Undecidability of Propositional Separation Logic and Its Neighbours

Abstract

In this article, we investigate the logical structure of memory models of theoretical and practical interest. Our main interest is in “the logic behind a fixed memory model”, rather than in “a model of any kind behind a given logical system”. As an effective language for reasoning about such memory models, we use the formalism of separation logic. Our main result is that for any concrete choice of heap-like memory model, validity in that model is undecidable even for purely propositional formulas in this language.

The main novelty of our approach to the problem is that we focus on validity in specific, concrete memory models, as opposed to validity in general classes of models.

Besides its intrinsic technical interest, this result also provides new insights into the nature of their decidable fragments. In particular, we show that, in order to obtain such decidable fragments, either the formula language must be severely restricted or the valuations of propositional variables must be constrained.

In addition, we show that a number of propositional systems that approximate separation logic are undecidable as well. In particular, this resolves the open problems of decidability for Boolean BI and Classical BI.

Moreover, we provide one of the simplest undecidable propositional systems currently known in the literature, called “Minimal Boolean BI”, by combining the purely positive implication-conjunction fragment of Boolean logic with the laws of multiplicative *-conjunction, its unit and its adjoint implication, originally provided by intuitionistic multiplicative linear logic. Each of these two components is individually decidable: the implication-conjunction fragment of Boolean logic is co-NP-complete, and intuitionistic multiplicative linear logic is NP-complete.

All of our undecidability results are obtained by means of a direct encoding of Minsky machines.

