Abstract
Wide reactive systems are rewriting systems specified by wide reaction rules, where redex and reactum are lists of terms (forests), i.e. rules of the form \(\langle l_1(\mathbf {x}_1),\dots ,l_n(\mathbf {x}_n) \rangle \Rightarrow \langle r_1(\mathbf {y}_1),\dots ,r_n(\mathbf {y}_n) \rangle \) such that \(\cup _i \mathbf {y}_i \subseteq \cup _i \mathbf {x}_i\). Wide reaction rules are particularly useful for process calculi for mobile and global computations, because they allow one to connect processes which can be at different places in the system, possibly crossing boundaries and firewalls. For instances, remote procedure calls can be modeled as a process in place \(i\) activating a reaction in a different place \(j\); code mobility can be modeled by instantiating variables in \(\mathbf {y}_i\) with terms using variables from \(\mathbf {x}_j\), for a different \(j\); etc.
In order to apply a wide reaction rule, we have to find a matching of the rule redex within the global state. This problem can be restated as follows: how to match a given forest (the redex) inside an unordered tree (the system), possibly finding the subtrees to be grafted at the forest’s leaves (i.e., instantiating the variables)? We show that, although the problem is NP-complete in general, the exponential explosion depends only on the number \(n\) of roots of the forest (the width of the redex), and not on the size of the global tree (the system state). In most practical cases, the width is constant and small (i.e., \(\le 3\)), hence our results show that the wide reaction systems can be actually used for process calculi.
This work is partially supported by MIUR PRIN project 2010LHT4KM, CINA.
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References
Bacci, G., Grohmann, D., Miculan, M.: Bigraphical models for protein and membrane interactions. In: Ciobanu, G. (ed.) Proc. MeCBIC. Electronic Proceedings in Theoretical Computer Science, vol. 11, pp. 3–18 (2009)
Bezem, M., Klop, J.W., de Vrijer, R.: Term rewriting systems. CUP (2003)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbius: fast subset convolution. In: Proc. STOC 2007, pp. 67–74. ACM (2007)
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On Problems without Polynomial Kernels (Extended Abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 563–574. Springer, Heidelberg (2008)
Boreale, M., Bruni, R., De Nicola, R., Loreti, M.: Sessions and Pipelines for Structured Service Programming. In: Barthe, G., de Boer, F.S. (eds.) FMOODS 2008. LNCS, vol. 5051, pp. 19–38. Springer, Heidelberg (2008)
Cardelli, L., Gordon, A.D.: Mobile Ambients. In: Nivat, M. (ed.) FOSSACS 1998. LNCS, vol. 1378, pp. 140–155. Springer, Heidelberg (1998)
Cook, S.A.: The complexity of theorem-proving procedures. In: Proc. STOC 1971, pp. 151–158. ACM (1971)
Damgaard, T.C., Glenstrup, A.J., Birkedal, L., Milner, R.: An inductive characterization of matching in binding bigraphs. Formal Aspects of Computing 25(2), 257–288 (2013)
Dörr, H.: Efficient graph rewriting and its implementation, vol. 922. LNCS. Springer (1995)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness I: Basic results. SIAM J. Comput. 24(4), 873–921 (1995)
Ferrari, G.-L., Hirsch, D., Lanese, I., Montanari, U., Tuosto, E.: Synchronised Hyperedge Replacement as a Model for Service Oriented Computing. In: de Boer, F.S., Bonsangue, M.M., Graf, S., de Roever, W.-P. (eds.) FMCO 2005. LNCS, vol. 4111, pp. 22–43. Springer, Heidelberg (2006)
Hopcroft, J.E., Tarjan, R.E.: A v\(^2\) algorithm for determining isomorphism of planar graphs. Inf. Process. Lett. 1(1), 32–34 (1971)
Hopcroft, J.E., Wong, J.K.: Linear time algorithm for isomorphism of planar graphs (preliminary report). In: Proc. STOC 1974, pp. 172–184. ACM (1974)
Jensen, O.H., Milner, R.: Bigraphs and transitions. In: Proc. POPL, pp. 38–49. ACM (2003)
Krivine, J., Milner, R., Troina, A.: Stochastic bigraphs. In: Proc. MFPS. Electronic Notes in Theoretical Computer Science, vol. 218, pp. 73–96 (2008)
Mansutti, A., Miculan, M., Peressotti, M.: Towards distributed bigraphical reactive systems. In: Echahed, R., Habel, A., Mosbah, M. (eds.) Proc. GCM 2014, p. 45 (2014)
Matula, D.W.: Subtree isomorphism in \(\mathop {O}(n^{5/2})\). Annals of Discrete Mathematics 2, 91–106 (1978)
Miculan, M., Peressotti, M.: A CSP implementation of the bigraph embedding problem. In: Proc. 1st International Workshop on Meta Models for Process Languages (MeMo) (2014)
Milner, R.: The Space and Motion of Communicating Agents. CUP (2009)
Serbanuta, T.-F., Rosu, G., Meseguer, J.: A rewriting logic approach to operational semantics. Inf. Comput. 207(2), 305–340 (2009)
Sevegnani, M., Unsworth, C., Calder, M.: A SAT based algorithm for the matching problem in bigraphs with sharing. Technical Report TR-2010-311, Department of Computer Science, University of Glasgow (2010)
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Bacci, G., Miculan, M., Rizzi, R. (2014). Finding a Forest in a Tree. In: Maffei, M., Tuosto, E. (eds) Trustworthy Global Computing. TGC 2014. Lecture Notes in Computer Science(), vol 8902. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45917-1_2
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DOI: https://doi.org/10.1007/978-3-662-45917-1_2
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