Abstract
We present a model for the dynamics of discrete deterministic systems, based on an extension of the Petri nets framework. Our model relies on the definition of a priority relation between conflicting transitions, which is encoded by orienting the edges of a transition conflict graph. We provide a characterization in terms of a local consistency condition of those deterministic systems whose dynamic behavior can be encoded using our approach. Finally, we consider the problem of recognizing when an orientation of the transition conflict graph is valid for encoding the dynamic behavior of a system.
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References
Adam NR, Atluri V, Huang WK (1998) Modeling and analysis of workflows using Petri nets. J Intell Inf Syst 10(2): 131–158. doi:10.1023/A:1008656726700
Balbo G (2002) Introducation to stochastic Petri nets. In: Lectures on formal methods and performance analysis: first EEF/Euro summer school on trends in computer science, Springer, New York, pp 84–155
Billington J, Diaz M, Rozenberg G (1999) Application of Petri nets to communication networks, advances in Petri nets. Springer, London
Caprotti O, Ferscha A, Hong H (1995) Reachability test in Petri nets by Gröbner bases. RISC-Linz Report Series 95–03, Research Institute for Symbolic Computation, Linz
Chandler A, Heyworth A (1999) Gröbner basis procedures for testing Petri nets. UWB Math preprint 99.11, University of Wales, Wales. arXiv:math/0002119v1
Chaouiya C (2007) Petri net modelling of biological networks. Briefings in Bioinformatics pp bbm029+. doi:10.1093/bib/bbm029
Chaouiya C, Remy E, Thieffry D (2008) Petri net modelling of biological regulatory networks. J Discret algorithms 6(2): 165–177. doi:10.1016/j.jda.2007.06.003
David R, Alla H (2005) Discrete, continuous, and hybrid Petri nets. Springer, Berlin
Gu T, Bahri PA (2002) A survey of Petri net applications in batch processes. Comput Ind 47(1): 99–111. doi:10.1016/S0166-3615(01)00142-7
Hardy S, Robillard PN (2004) Modeling and simulation of molecular biology systems using Petri nets: modeling goals of various approaches. J Bioinformatics Comput Biol 2(4): 619–637
Jensen K (1997) Coloured Petri nets: basic concepts, analysis methods and practical use, vol 3. Springer, New York
Koch I, Heiner M, (2008) Petri nets In: Junker BH, Schreiber F (eds) Analysis of biological networks, Wiley book series in bioinformatics, pp 139–180
Lipton R (1976) The reachability problem requires exponential space. Research Report 62, Computer Science Department, Yale University, USA
Marsan M, Balbo G, Donatelli S, Franceschinis G, Conte G (1995) Modelling with generalized stochastic Petri nets, Wiley series in parallel computing
Marwan W (2003) Theory of time-resolved somatic complementation and its use for the analysis of the sporulation control network of physarum polycephalum. Genetics 164: 105–115
Marwan W, Starostzik C (2002) The sequence of regulatory events in the sporulation control network of physarum polycephalum analysed by time-resolved somatic complementation of mutants. Protist 153: 391–400
Marwan W, Sujatha A, Starostzik C (2005) Reconstructing the regulatory network controlling commitment and sporulation in physarum polycephalum based on hierarchical Petri net modeling and simulation. J Theor Biol 236: 349–365
Marwan W, Wagler A, Weismantel R (2008) A mathematical approach to solve the network reconstruction problem. Math Methods Oper Res 67: 117–132
Matsuno H, Aoshima H, Doi A, Tanaka Y, Matsui M (2003) Biopathways representation and simulation on hybrid functional Petri net. In Silico Biol. 3(3): 389–404
Mayr EW (1981) An algorithm for the general Petri net reachability problem. In: Proceedings of the 13th Ann. ACM Symposium on theory of computing, ACM Press, New York, pp 238–246
Mayr EW (1984) An algorithm for the general Petri net reachability problem. SIAM J Comput 13(3): 441–460
Mayr EW (1995) On polynomial ideals, their complexity and applications. In: Reichel H (eds) Fundamentals of computation theory, no. 965 in Lecture Notes in Computer Science. Springer, Berlin, pp 89–105
Reisig W (1985) Petri nets: an introduction. Springer, New York
Reisig W (1998) Elements of distributed algorithms: modeling and analysis with Petri nets. Springer, New York
Runge T (2004) Application of coloured Petri nets in systems biology. In: Proceedings of the 5th workshop on CPN. University of Aarhus, Aarhus, pp 77–95
Torres LM, Wagler A (2010) Model reconstruction for discrete deterministic systems. Electron notes Discret Math 36: 175–182
Wagler A, Weismantel R (2009) The combinatorics of modeling and analyzing biological systems. Nat Comput (to appear). doi:10.1007/S11047-009-9165-5
Yakovlev A, Koelmans A, Semenov A, Kinniment D (1996) Modelling, analysis and synthesis of asynchronous control circuits using Petri nets. Integr VLSI J 21(3): 143–170. doi:10.1016/S0167-9260(96)00010-7
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Torres, L.M., Wagler, A.K. Encoding the dynamics of deterministic systems. Math Meth Oper Res 73, 281–300 (2011). https://doi.org/10.1007/s00186-011-0353-6
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DOI: https://doi.org/10.1007/s00186-011-0353-6