References

1. A. Ahmed, L. Jia, and D. Walker. 2003. Reasoning about hierarchical storage. In Proceedings of LICS-18. IEEE Computer Society, 33--44.
2. A. V. Aho, J. E. Hopcroft, and J. D. Ullman. 1974. The Design and Analysis of Computer Algorithms. Addison-Wesley.
3. G. E. Andrews. 1976. The Theory of partitions. Encyclopedia of Mathematics and Its Applications. Addison-Wesley.
4. N. P. Benton, G. M. Bierman, V. de Paiva, and M. Hyland. 1993. A term calculus for intuitionistic linear logic. In Proceedings of TLCA-1. Springer, 75--90.
5. J. Berdine, C. Calcagno, and P. O'Hearn. 2004. A decidable fragment of separation logic. In Proceedings of FSTTCS-24. Springer, 97--109.
6. R. Bornat, C. Calcagno, P. O'Hearn, and M. Parkinson. 2005. Permission accounting in separation logic. In Proceedings of POPL-32. 59--70.
7. J. Brotherston. 2012. Bunched logics displayed. Studia Logica 100, 6, 1223--1254.
8. J. Brotherston and C. Calcagno. 2010. Classical BI: Its semantics and proof theory. Logical Meth. Comput. Sci. 6, 3.
9. J. Brotherston and M. Kanovich. 2010. Undecidability of propositional separation logic and its neighbours. In Proceedings of LICS-25. IEEE Computer Society, 137--146.
10. J. Brotherston and J. Villard. 2014. Parametric completeness for separation theories. In Proceedings of POPL-41. ACM. 453--464.
11. C. Calcagno, D. Distefano, P. O'Hearn, and H. Yang. 2011. Compositional shape analysis by means of bi-abduction. J. ACM 58, 6.
12. C. Calcagno, P. O'Hearn, and H. Yang. 2007. Local action and abstract separation logic. In Proceedings of LICS-22. IEEE Computer Society, 366--378.
13. C. Calcagno, H. Yang, and P. W. O'Hearn. 2001. Computability and complexity results for a spatial assertion language for data structures. In Proceedings of FSTTCS-21. Springer, 108--119.
14. D. Distefano and M. Parkinson. 2008. jStar: Towards practical verification for Java. In Proceedings of OOPSLA-23. ACM, 213--226.
15. M. Dodds, X. Feng, M. Parkinson, and V. Vafeiadis. 2009. Deny-guarantee reasoning. In Proceedings of ESOP-18. Springer, 363--377.
16. D. Foulis and M. Bennett. 1994. Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331--1352.
17. D. Galmiche and D. Larchey-Wendling. 2006. Expressivity properties of Boolean BI through relational models. In Proceedings of FSTTCS-26. Springer, 357--368.
18. D. Galmiche, D. Méry, and D. Pym. 2005. The semantics of BI and resource tableaux. Math. Struct. Comput. Sci. 15, 1033--1088.
19. P. Gardner, S. Maffeis, and G. D. Smith. 2012. Towards a program logic for JavaScript. In Proceedings of POPL-39. 31--44.
20. J.-Y. Girard and Y. Lafont. 1987. Linear logic and lazy computation. In Proceedings of TAPSOFT'87. Springer-Verlag, 52--66.
21. A. Gotsman, B. Cook, M. Parkinson, and V. Vafeiadis. 2009. Proving that non-blocking algorithms don't block. In Proceedings of POPL-36. ACM, 16--28.
22. S. Ishtiaq and P. W. O'Hearn. 2001. BI as an assertion language for mutable data structures. In Proceedings of POPL-28. ACM, 14--26.
23. E. Jipsen. 1992. Computer aided investigations of relation algebras. Ph.D. dissertation, Vanderbilt University, Nashville, Tennessee.
24. M. Kanovich. 1992. Horn programming in linear logic is NP-complete. In Proceedings of LICS-7. IEEE Computer Society, 200--210.
25. M. Kanovich. 1995. The direct simulation of Minsky machines in linear logic. In Advances in Linear Logic, London Mathematical Society Lecture Notes Series, vol. 222, Cambridge University Press, 123--145.
26. A. Kurucz, I. Nemeti, I. Sain, and A. Simon. 1995. Decidable and undecidable modal logics with a binary modality. J. Logic, Lang. Inf. 4, 191--206.
27. A. Kurucz. 1997. Decision problems in algebraic logic. Ph.D. dissertation, Hungarian Academy of Sciences.
28. D. Larchey-Wendling and D. Galmiche. 2010. The undecidability of Boolean BI through phase semantics. In Proceedings of LICS-25. IEEE Computer Society, 140--149.
29. D. Larchey-Wendling and D. Galmiche. 2013. Nondeterministic phase semantics and the undecidability of Boolean BI. ACM Trans. Comput. Logic 14, 1, 6.
30. M. Minsky. 1967. Computation: Finite and Infinite Machines. Prentice-Hall, Inc.
31. T. Murata. 1989. Petri nets: Properties, analysis and applications. Proc. IEEE 77, 4, 541--580.
32. P. W. O'Hearn and D. J. Pym. 1999. The logic of bunched implications. Bull. Symb. Logic 5, 2, 215--244.
33. M. Parkinson and G. Bierman. 2008. Separation logic, abstraction and inheritance. In Proceedings of POPL-35. ACM, 75--86.
34. M. Parkinson, R. Bornat, and C. Calcagno. 2006. Variables as resource in Hoare logics. In Proceedings of LICS-21. IEEE Computer Society, 137--146.
35. J. L. Peterson. 1981. Petri Net Theory and the Modeling of Systems. Prentice-Hall.
36. D. Pym. 2002. The Semantics and Proof Theory of the Logic of Bunched Implications. Applied Logic Series. Kluwer.
37. D. Pym, P. O'Hearn, and, H. Yang. 2004. Possible worlds and resources: The semantics of BI. Theoret. Comput. Sci. 315, 1, 257--305.
38. J. C. Reynolds. 2002. Separation logic: A logic for shared mutable data structures. In Proceedings of LICS-17. IEEE Computer Society, 55--74.
39. R. Statman. 1979. Intuitionistic propositional logic is polynomial-space complete. Theoret. Comput. Sci. 9, 67--72.
40. H. Yang, O. Lee, J. Berdine, C. Calcagno, B. Cook, D. Distefano, and P. O'Hearn. 2008. Scalable shape analysis for systems code. In Proceedings of CAV-20. Springer, 385--398.

Index Terms

• Computing methodologies

  • Symbolic and algebraic manipulation

    • Symbolic and algebraic algorithms

      • Theorem proving algorithms

• Theory of computation

  • Logic

    • Finite Model Theory

    • Programming logic

    • Proof theory

  • Semantics and reasoning

    • Program reasoning

      • Assertions