Abstract
Starting from the algebraic view of Petri nets as monoids (as advocated by Meseguer and Montanari in [MM90]) we present the marking graphs of place transition nets as free monoid graphs and the marking graphs of specific elementary nets as powerset graphs. These are two important special cases of a general categorical version of Petri nets based on a functor M, called M-nets. These nets have a compositional marking graph semantics in terms of F-graphs, a generalization of free monoid and powerset graphs. Moreover we are able to characterize those F-graphs, called reflexive F-graphs, which are realizable by corresponding M-nets. The main result shows that the behavior and realization constructions are adjoint functors leading to an equivalence of the categories MNet of M-nets and RFGraph of reflexive F-graphs. This implies that the behavior construction preserves colimits so that the marking graph construction using F-graphs is compositional.
In addition to place transition nets and elementary nets we provide other interesting applications of M-nets and F-graphs. Moreover we discuss the relation to classical elementary net systems. The behavior and realization constructions we have introduced are compatible with corresponding constructions for elementary net systems (with initial state) and elementary transition systems in the sense of [NRT92].
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References
M. A. Bednarczyk and A. M. Borzyszkowski. On Concurrent Realization of Reactive Systems and Their Morphisms. In H. Ehrig, G. Juhás, J. Padberg, and G. Rozenberg, editors, Advances in Petri Nets: Unifying Petri Nets, LNCS. Springer, 2001.
E. Badouel and Ph. Darondeau. Theory of regions. In W. Reisig and G. Rozenberg, editors, Lectures on Petri Nets: Basic Models, pages 529–586. Springer, LNCS 1491, 1998.
E. Best, H. Fleischhack, W. Fraczak, R.P. Hopkins, H. Klaudel, and E. Pelz. An M-Net Semantics of B(PN) 2. In J. Desel, editor, Structures in Concurrency Theory, pages 85–100. Springer-Verlag, 1995.
P. Darondeau. Region Based Synthesis of P/T-Nets and Its Potential Applications. In M. Nielsen and D. Simpson, editors, 21st International Conference on Application and Theory of Petri Nets (ICATPN 2000), volume 1825 of Lecture Notes in Computer Science, pages 16–23. Springer-Verlag, 2000.
J. Desel, G. Juhás, and R. Lorenz. Petri Nets over Partial Algebras. In H. Ehrig, G. Juhás, J. Padberg, and G. Rozenberg, editors, Advances in Petri Nets: Unifying Petri Nets, LNCS. Springer, 2001.
C. Diamantini, S. Kasangian, L. Pomello, and C. Simone. Elementary Nets and 2-Categories. In E. Best and et al., editors, GMD-Studien Nr. 191; Hildesheimer Informatik-Berichte 6/91; 3rd Workshop on Concurrency and Compositionality, 1991, Goslar, Germany, pages 83–85. Gesellschaft für Mathematik und Datenverarbeitung mbH — Universität Hildesheim (Germany), Institut für Informatik, 1991.
J. Desel and W. Reisig. The synthesis problem of Petri nets. Acta Informatica, 33:297–315, 1996.
H. Ehrig and J. Padberg. A Uniform Approach to Petri Nets. In Ch. Freksa, M. Jantzen, and R. Valk, editors, Foundations of Computer Science: Potential-Theory-Cognition. Springer, LNCS 1337, 1997.
H. Ehrig, J. Padberg, and G. Rozenberg. Behaviour and Realization Construction for Petri NetsBased on Free Monoid and Power Set Graphs. Technical report, Technical University Berlin TR 94-15, 1994.
A. Ehrenfeucht and G. Rozenberg. Partial (Set) 2-Structures, Part I and II. Acta Informatica, 27:315–368, 1990.
J.A. Goguem. Realization is universal. Mathematical systems Theory, 6, 1973.
Gabriel Juhás. Reasoning about algebraic generalisation of Petri nets. In Proc. of 20th Conference on Theory and Application of Petri nets, pages 324–343. Springer, LNCS 1639, 1999.
J. Meseguer and U. Montanari. Petri Nets are Monoids. Information and Computation, 88(2):105–155, 1990.
U. Montanari and F. Rossi. Contextual occurrence nets and concurrent constraint programming. In H.-J. Schneider and H. Ehrig, editors, Proceedings of the Dagstuhl Seminar 9301 on Graph Transformations in Computer Science, volume 776 of Lecture Notes in Computer Science. Springer Verlag, 1994.
U. Montanari and F. Rossi. Contextual nets. Acta Informatica, 32, 1995. Also as Technical Report TR 4-93, Department of Computer Science, University of Pisa, February 1993.
M. Nielsen, G. Rozenberg, and P.S. Thiagarajan. Elementary transition systems. TCS, 96:3–33, 1992.
J. Padberg. Abstract Petri Nets: A Uniform Approach and Rule-Based Refinement. PhD thesis, Technical University Berlin, 1996. Shaker Verlag.
J. Padberg and H. Ehrig. Introduction to Parametrized Net Classes. In H. Ehrig, G. Juhás, J. Padberg, and G. Rozenberg, editors, Advances in Petri Nets: Unifying Petri Nets, LNCS. Springer, 2001.
G. Rozenberg. Behaviour of elementary net systems. In W. Brauer, W. Reisig, and G. Rozenberg, editors, Advances in Petri nets 1986, pages 60–94. Springer Verlag Berlin, 1987.
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Padberg, J., Ehrig, H., Rozenberg, G. (2001). Behavior and Realization Construction for Petri Nets Based on Free Monoid and Power Set Graphs. In: Ehrig, H., Padberg, J., Juhás, G., Rozenberg, G. (eds) Unifying Petri Nets. Lecture Notes in Computer Science, vol 2128. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45541-8_8
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DOI: https://doi.org/10.1007/3-540-45541-8_8
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