Skip to main content Accessibility help
×
×
Home

Constructing weak simulations from linear implications for processes with private names

  • Ross Horne (a1) (a2) and Alwen Tiu (a3)
Abstract

This paper clarifies that linear implication defines a branching-time preorder, preserved in all contexts, when used to compare embeddings of process in non-commutative logic. The logic considered is a first-order extension of the proof system BV featuring a de Morgan dual pair of nominal quantifiers, called BV1. An embedding of π-calculus processes as formulae in BV1 is defined, and the soundness of linear implication in BV1 with respect to a notion of weak simulation in the π-calculus is established. A novel contribution of this work is that we generalise the notion of a ‘left proof’ to a class of formulae sufficiently large to compare embeddings of processes, from which simulating execution steps are extracted. We illustrate the expressive power of BV1 by demonstrating that results extend to the internal π-calculus, where privacy of inputs is guaranteed. We also remark that linear implication is strictly finer than any interleaving preorder.

Copyright
Corresponding author
*Corresponding author. Email: ross.horne@uni.lu
References
Hide All
Abramsky, S. (1994). Proofs as processes. Theoretical Computer Science 135 (1) 59.
Ahn, K. Y., Horne, R. and Tiu, A. (2017). A characterisation of open bisimilarity using an intuitionistic modal logic. In: Meyer, R., and Nestmann, U. (eds.) 28th International Conference on Concurrency Theory (CONCUR 2017), Leibniz International Proceedings in Informatics (LIPIcs), vol. 85, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 7:1–7:17.
Andreoli, J.-M. (1992). Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2 (3) 297347.
Bellin, G. and Scott, P. J. (1994). On the pi-calculus and linear logic. Theoretical Computer Science 135 (1) 1165.
Bengtson, J. and Parrow, J. (2009). Formalising the pi-calculus using nominal logic. Logical Methods in Computer Science 5 (2), 136.
Bernardi, G. and Hennessy, M. (2013). Mutually testing processes. In: International Conference on Concurrency Theory, Lecture Notes in Computer Science, vol. 8052, Springer, pp. 6175.
Brünnler, K. and Tiu, A. F. (2001). A local system for classical logic. In: Nieuwenhuis, R. and Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning, Lecture Notes in Computer Science, Springer, 347361.
Bruscoli, P. (2002). A purely logical account of sequentiality in proof search. In: Logic Programming, 18th International Conference, ICLP 2002, Copenhagen, Denmark, July 29–August 1, Proceedings, Lecture Notes in Computer Science, vol. 2401, Springer, 302316.
Caires, L., Pfenning, F. and Toninho, B. (2016). Linear logic propositions as session types. Mathematical Structures in Computer Science 26 (3) 367423.
Chaudhuri, K., Guenot, N. and Straßburger, L. (2011). The focused calculus of structures. In: Bezem, M. (ed.) Computer Science Logic (CSL 2011) - 25th International Workshop/20th Annual Conference of the EACSL, Leibniz International Proceedings in Informatics, vol. 12, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 159173.
Ciobanu, G. and Horne, R. (2015). Behavioural analysis of sessions using the calculus of structures. In: Mazzara, M. and Voronkov, A. (eds.) Perspectives of System Informatics, Springer International Publishing, 91106.
Deng, Y., Van Glabbeek, R., Hennessy, M., Morgan, C. and Zhang, C. (2007). Characterising testing preorders for finite probabilistic processes. In: 22nd Annual IEEE Symposium on Logic in Computer Science, LICS 2007, IEEE, 313325.
Deniélou, P. and Yoshida, N. (2013). Multiparty compatibility in communicating automata: characterisation and synthesis of global session types. In: Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8–12, Springer, 174186.
Gacek, A., Miller, D. and Nadathur, G. (2011). Nominal abstraction. Information and Computation 209 (1) 4873.
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50 (1) 1112.
Gischer, J. L. (1988). The equational theory of pomsets. Theoretical Computer Science 61 199224.
Guglielmi, A. (2007). A system of interaction and structure. ACM Transactions on Computational Logic 8 (1) 164.
