.: Checking beliefs in dynamic networks. In: NSDI, May 2015

    Google Scholar 

  52. Manna, Z., Pnueli, A.: Temporal Verification of Reactive Systems: Safety. Springer, Berlin (1995)

    CrossRef  MATH  Google Scholar 

  53. McCarthy, J.: Towards a mathematical science of computation. In: IFIP Congress, pp. 21–28 (1962)

    Google Scholar 

  54. McMillan, K.L.: Lazy annotation revisited. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 243–259. Springer, Heidelberg (2014)

    Google Scholar 

  55. Oppen, D.C.: Complexity, convexity and combinations of theories. Theor. Comput. Sci. 12, 291–302 (1980)

    MathSciNet  CrossRef  MATH  Google Scholar 

  56. Navarro Pérez, J.A., Rybalchenko, A.: Separation logic modulo theories. In: Shan, C. (ed.) APLAS 2013. LNCS, vol. 8301, pp. 90–106. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  57. Pettorossi, A., Proietti, M.: Synthesis and transformation of logic programs using unfold/fold proofs. Technical report 457, Universitá di Roma Tor Vergata (1997)

    Google Scholar 

  58. Pudl’ak, P.: Lower bounds for resolution and cutting planes proofs and monotone computations. J. Symbolic Logic 62(3), 981–998 (1995)

    MathSciNet  CrossRef  Google Scholar 

  59. Ramsay, S.J., Neatherway, R.P., Luke Ong, C.-H.: A type-directed abstraction refinement approach to higher-order model checking. In: POPL, pp. 61–72 (2014)

    Google Scholar 

  60. Reps, T.W., Horwitz, S., Sagiv, S.: Precise interprocedural dataflow analysis via graph reachability. In: POPL, pp. 49–61 (1995)

    Google Scholar 

  61. Revesz, P.Z.: Safe datalog queries with linear constraints. In: Maher, M.J., Puget, J.-F. (eds.) CP 1998. LNCS, vol. 1520, pp. 355–369. Springer, Heidelberg (1998)

    CrossRef  Google Scholar 

  62. Rondon, P.M., Kawaguchi, M., Jhala, R.: Liquid types. In: PLDI, pp. 159–169 (2008)

    Google Scholar 

  63. Rümmer, P., Hojjat, H., Kuncak, V.: Disjunctive interpolants for horn-clause verification. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 347–363. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  64. Sheeran, M., Singh, S., Stålmarck, G.: Checking safety properties using induction and a SAT-solver. In: Johnson, S.D., Hunt Jr., W.A. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 108–125. Springer, Heidelberg (2000)

    CrossRef  Google Scholar 

  65. Tamaki, H., Sato, T.: Unfold/fold transformation of logic programs. In: Proceedings of the Second International Conference on Logic Programming (1984)

    Google Scholar 

  66. Turchin, V.F.: The concept of a supercompiler. ACM TOPLAS 8(3), 292–325 (1986)

    MathSciNet  CrossRef  MATH  Google Scholar 

  67. van Emden, M.H., Kowalski, R.A.: The semantics of predicate logic as a programming language. J. ACM 23(4), 733–742 (1976)

    MathSciNet  CrossRef  MATH  Google Scholar 

  68. Warren, D.S.: Memoing for logic programs. Commun. ACM 35(3), 93–111 (1992)

    CrossRef  Google Scholar 

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Bjørner, N., Gurfinkel, A., McMillan, K., Rybalchenko, A. (2015). Horn Clause Solvers for Program Verification. In: Beklemishev, L., Blass, A., Dershowitz, N., Finkbeiner, B., Schulte, W. (eds) Fields of Logic and Computation II. Lecture Notes in Computer Science(), vol 9300. Springer, Cham. https://doi.org/10.1007/978-3-319-23534-9_2

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