Abstract
Predicate abstraction and canonical abstraction are two finitary abstractions used to prove properties of programs. We study the relationship between these two abstractions by considering a very limited case: abstraction of (potentially cyclic) singly-linked lists.
We provide a new and rather precise family of abstractions for potentially cyclic singly-linked lists. The main observation behind this family of abstractions is that the number of shared nodes in linked lists can be statically bounded. Therefore, the number of possible “heap shapes” is also bounded. We present the new abstraction in both predicate abstraction form as well as in canonical abstraction form.
As we illustrate in the paper, given any canonical abstraction, it is possible to define a predicate abstraction that is equivalent to the canonical abstraction. However, with this straightforward simulation, the number of predicates used for the predicate abstraction is exponential in the number of predicates used by the canonical abstraction.
An important feature of the family of abstractions we present in this paper is that the predicate abstraction representation we define is far more practical as it uses a number of predicates that is quadratic in the number of predicates used by the corresponding canonical abstraction representation. In particular, for the most abstract abstraction in this family, the number of predicates used by the canonical abstraction is linear in the number of program variables, while the number of predicates used by the predicate abstraction is quadratic in the number of program variables.
We have encoded this particular predicate abstraction and corresponding transformers in TVLA, and used this implementation to successfully verify safety properties of several list manipulating programs, including programs that were not previously verified using predicate abstraction or canonical abstraction.
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References
Ball, T., Majumdar, R., Millstein, T., Rajamani, S.: Automatic predicate abstraction of C programs. In: Proc. Conf. on Prog. Lang. Design and Impl., pp. 203–213 (June 2001)
Ball, T., Rajamani, S.: Generating abstract explanations of spurious counterexamples in c programs. Report MSR-TR-2002-09, Microsoft Research, Microsoft Redmond (January 2002), http://research.microsoft.com/slam/
Benedikt, M., Reps, T., Sagiv, M.: A decidable logic for describing linked data structures. In: Proceedings of the 1999 European Symposium On Programming, pp. 2–19 (March 1999)
Blanchet, B., Cousot, P., Cousot, R., Feret, J., Mauborgne, L., Mine, M., Monniaux, D., Rival, X.: A static analyzer for large safety-critical software. In: Fenwick, J.J.B., Norris, C. (eds.) Proceedings of the ACM SIGPLAN 2003 Conference on Programming Language Design and Implementation (PLDI 2003), June 9–11. ACMSIGPLAN Notices, vol. 38(5), pp. 196–207. ACM Press, New York (2003)
Chakaravarthy, V.T.: New results on the computability and complexity of points–to analysis. In: Proceedings of the 30th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, pp. 115–125. ACM Press, New York (2003)
Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 154–169. Springer, Heidelberg (2000)
Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Proc. Symp. on Principles of Prog. Languages, pp. 269–282. ACM Press, New York (1979)
Dams, D., Namjoshi, K.S.: Shape analysis through predicate abstraction and model checking. In: Zuck, L.D., Attie, P.C., Cortesi, A., Mukhopadhyay, S. (eds.) VMCAI 2003. LNCS, vol. 2575, pp. 310–324. Springer, Heidelberg (2002)
Deutsch, A.: Interprocedural may-alias analysis for pointers: Beyond k-limiting. In: Proc. Conf. on Prog. Lang. Design and Impl., pp. 230–241. ACM Press, New York (1994)
Gopan, D., DiMaio, F., Dor, N., Reps, T., Sagiv, M.: Numeric domains with summarized dimensions. In: Tools and Algs. for the Construct. and Anal. of Syst., pp. 512–529 (2004)
Graf, S., Saidi, H.: Construction of abstract state graphs with PVS. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 72–83. Springer, Heidelberg (1997)
Henzinger, T., Jhala, R., Majumdar, R., Sutre, G.: Lazy abstraction. In: Symposium on Principles of Programming Languages, pp. 58–70 (2002)
Immerman, N., Rabinovich, A., Reps, T., Sagiv, M., Yorsh, G.: The boundary between decidability and undecidability for transitive closure logics. In: Proc. Computer Science Logic (2004) (to appear)
Ishtiaq, S.S., O’Hearn, P.W.: BI as an assertion language for mutable data structures. ACM SIGPLAN Notices 36(3), 14–26 (2001)
Jensen, J., Joergensen, M., Klarlund, N., Schwartzbach, M.: Automatic verification of pointer programs using monadic second-order logic. In: Proc. Conf. on Prog. Lang. Design and Impl. (1997)
Jones, N., Muchnick, S.: Flow analysis and optimization of Lisp-like structures. In: Muchnick, S., Jones, N. (eds.) Program Flow Analysis: Theory and Applications, ch. 4, pp. 102–131. Prentice-Hall, Englewood Cliffs (1981)
Lev-Ami, T., Sagiv, M.: TVLA: A framework for Kleene based static analysis. In: Palsberg, J. (ed.) SAS 2000. LNCS, vol. 1824, pp. 280–301. Springer, Heidelberg (2000)
Manevich, R., Yahav, E., Ramalingam, G., Sagiv, M.: Predicate abstraction and canonical abstraction for singly-linked lists. Technical Report TR-2005-01-191212, Tel Aviv University (2005)
Microsoft Research. The SLAM project (2001), http://research.microsoft.com/slam/
Rabin, M.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc. 141(1), 1–35 (1969)
Ramalingam, G., Warshavsky, A., Field, J., Goyal, D., Sagiv, M.: Deriving specialized program analyses for certifying component-client conformance. In: Proc. Conf. on Prog. Lang. Design and Impl., vol. 37(5), pp. 83–94 (June 2002)
Reps, T., Sagiv, M., Yorsh, G.: Symbolic implementation of the best transformer. In: Proc. Verification, Model Checking, and Abstract Interpretation, pp. 252–266. Springer, Heidelberg (2004)
Sagiv, M., Reps, T., Wilhelm, R.: Parametric shape analysis via 3-valued logic. ACM Transactions on Programming Languages and Systems (TOPLAS) 24(3), 217–298 (2002)
Shaham, R., Yahav, E., Kolodner, E.K., Sagiv, M.: Establishing local temporal heap safety properties with applications to compile-time memory management. In: Cousot, R. (ed.) SAS 2003. LNCS, vol. 2694. Springer, Heidelberg (2003)
Yahav, E., Ramalingam, G.: Verifying safety properties using separation and heterogeneous abstractions. In: Proceedings of the ACM SIGPLAN 2004 conference on Programming language design and implementation, pp. 25–34. ACM Press, New York (2004)
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Manevich, R., Yahav, E., Ramalingam, G., Sagiv, M. (2005). Predicate Abstraction and Canonical Abstraction for Singly-Linked Lists. In: Cousot, R. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2005. Lecture Notes in Computer Science, vol 3385. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30579-8_13
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DOI: https://doi.org/10.1007/978-3-540-30579-8_13
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