Guglielmi, A. and Straßburger, L. (2001). Non-commutativity and MELL in the calculus of structures. In: Fribourg, L. (ed.) Computer Science Logic, Lecture Notes in Computer Science, Springer, 5468.
Guglielmi, A. and Straßburger, L. (2011). A system of interaction and structure V: the exponentials and splitting. Mathematical Structures in Computer Science 21 (03) 563584.
Horne, R. (2015). The consistency and complexity of multiplicative additive system virtual. Scientific Annals of Computer Science 25 (2) 245316.
Horne, R., Mauw, S. and Tiu, A. (2017). Semantics for specialising attack trees based on linear logic. Fundamenta Informaticae 153 5786.
Horne, R., Tiu, A., Aman, B. and Ciobanu, G. (2016). Private names in non-commutative logic. In: Desharnais, J. and Jagadeesan, R. (eds.) 27th International Conference on Concurrency Theory (CONCUR 2016), Leibniz International Proceedings in Informatics (LIPIcs), vol. 59, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 31:1–31:16.
Horne, R., Tiu, A., Aman, B. and Ciobanu, G. (2018). Polarised Nominal Quantifiers Model Private Names in Non-commutative Logic. Technical Report 1502. ISSN 1842-1490, extended version of above supporting submission to TOCL. http://iit.iit.tuiasi.ro/TR/reports/fml1502.pdf.
McDowell, R., Miller, D. and Palamidessi, C. (2003). Encoding transition systems in sequent calculus. Theoretical Computer Science 294 (3) 411437.
Miller, D. (1993). The pi-calculus as a theory in linear logic: preliminary results. In: Extensions of Logic Programming, Third International Workshop, ELP 1992, Bologna, Italy, February 26–28, 1992, Proceedings, Lecture Notes in Computer Science, vol. 660, Springer, 242264.
Miller, D., Nadathur, G., Pfenning, F. and Scedrov, A. (1991). Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic 51 (1) 125157.
Miller, D. and Tiu, A. (2005). A proof theory for generic judgements. ACM Transactions on Computational Logic (TOCL) 6 (4) 749783.
Milner, R. (1989). Communication and Concurrency, Prentice-Hall International.
Milner, R., Parrow, J. and Walker, D. (1992). A calculus of mobile processes, I and II. Information and Computation 100 (1) 177.
Pitts, A. (2003). Nominal logic, a first order theory of names and binding. Information and Computation 186 (2) 165193.
Retoré, C. (1997). Pomset logic: a non-commutative extension of classical linear logic. In: de Groote, P. and Roger Hindley, J. (eds.) Typed Lambda Calculi and Applications, Lecture Notes in Computer Science, Springer, 300318.
Roversi, L. (2016). A deep inference system with a self-dual binder which is complete for linear lambda calculus. Journal of Logic and Computation 26 (2) 677.
Sangiorgi, D. (1996a). π-calculus, internal mobility, and agent-passing calculi. Theoretical Computer Science 167 (1) 235274.
Sangiorgi, D. (1996b). A theory of bisimulation for the π-calculus. Acta Informatica 33 (1) 6997.
Sassone, V., Nielsen, M. and Winskel, G. (1996). Models for concurrency: towards a classification. Theoretical Computer Science 170 (1–2) 297348.
Straßburger, L. and Guglielmi, A. (2011). A system of interaction and structure IV: the exponentials and decomposition. ACM Transactions on Computational Logic (TOCL) 12 (4) 23.
Tiu, A. (2006). A system of interaction and structure II: the need for deep inference. Logical Methods in Computer Science 2 (2:4) 124.
Tiu, A. and Miller, D. (2010). Proof search specifications of bisimulation and modal logics for the π-calculus. ACM Transactions on Computational Logic 11 (2) 13.
van Glabbeek, R. (1990). The linear time-branching time spectrum (extended abstract). In: CONCUR 1990, Amsterdam, The Netherlands, August 27–30, Lecture Notes in Computer Science, vol. 458, Springer, 278297.
van Glabbeek, R. and Goltz, U. (2001). Refinement of actions and equivalence notions for concurrent systems. Acta Informatica 37 (4–5) 229327.
van Glabbeek, R. and Vaandrager, F. (1987). Petri net models for algebraic theories of concurrency. In: de Bakker, J. W., Nijman, A. J., and Treleaven, P. C. (eds.) PARLE Parallel Architectures and Languages Europe, Lecture Notes in Computer Science, volume 259, Springer, 224242.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed