Abstract
In this paper, we close the logical gap between provability in the logic BBI, which is the propositional basis for separation logic, and validity in an intended class of separation models, as employed in applications of separation logic such as program verification. An intended class of separation models is usually specified by a collection of axioms describing the specific model properties that are expected to hold, which we call a separation theory.
Our main contributions are as follows. First, we show that several typical properties of separation theories are not definable in BBI. Second, we show that these properties become definable in a suitable hybrid extension of BBI, obtained by adding a theory of naming to BBI in the same way that hybrid logic extends normal modal logic. The binder-free extension captures most of the properties we consider, and the full extension HyBBI(↓) with the usual ↓ binder of hybrid logic covers all these properties. Third, we present an axiomatic proof system for our hybrid logic whose extension with any set of "pure" axioms is sound and complete with respect to the models satisfying those axioms. As a corollary of this general result, we obtain, in a parametric manner, a sound and complete axiomatic proof system for any separation theory from our considered class. To the best of our knowledge, this class includes all separation theories appearing in the published literature.
Supplemental Material
- P. Blackburn, M. de Rijke, and Y. Venema. Modal Logic. Cambridge University Press, 2001. Google Scholar
Digital Library
- P. Blackburn and B. ten Cate. Pure extensions, proof rules, and hybrid axiomatics. Studia Logica, 84(2), 2006.Google Scholar
- G. E. P. Box and N. R. Draper. Empirical model-building and response surfaces. Wiley series in probability and mathematical statistics. John Wiley & Sons, Inc., 1987. Google Scholar
Digital Library
- J. Brotherston and C. Calcagno. Classical BI: Its semantics and proof theory. Logical Methods in Computer Science, 6(3), 2010.Google Scholar
- J. Brotherston and M. Kanovich. Undecidability of propositional separation logic and its neighbours. To appear in phJournal of the ACM.Google Scholar
- C. Calcagno, D. Distefano, P. O'Hearn, and H. Yang. Compositional shape analysis by means of bi-abduction. Journal of the ACM, 58(6), 2011. Google Scholar
Digital Library
- C. Calcagno, P. Gardner, and U. Zarfaty. Context logic as modal logic: Completeness and parametric inexpressivity. In POPL-34. ACM, 2007. Google Scholar
Digital Library
- C. Calcagno, P. O'Hearn, and H. Yang. Local action and abstract separation logic. In LICS-22. IEEE Computer Society, 2007. Google Scholar
Digital Library
- R. Cherini and J. O. Blanco. Local reasoning for abstraction and sharing. In SAC-24. ACM, 2009. Google Scholar
Digital Library
- C. David and W.-N. Chin. Immutable specifications for more concise and precise verification. In OOPSLA-11. ACM, 2011. Google Scholar
Digital Library
- T. Dinsdale-Young, L. Birkedal, P. Gardner, M. J. Parkinson, and H. Yang. Views: compositional reasoning for concurrent programs. In POPL-40. ACM, 2013. Google Scholar
Digital Library
- R. Dockins, A. Hobor, and A. W. Appel. A fresh look at separation algebras and share accounting. In APLAS-7. Springer, 2009. Google Scholar
Digital Library
- G. J. Duck, J. Jaffar, and N. C. H. Koh. Constraint-based program reasoning with heaps and separation. 2013. To appear in CP-19.Google Scholar
- D. Galmiche and D. Larchey-Wendling. Expressivity properties of Boolean BI through relational models. In FSTTCS-26. Springer, 2006. Google Scholar
Digital Library
- K. Gödel. On formally undecidable propositions of Principia Mathematica and related systems. 1962. English translation by B. Meltzer.Google Scholar
- A. Hobor and J. Villard. The ramifications of sharing in data structures. In POPL-40. ACM, 2013. Google Scholar
Digital Library
- Z. Hou, A. Tiu, and R. Goré. A labelled sequent calculus for Boolean BI: Proof theory and proof search. To appear in TABLEAUX-22, 2013.Google Scholar
- S. Ishtiaq and P. W. O'Hearn. BI as an assertion language for mutable data structures. In POPL-28. ACM, 2001. Google Scholar
Digital Library
- D. Larchey-Wendling. The formal strong completeness of Boolean BI. Submitted, 2012.Google Scholar
- D. Larchey-Wendling and D. Galmiche. Exploring the relation between intuitionistic BI and Boolean BI: An unexpected embedding. Mathematical Structures in Computer Science, 19(3), 2009. Google Scholar
Digital Library
- D. Larchey-Wendling and D. Galmiche. The undecidability of Boolean BI through phase semantics. In LICS-25. IEEE Computer Society, 2010. Google Scholar
Digital Library
- J. Park, J. Seo, and S. Park. A theorem prover for Boolean BI. In POPL-40. ACM, 2013. Google Scholar
Digital Library
- D. Pym. The Semantics and Proof Theory of the Logic of Bunched Implications. Applied Logic Series. Kluwer, 2002.Google Scholar
- J. C. Reynolds. Separation logic: A logic for shared mutable data structures. In LICS-17. IEEE Computer Society, 2002. Google Scholar
Digital Library
- J. C. Reynolds. A short course on separation logic. http://www.cs.cmu.edu/afs/cs.cmu.edu/project/fox-19/member/jcr/wwwaac2003/aac.html, 2003.Google Scholar
- H. Yang, O. Lee, J. Berdine, C. Calcagno, B. Cook, D. Distefano, and P. O'Hearn. Scalable shape analysis for systems code. In CAV-20. Springer, 2008. Google Scholar
Digital Library
Index Terms
Parametric completeness for separation theories
Recommendations
Undecidability of Propositional Separation Logic and Its Neighbours
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 ...
Expressive Completeness of Separation Logic with Two Variables and No Separating Conjunction
Separation logic is used as an assertion language for Hoare-style proof systems about programs with pointers, and there is an ongoing quest for understanding its complexity and expressive power. Herein, we show that first-order separation logic with one ...
Parametric completeness for separation theories
POPL '14: Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming LanguagesIn this paper, we close the logical gap between provability in the logic BBI, which is the propositional basis for separation logic, and validity in an intended class of separation models, as employed in applications of separation logic such as program ...







Comments