Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-11T00:24:41.528Z Has data issue: false hasContentIssue false

Petri nets based on Lawvere theories

Published online by Cambridge University Press:  09 November 2020

Jade Master*
Affiliation:
University of California Riverside, Department of Mathematics, Riverside, CAUSA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a definition of Q-net, a generalization of Petri nets based on a Lawvere theory Q, for which many existing variants of Petri nets are a special case. This definition is functorial with respect to change in Lawvere theory, and we exploit this to explore the relationships between different kinds of Q-nets. To justify our definition of Q-net, we construct a family of adjunctions for each Lawvere theory explicating the way in which Q-nets present free models of Q in Cat. This gives a functorial description of the operational semantics for an arbitrary category of Q-nets. We show how this can be used to construct the semantics for Petri nets, pre-nets, integer nets, and elementary net systems.

Type
Paper
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - SA
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Adámek, J. and Porst, H.-E. (1998). Algebraic theories of quasivarieties. Journal of Algebra 208 (2) 379398. Available at http://www.math.uni-bremen.de/porst/dvis/AlgTheocorr.pdf.CrossRefGoogle Scholar
Adámek, J. and Rosický, J. (1994). Locally Presentable and Accessible Categories, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Baez, J. C. and Master, J. (2020). Open petri nets. Mathematical Structures in Computer Science 30 (3) 314341. Available at http://dx.doi.org/10.1017/S0960129520000043.CrossRefGoogle Scholar
Barr, M. and Wells, C. (1985). Toposes, Triples and Theories, Springer-Verlag, New York.CrossRefGoogle Scholar
Bartoletti, M., Cimoli, T. and Michele Pinna, G. (2015). Lending petri nets. Science of Computer Programming 112 75101. Available at https://www.sciencedirect.com/science/article/pii/S0167642315001021.CrossRefGoogle Scholar
Bartoletti, M. and Zunino, R. (2010). A calculus of contracting processes. In: 25th Annual IEEE Symposium on Logic in Computer Science, IEEE, 332341.Google Scholar
Baues, H.-J., Jibladze, M. and Tonks, A. (1997). Cohomology of monoids in monoidal categories. Contemporary Mathematics 202 137166.CrossRefGoogle Scholar
Betti, R. (1996) Formal theory of internal categories. Le Matematiche 51 (3) 3552. Available at https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/456.Google Scholar
Borceux, F. (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures, Cambridge University Press, Cambridge.Google Scholar
Bruni, R., Meseguer, J., Montanari, U. and Sassone, V. (2001). Functorial models for petri nets. Information and Computation 170 (2) 207236.CrossRefGoogle Scholar
Buckley, M. (2008). Lawvere theories. Available at http://web.science.mq.edu.au/street/MitchB.pdf.Google Scholar
Degano, P., Meseguer, J. and Montanari, U. (1989). Axiomatizing net computations and processes. In: Fourth Annual Symposium on Logic in Computer Science, 1989. LICS’89, Proceedings, IEEE, 175185.CrossRefGoogle Scholar
Dubuc, E. (1968). Adjoint triangles. In: Reports of the Midwest Category Seminar II, Springer, 6991.CrossRefGoogle Scholar
Gabriel, P. and Ulmer, F. (2006). Lokal Präsentierbare Kategorien, Springer-Verlag, New York.Google Scholar
Genovese, F. R. and Herold, J. (2018). Executions in (semi-)integer petri nets are compact closed categories. Available at arXiv:1805.05988.Google Scholar
Joyal, A. and Street, R. (1991). The geometry of tensor calculus, I. Advances in Mathematics 88 (1) 55112.CrossRefGoogle Scholar
Lack, S. (2010). Note on the construction of free monoids. Applied Categorical Structures 18 (1) 1729.CrossRefGoogle Scholar
Linton, F. E. J. (1966). Some aspects of equational categories. In: Proceedings of the Conference on Categorical Algebra, Springer, 8494.CrossRefGoogle Scholar
Master, J. (2019). Generalized petri nets. The n-Category Café. Available at https://golem.ph.utexas.edu/category/2019/04/generalized_petri_nets.html.Google Scholar
Meseguer, J. and Montanari, U. (1990). Petri nets are monoids. Information and Computation 88 (2) 105155.CrossRefGoogle Scholar
Petri, C. A. (1966). Communication with Automata. Phd thesis, Universität Hamburg.Google Scholar
Rozenberg, G. and Thiagarajan, P. S. (1986). Petri nets: Basic notions, structure, behaviour. In: Current Trends in Concurrency, Springer, 585668.CrossRefGoogle Scholar
Rydeheard, D. E. and Burstall, R. M. (1988). Computational Category Theory, Prentice Hall, New Jersey.Google Scholar
Sassone, V. (1995). On the category of petri net computations. In: Colloquium on Trees in Algebra and Programming, Springer, 334348.Google Scholar
Sassone, V. (1996). An axiomatization of the algebra of petri net concatenable processes. Theoretical Computer Science 170 (1–2) 277296.CrossRefGoogle Scholar
The Petri Net bibliography. (2019). Accessed: 2019-04-04. Available at http://www.informatik.uni-hamburg.de/TGI/PetriNets/bibliographies/aboutpnbibl.html.Google Scholar
William Lawvere, F. (1963). Functorial Semantics of Algebraic Theories . Phd thesis, Columbia University. Available at http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html.Google Scholar