Abstract
This paper proposes a new interpretation of the logical contents of programs in the context of concurrent interaction, wherein proofs correspond to valid executions of a processes. A type system based on linear logic is used, in which a given process has many different types, each typing corresponding to a particular way of interacting with its environment and cut elimination corresponds to executing the process in a given interaction scenario. A completeness result is established, stating that every lock-avoiding execution of a process in some environment corresponds to a particular typing. Besides traces, types contain precise information about the flow of control between a process and its environment, and proofs are interpreted as composable schedulings of processes. In this interpretation, logic appears as a way of making explicit the flow of causality between interacting processes.
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References
Abramsky, S.: Proofs as processes. TCS 135(1), 5–9 (1994)
Beffara, E.: A concurrent model for linear logic. ENTCS 155, 147–168 (2006)
Beffara, E., Maurel, F.: Concurrent nets: a study of prefixing in process calculi. TCS 356(3), 356–373 (2006)
Beffara, E., Mogbil, V.: Proofs as executions. Technical Report 00586459, HAL (July 2012), http://hal.archives-ouvertes.fr/hal-00586459
Bruscoli, P.: A Purely Logical Account of Sequentiality in Proof Search. In: Stuckey, P.J. (ed.) ICLP 2002. LNCS, vol. 2401, pp. 302–316. Springer, Heidelberg (2002)
Caires, L., Pfenning, F.: Session Types as Intuitionistic Linear Propositions. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 222–236. Springer, Heidelberg (2010)
Danos, V., Regnier, L.: The structure of multiplicatives. Archive for Math. Logic 28(3), 181–203 (1989)
Jacobé de Naurois, P., Mogbil, V.: Correctness of linear logic proof structures is NL-complete. TCS 412(20), 1941–1957 (2011)
Ehrhard, T., Laurent, O.: Interpreting a Finitary π-calculus in Differential Interaction Nets. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 333–348. Springer, Heidelberg (2007)
Ehrhard, T., Regnier, L.: Differential interaction nets. TCS 364(2), 166–195 (2006)
Girard, J.-Y.: Linear logic. TCS 50(1), 1–102 (1987)
Girard, J.-Y.: Proof-nets: the parallel syntax for proof theory. Logic and Algebra 180 (1996)
Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. Journal of the ACM 32(1), 137–161 (1985)
Honda, K., Laurent, O.: An exact correspondence between a typed π-calculus and polarised proof-nets. TCS 411(22–24), 2223–2238 (2010)
Laneve, C., Victor, B.: Solos in Concert. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 513–523. Springer, Heidelberg (1999)
Mairson, H.G., Terui, K.: On the Computational Complexity of Cut-Elimination in Linear Logic. In: Blundo, C., Laneve, C. (eds.) ICTCS 2003. LNCS, vol. 2841, pp. 23–36. Springer, Heidelberg (2003)
Maurel, F.: Nondeterministic Light Logics and NP-Time. In: Hofmann, M.O. (ed.) TLCA 2003. LNCS, vol. 2701, pp. 241–255. Springer, Heidelberg (2003)
Mazurak, K., Zdancewic, S.: Lolliproc: to concurrency from classical linear logic via curry-howard and control. In: ICFP, pp. 39–50 (2010)
Miller, D.: The π-Calculus as a Theory in Linear Logic: Preliminary Results. In: Lamma, E., Mello, P. (eds.) ELP 1992. LNCS, vol. 660, pp. 242–264. Springer, Heidelberg (1993)
Milner, R.: Communication and concurrency. Prentice Hall (1989)
Mogbil, V.: Non-deterministic Boolean Proof Nets. In: van Eekelen, M., Shkaravska, O. (eds.) FOPARA 2009. LNCS, vol. 6324, pp. 131–145. Springer, Heidelberg (2010)
Tiu, A., Miller, D.: A proof search specification of the π-calculus. ENTCS 138(1), 79–101 (2005)
Winskel, G.: Event structures. In: Advances in Petri nets: Applications and Relationships to Other Models of Concurrency, pp. 325–392. Springer, Heidelberg (1987)
Yoshida, N., Berger, M., Honda, K.: Strong normalisation in the π-calculus. In: LICS, pp. 311–322 (2001)
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Beffara, E., Mogbil, V. (2012). Proofs as Executions. In: Baeten, J.C.M., Ball, T., de Boer, F.S. (eds) Theoretical Computer Science. TCS 2012. Lecture Notes in Computer Science, vol 7604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33475-7_20
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DOI: https://doi.org/10.1007/978-3-642-33475-7_20
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