Abstract
Abstract interpretation provides an over-approximation of program behaviours that is used to prove the absence of bugs. When the computed approximation in the chosen abstract domain is as precise as possible, we say the analysis is complete and false alarms cannot arise. Unfortunately for any non trivial abstract domain there is some program whose analysis is incomplete. In this paper we want to characterize the classes of complete programs on some non-trivial abstract domains for studying their expressiveness. To this aim we introduce the notion of bounded domains for posets with ascending chains of bounded length only. We show that any complete program on bounded domains can be rewritten in an equivalent canonical form without nontrivial loops. This result proves that program termination on the class of complete programs on bounded domain is decidable. Moreover, semantic equivalence between programs in the above class can be reduced to determining the equivalence of a set of guarded statements. We show how our approach can be applied to a quite large class of programs. Indeed, abstract domains defined on Boolean abstractions that are complete for the same functions can be composed by preserving boundedness and completeness also w.r.t. any expressible guard. This suggests that new complete bounded abstract domains can be tailored on the guards and functions appearing in the program. Their existence is sufficient to prove decidability of termination and program equivalence for such programs.
Research supported by MIUR PRIN Project 201784YSZ5 ASPRA–Analysis of Program Analyses. Partially funded by RYC-2016-20281, by ESF Investing in your future and by Madrid regional government as part of the program S2018/TCS-4339 co-funded by EIE Funds of the European Union.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Namely, the identical abstraction, making abstract and concrete semantics the same, and the top abstraction, making all programs equivalent by abstract semantics.
- 2.
Note that this is different from establishing termination for all input in S, which should be addressed separately.
References
Ball, T., Podelski, A., Rajamani, S.K.: Boolean and Cartesian abstraction for model checking C programs. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 268–283. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45319-9_19
Barringer, H.: A Survey of Verification Techniques for Parallel Programs. Springer, Heidelberg (1985). https://doi.org/10.1007/3-540-15239-3
Bruni, R., Giacobazzi, R., Gori, R., Garcia-Contreras, I., Pavlovic, D.: Abstract extensionality: on the properties of incomplete abstract interpretations. Proc. ACM on Program. Lang. 4(POPL), 1–28 (2019)
Bruni, R., Giacobazzi, R., Gori, R., Ranzato, F.: A logic for locally complete abstract interpretations. In: Proceedings of LICS 2021, 36th Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 1–13. IEEE (2021). https://doi.org/10.1109/LICS52264.2021.9470608, distinguished paper
Cousot, P.: Principles of Abstract Interpretation. MIT Press, Cambridge (2021)
Cousot, P.: Abstract interpretation based formal methods and future challenges. In: Wilhelm, R. (ed.) Informatics. LNCS, vol. 2000, pp. 138–156. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44577-3_10
Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proceedings of the 4th ACM SIGACT-SIGPLAN Symposium on Principles of programming Languages, pp. 238–252 (1977)
Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Proceedings of the 6th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, pp. 269–282 (1979)
Cousot, P., Cousot, R.: Basic concepts of abstract interpretation. In: Jacquart, R. (ed.) Building the Information Society. IIFIP, vol. 156, pp. 359–366. Springer, Boston, MA (2004). https://doi.org/10.1007/978-1-4020-8157-6_27
Das, M., Lerner, S., Seigle, M.: ESP: path-sensitive program verification in polynomial time. In: Proceedings of the ACM SIGPLAN 2002 Conference on Programming Language Design and Implementation, pp. 57–68 (2002)
Dijkstra, E.W.: A Discipline Of programming. Series in Automatic Computation, Prentice-Hall, Hoboken (1976)
D’silva, V., Kroening, D., Weissenbacher, G.: A survey of automated techniques for formal software verification. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 27(7), 1165–1178 (2008)
Giacobazzi, R., Ranzato, F., Scozzari., F.: Making abstract interpretation complete. J. ACM 47(2), 361–416 (2000). https://doi.org/10.1145/333979.333989
Giacobazzi, R., Logozzo, F., Ranzato, F.: Analyzing program analyses. ACM SIGPLAN Not. 50(1), 261–273 (2015)
Kozen, D.: Kleene algebra with tests. ACM Trans. Program. Lang. Syst. 19(3), 427–443 (1997). https://doi.org/10.1145/256167.256195
Miné, A.: Tutorial on static inference of numeric invariants by abstract interpretation. Found. Trends Program. Lang. 4(3–4), 120–372 (2017)
de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24
Nelson, C.G.: Techniques for Program Verification. Stanford University, Stanford (1980)
Nielson, F., Nielson, H., Hankin, C.: Principles of Program Analysis. Springer, Cham (2010). https://doi.org/10.1007/978-3-662-03811-6
Rice, H.G.: Classes of recursively enumerable sets and their decision problems. Trans. Am. Math. Soc. 74(2), 358–366 (1953)
Rival, X., Yi, K.: Introduction to Static Analysis - An Abstract Interpretation Perspective. MIT Press, Cambridge (2020)
Turing, A.M.: On computable numbers, with an application to the entscheidungsproblem. A correction. Proc. Lond. Math. Soc. 2(1), 544–546 (1938)
Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification. In: Proceedings of the First Symposium on Logic in Computer Science, pp. 322–331. IEEE Computer Society (1986)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Bruni, R., Gori, R., Manini, N. (2022). Deciding Program Properties via Complete Abstractions on Bounded Domains. In: Singh, G., Urban, C. (eds) Static Analysis. SAS 2022. Lecture Notes in Computer Science, vol 13790. Springer, Cham. https://doi.org/10.1007/978-3-031-22308-2_9
Download citation
DOI: https://doi.org/10.1007/978-3-031-22308-2_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-22307-5
Online ISBN: 978-3-031-22308-2
eBook Packages: Computer ScienceComputer Science (R0)