Formalizing Counterexample-Driven Refinement with Weakest Preconditions

  • Thomas Ball
Part of the NATO Science Series book series (NAII, volume 195)

Abstract

To check a safety property of a program, it is sufficient to check the property on an abstraction that has more behaviors than the original program. If the safety property holds of the abstraction then it also holds of the original program.

However, if the property does not hold of the abstraction along some trace t (a counterexample), it may or may not hold of the original program on trace t. If it can be proved that the property does not hold in the original program on trace t then it makes sense to refine the abstraction to eliminate the “spurious counterexample” t (rather than a report a known false negative to the user).

The SLAM tool developed at Microsoft Research implements such an automated abstraction-refinement process. In this paper, we reformulate this process for a tiny while language using the concepts of weakest preconditions, bounded model checking and Craig interpolants. This representation of SLAM simplifies and distills the concepts of counterexample-driven refinement in a form that should be suitable for teaching the process in a few lectures of a graduate-level course.

Keywords

Hoare logic weakest preconditions predicate abstraction abstract interpretation inductive invariants symbolic model checking automatic theorem proving Craig interpolants 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ball, T., Podelski, A., and Rajamani, S. K. (2001). Boolean and cartesian abstractions for model checking C programs. In TACAS 01: Tools and Algorithms for Construction and Analysis of Systems, LNCS 2031, pages 268–283. Springer-Verlag.Google Scholar
  2. Ball, T. and Rajamani, S. K. (2000). Boolean programs: A model and process for software analysis. Technical Report MSR-TR-2000-14, Microsoft Research.Google Scholar
  3. Ball, T. and Rajamani, S. K. (2001). Automatically validating temporal safety properties of interfaces. In SPIN 01: SPIN Workshop, LNCS 2057, pages 103–122. Springer-Verlag.Google Scholar
  4. Bryant, R. (1986). Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers, C-35(8):677–691.Google Scholar
  5. Burch, J., Clarke, E., McMillan, K., Dill, D., and Hwang, L. (1992). Symbolic model checking: 1020 states and beyond. Information and Computation, 98(2):142–170.CrossRefMathSciNetGoogle Scholar
  6. Clarke, E., Grumberg, O., Jha, S., Lu, Y., and Veith, H. (2000). Counterexample-guided abstraction refinement. In CAV 00: Computer Aided Verification, LNCS 1855, pages 154–169. Springer-Verlag.Google Scholar
  7. Clarke, E. M. and Emerson, E. A. (1981). Synthesis of synchronization skeletons for branching time temporal logic. In Logic of Programs, LNCS 131, pages 52–71. Springer-Verlag.Google Scholar
  8. Cousot, P. and Cousot, R. (1977). Abstract interpretation: a unified lattice model for the static analysis of programs by construction or approximation of fixpoints. In POPL 77: Principles of Programming Languages, pages 238–252. ACM.Google Scholar
  9. Cousot, P. and Cousot, R. (1978). Static determination of dynamic properties of recursive procedures. In Neuhold, E., editor, Formal Descriptions of Programming Concepts, (IFIP WG 2.2, St. Andrews, Canada, August 1977), pages 237–277. North-Holland.Google Scholar
  10. Craig, W. (1957). Linear reasoning. a new form of the herbrand-gentzen theorem. J. Symbolic Logic, 22:250–268.MATHMathSciNetGoogle Scholar
  11. Das, M. (2000). Unification-based pointer analysis with directional assignments. In PLDI 00: Programming Language Design and Implementation, pages 35–46. ACM.Google Scholar
  12. Detlefs, D., Nelson, G., and Saxe, J. B. (2003). Simplify: A theorem prover for program checking. Technical Report HPL-2003-148, HP Labs.Google Scholar
  13. Dijkstra, E. (1976). A Discipline of Programming. Prentice-Hall.Google Scholar
  14. Flanagan, C., Leino, K. R. M., Lillibridge, M., Nelson, G., Saxe, J. B., and Stata, R. (2002). Extended static checking for java. In PLDI 02: Programming Language Design and Implementation, pages 234–245. ACM.Google Scholar
  15. Graf, S. and Saidi, H. (1997). Construction of abstract state graphs with PVS. In CAV 97: Computer-aided Verification, LNCS 1254, pages 72–83. Springer-Verlag.Google Scholar
  16. Henzinger, T. A., Jhala, R., Majumdar, R., and McMillan, K. L. (2004). Abstractions from proofs. In POPL 04: Principles of Programming Languages, pages 232–244. ACM.Google Scholar
  17. Henzinger, T. A., Jhala, R., Majumdar, R., and Sutre, G. (2002). Lazy abstraction. In POPL’ 02, pages 58–70. ACM.Google Scholar
  18. Hoare, C. A. R. (1969). An axiomatic basis for computer programming. Communications of the ACM, 12(10):576–583.CrossRefMATHGoogle Scholar
  19. Knoop, J. and Steffen, B. (1992). The interprocedural coincidence theorem. In CC 92: Compiler Construction, pages 125–140.Google Scholar
  20. Kurshan, R. (1994). Computer-aided Verification of Coordinating Processes. Princeton University Press.Google Scholar
  21. McMillan, K. (1993). Symbolic Model Checking: An Approach to the State-Explosion Problem. Kluwer Academic Publishers.Google Scholar
  22. McMillan, K. (2003). Interpolation and sat-based model checking. In CAV 03: Computer-Aided Verification, LNCS 2725, pages 1–13. Springer-Verlag.Google Scholar
  23. Morris, J. M. (1982). A general axiom of assignment. In Theoretical Foundations of Programming Methodology, Lecture Notes of an International Summer School, pages 25–34. D. Reidel Publishing Company.Google Scholar
  24. Nelson, G. and Oppen, D. C. (1979). Simplification by cooperating decision procedures. ACM Transactions on Programming Languages and Systems, 1(2):245–257.CrossRefGoogle Scholar
  25. Queille, J. and Sifakis, J. (1981). Specification and verification of concurrent systems in Cesar. In Proc. 5th International Symp. on Programming, volume 137 of Lecture Notes in Computer Science, pages 337–351. Springer-Verlag.MathSciNetGoogle Scholar
  26. Reps, T., Horwitz, S., and Sagiv, M. (1995). Precise interprocedural dataflow analysis via graph reachability. In POPL 95: Principles of Programming Languages, pages 49–61. ACM.Google Scholar
  27. Sagiv, M., Reps, T., and Wilhelm, R. (1999). Parametric shape analysis via 3-valued logic. In POPL 99: Principles of Programming Languages, pages 105–118. ACM.Google Scholar
  28. Sharir, M. and Pnueli, A. (1981). Two approaches to interprocedural data flow analysis. In Program Flow Analysis: Theory and Applications, pages 189–233. Prentice-Hall.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Thomas Ball
    • 1
  1. 1.Microsoft ResearchUK

Personalised recommendations