Abstract
Research results from so-called “classical” separation logics are not easily ported to so-called “intuitionistic” separation logics, and vice versa. Basic questions like, “Can the frame rule be proved independently of whether the programming language is garbage-collected?” “Can amortized resource analysis be ported from one separation logic to another?” should be straightforward. But they are not. Proofs done in a particular separation logic are difficult to generalize. We argue that this limitation is caused by incompatible semantics. For example, emp sometimes holds everywhere and sometimes only on units.
In this paper, we introduce a unifying semantics and build a framework that allows to reason parametrically over all separation logics. Many separation algebras in the literature are accompanied, explicitly or implicitly, by a preorder. Our key insight is to axiomatize the interaction between the join relation and the preorder. We prove every separation logic to be sound and complete with respect to this unifying semantics. Further, our framework enables us to generalize the sound0.ness proofs for the frame rule and CSL. It also reveals a new world of meaningful intermediate separation logics between “intuitionistic” and “classical”.
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Notes
- 1.
Pottier also adds a passive execution order which constitutes what he calls a monotonic separation algebra. The idea is similar but goes in a different direction, aiming for a type system and not a separation logic.
- 2.
Coq development: https://github.com/QinxiangCao/UnifySL. Appendix: http://www.cs.princeton.edu/~appel/papers/bringing-order-appendix.pdf.
References
Appel, A.W., Dockins, R., Hobor, A., Beringer, L., Dodds, J., Stewart, G., Blazy, S., Leroy, X.: Program Logics for Certified Compilers, Cambridge (2014)
Appel, A.W., McAllester, D.A.: An indexed model of recursive types for foundational proof-carrying code. ACM Trans. Program. Lang. Syst. 23(5), 657–683 (2001)
Appel, A.W., Melliès, P.-A., Richards, C.D., Vouillon, J.: A very modal model of a modern, major, general type system. In: Proceedings of the 34th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2007)
Atkey, R.: Amortised resource analysis with separation logic. Logical Methods Comput. Sci. 7(2) (2011)
Bengtson, J., Jensen, J.B., Sieczkowski, F., Birkedal, L.: Verifying object-oriented programs with higher-order separation logic in Coq. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds.) ITP 2011. LNCS, vol. 6898, pp. 22–38. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22863-6_5
Birkedal, L., Reus, B., Schwinghammer, J., Støvring, K., Thamsborg, J., Yang, H.: Step-indexed Kripke models over recursive worlds. In: Proceedings of the 38th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2011)
Brookes, S.: A semantics for concurrent separation logic. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 16–34. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-28644-8_2
Brotherston, J., Kanovich, M.: Undecidability of propositional separation logic and its neighbours. In: 2010 25th Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 130–139. IEEE (2010)
Brotherston, J., Villard, J.: Parametric completeness for separation theories. In: The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2014)
Calcagno, C., O’Hearn, P.W., Yang, H.: Local action and abstract separation logic. In: Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science, LICS 2007, pp. 366–378, Washington, DC, USA. IEEE Computer Society (2007)
Chen, H., Ziegler, D., Chajed, T., Chlipala, A., Kaashoek, M.F., Zeldovich, N.: Using crash hoare logic for certifying the FSCQ file system. In: Miller, E.L., Hand, S. (eds.) Proceedings of the 25th Symposium on Operating Systems Principles, SOSP 2015, Monterey, CA, USA, 4–7 October 2015, pp. 18–37. ACM (2015)
Dinsdale-Young, T., Birkedal, L., Gardner, P., Parkinson, M.J., Yang, H.: Views: compositional reasoning for concurrent programs. In: The 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2013)
