Logic for Programming, Artificial Intelligence, and Reasoning

Logic for Programming, Artificial Intelligence, and Reasoning pp 146-161

An Adequate Compositional Encoding of Bigraph Structure in Linear Logic with Subexponentials

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9450)

Abstract

In linear logic, formulas can be split into two sets: classical (those that can be used as many times as necessary) or linear (those that are consumed and no longer available after being used). Subexponentials generalize this notion by allowing the formulas to be split into many sets, each of which can then be specified to be classical or linear. This flexibility increases its expressiveness: we already have adequate encodings of a number of other proof systems, and for computational models such as concurrent constraint programming, in linear logic with subexponentials (
). Bigraphs were proposed by Milner in 2001 as a model for ubiquitous computing, subsuming models of computation such as CCS and the \(\pi \)-calculus and capable of modeling connectivity and locality at the same time. In this work we present an encoding of the bigraph structure in
, thus giving an indication of the expressive power of this logic, and at the same time providing a framework for reasoning and operating on bigraphs. Our encoding is adequate and therefore the operations of composition and juxtaposition can be performed on the logical level. Moreover, all the proof-theoretical tools of
become available for querying and proving properties of bigraph structures.

References

  1. 1.
    Beauquier, M., Schürmann, C.: A bigraph relational model. In: LFMTP, vol. 71 of EPTCS, pp. 14–28 (2011)Google Scholar
  2. 2.
    Cardelli, L.: Brane calculi. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 257–278. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  3. 3.
    Cardelli, L., Gordon, A.D.: Mobile ambients. In: Nivat, M. (ed.) FOSSACS 1998. LNCS, vol. 1378, p. 140. Springer, Heidelberg (1998) CrossRefGoogle Scholar
  4. 4.
    Chaudhuri, K.: Classical and intuitionistic subexponential logics are equally expressive. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 185–199. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  5. 5.
    Chaudhuri, K.: Undecidability of multiplicative subexponential logic. In: 3rd LINEARITY, vol. 176 of EPTCS, pp. 1–8, July 2014Google Scholar
  6. 6.
    Danos, V., Joinet, J.-B., Schellinx, H.: The structure of exponentials: Uncovering the dynamics of linear logic proofs. In: Gottlob, G., Leitsch, A., Mundici, D. (eds.) Computational Logic and Proof Theory. LNCS, vol. 713, pp. 159–171. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  7. 7.
    Grohmann, D., Miculan, M.: Directed bigraphs. ENTCS 173, 121–137 (2007)Google Scholar
  8. 8.
    Jensen, O.H., Milner, R.: Bigraphs and mobile processes (revised). Technical Report UCAM-CL-TR-580, University of Cambridge, February 2004Google Scholar
  9. 9.
    Milner, R.: Pure bigraphs: Structure and dynamics. Inf. Comput. 204(1), 60–122 (2006)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Milner, R.: The Space and Motion of Communicating Agents. Cambridge University Press, Cambridge (2009)MATHCrossRefGoogle Scholar
  11. 11.
    Nigam, V.: Exploiting non-canonicity in the sequent calculus. Ph.D. thesis, Ecole Polytechnique, September 2009Google Scholar
  12. 12.
    Nigam, V., Miller, D.: Algorithmic specifications in linear logic with subexponentials. In: PPDP, pp. 129–140 (2009)Google Scholar
  13. 13.
    Nigam, V., Olarte, C., Pimentel, E.: On subexponentials, focusing and modalities in concurrent systems. Draft Manuscript submitted for publication (2015)Google Scholar
  14. 14.
    Nigam, V., Pimentel, E., Reis, G.: An extended framework for specifying and reasoning about proof systems. J. of Logic Comput. (2014). doi:10.1093/logcom/exu029, http://logcom.oxfordjournals.org/content/early/2014/06/06/logcom.exu029
  15. 15.
    Păun, G.: Membrane computing. Handbook of Natural Computing, pp. 1355–1377. Springer, Heidelberg (2012)MATHGoogle Scholar
  16. 16.
    Sevegnani, M., Calder, M.: Bigraphs with sharing. Theor. Comput. Sci. 577, 43–73 (2015)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Inria & LIX/École PolytechniquePalaiseauFrance

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