Dockins, R., Hobor, A., Appel, A.W.: A fresh look at separation algebras and share accounting. In: Hu, Z. (ed.) APLAS 2009. LNCS, vol. 5904, pp. 161–177. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10672-9_13
Galmiche, D., Larchey-Wendling, D.: Expressivity properties of Boolean BI through relational models. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 357–368. Springer, Heidelberg (2006). https://doi.org/10.1007/11944836_33
Galmiche, D., Méry, D., Pym, D.J.: The semantics of BI and resource tableaux. Mathe. Struct. Comput. Sci. 15(6), 1033–1088 (2005)
Gotsman, A., Berdine, J., Cook, B.: Precision and the conjunction rule in concurrent separation logic. Electr. Notes Theor. Comput. Sci. 276, 171–190 (2011)
Hobor, A., Dockins, R., Appel, A.W.: A theory of indirection via approximation. In: Proceedings of the 37th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2010)
Hur, C.-K., Dreyer, D., Vafeiadis, V.: Separation logic in the presence of garbage collection. In: Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science, LICS 2011, 21–24 June 2011, Toronto, Ontario, Canada, pp. 247–256 (2011)
Ishtiaq, S.S., O’Hearn, P.W.: BI as an assertion language for mutable data structures. In: Conference Record of POPL 2001: The 28th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2001)
Jensen, J.B.: Techniques for model construction in separation logic. Ph.D. thesis, IT University of Copenhagen, March 2014
Jensen, J.B., Birkedal, L.: Fictional separation logic. In: Programming Languages and Systems - 21st European Symposium on Programming (2012)
Jung, R., Swasey, D., Sieczkowski, F., Svendsen, K., Turon, A., Birkedal, L., Dreyer, D.: Iris: Monoids and invariants as an orthogonal basis for concurrent reasoning. In: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2015)
Kripke, S.A.: Semantical analysis of intuitionistic logic i. Studies Logic Found. Mathe. 50, 92–130 (1965)
Larchey-Wendling, D., Galmiche, D.: Exploring the relation between intuitionistic BI and boolean BI: an unexpected embedding. Mathe. Struct. Comput. Sci. 19(3), 435–500 (2009)
O’Hearn, P.W., Pym, D.J.: The logic of bunched implications. Bull. Symbolic Logic 5(2), 215–244 (1999)
O’Hearn, P.W., Yang, H., Reynolds, J.C.: Separation and information hiding. In: Proceedings of the 31st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2004, Venice, Italy, 14–16 January 2004, pp. 268–280 (2004)
Parkinson, M.: The next 700 separation logics. In: Leavens, G.T., O’Hearn, P., Rajamani, S.K. (eds.) VSTTE 2010. LNCS, vol. 6217, pp. 169–182. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15057-9_12
Parkinson, M.J., Summers, A.J.: The relationship between separation logic and implicit dynamic frames. In: Barthe, G. (ed.) ESOP 2011. LNCS, vol. 6602, pp. 439–458. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19718-5_23
Pilkiewicz, A., Pottier, F.: The essence of monotonic state. In: Proceedings of TLDI 2011: 2011 ACM SIGPLAN International Workshop on Types in Languages Design and Implementation, pp. 73–86 (2011)
Pottier, F.: Syntactic soundness proof of a type-and-capability system with hidden state. J. Funct. Program. 23(1), 38–144 (2013)
Pym, D.J., O’Hearn, P.W., Yang, H.: Possible worlds and resources: the semantics of BI. Theor. Comput. Sci. 315(1), 257–305 (2004)
Reynolds, J.C.: Intuitionistic reasoning about shared mutable data structure. In: Millennial Perspectives in Computer Science, pp. 303–321. Palgrave (2000)
Simpson, A.K.: The proof theory and semantics of intuitionistic modal logic. Technical report, University of Edinburgh, College of Science and Engineering, School of Informatics (1994)
Acknowledgment
This research was supported in part by NSF Grant CCF-1521602.
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Cao, Q., Cuellar, S., Appel, A.W. (2017). Bringing Order to the Separation Logic Jungle. In: Chang, BY. (eds) Programming Languages and Systems. APLAS 2017. Lecture Notes in Computer Science(), vol 10695. Springer, Cham. https://doi.org/10.1007/978-3-319-71237-6_10
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