Calculi of net structures and sets are similar
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 42 (2008) no. 2, p. 271-284
Three basic operations on labelled net structures are proposed: synchronised union, synchronised intersection and synchronised difference. The first of them is a version of known parallel composition with synchronised actions identically labelled. The operations work analogously to the ordinary union, intersection and difference on sets. It is shown that the universe of net structures with these operations is a distributive lattice and - if infinite pre/post sets of transitions are allowed - even a Boolean algebra. As a consequence, some representation theorems of this algebra are stated. The primitive objects are atomic net structures containing one transition with at most one pre-place or post-place (but not both). A simple example of a production system constructed by making use of the operations (and its transformations) is given. Some remarks on behavioural properties of compound nets are stated, in particular, how some constructing strategies may help to infer liveness. The latter issue is limited to semantics of place/transition nets without weights on arrows and with unbounded capacity of places and is not extensively investigated, since the main objective is focused on a calculus of net structures.
DOI : https://doi.org/10.1051/ita:2007033
Classification:  68Q85
Keywords: net structures, synchronised operations, distributive lattice, boolean algebra, representation theorems
@article{ITA_2008__42_2_271_0,
     author = {Czaja, Ludwik},
     title = {Calculi of net structures and sets are similar},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {2},
     year = {2008},
     pages = {271-284},
     doi = {10.1051/ita:2007033},
     zbl = {1144.68042},
     mrnumber = {2401262},
     language = {en},
     url = {http://www.numdam.org/item/ITA_2008__42_2_271_0}
}
Czaja, Ludwik. Calculi of net structures and sets are similar. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 42 (2008) no. 2, pp. 271-284. doi : 10.1051/ita:2007033. http://www.numdam.org/item/ITA_2008__42_2_271_0/

[1] G. Berthelot, Checking properties of nets using transformations, in Advances in Petri Nets, edited by G. Goos and J. Hartmanis. Lect. Notes Comput. Sci. 222 (1985). | MR 864510 | Zbl 0606.68055

[2] E. Best, R. Devillers, M. Koutny, The box algebra = Petri nets + process expressions. Inform. Comput. 178 (2002) 44-100. | MR 1931737 | Zbl 1012.68117

[3] L. Czaja, Making Nets Abstract and Structured, in Advances in Petri Nets, edited by G. Goos and J. Hartmanis. Lect. Notes Comput. Sci. 222 (1985) 181-202. | MR 864518 | Zbl 0612.68054

[4] L. Czaja, Equations for message passing. Fund. Inform. 72 (2006) 81-93. | MR 2265155 | Zbl 1097.68088

[5] L. Czaja, Interpreted nets. Fund. Inform. 79 (2007) 283-293. | MR 2346248 | Zbl 1124.68067

[6] P. Degano, J. Meseguer and U. Montanari, Axiomatising the algebra of net computations and processes. Acta Inform. 33 (1996) 641-667. | MR 1412418 | Zbl 0849.68087

[7] J. Engelfriet, Branching processes of Petri nets. Acta Inform. 28 (1991) 575-591. | MR 1120112 | Zbl 0743.68106

[8] R. Gorrieri, Refinement, atomicity and transactions for process description languages. Ph.D. Thesis. Dipartimento di Informatica, Universita di Pisa, TD - 2/91 (1991).

[9] C.A.R. Hoare, Notes on Communicating Sequential Processes. Oxford University Computing Laboratory Technical Monograph PRG-33 (1983).

[10] K. Kuratowski and A. Mostowski, Set Theory. North Holland, Amsterdam, PWN, Warsaw (1967). | Zbl 0165.01701

[11] A. Mazurkiewicz, Semantics of concurrent systems: a modular fixed point trace approach. Internal Report, Institute of Applied Mathematics and Computer Science, University of Leiden, The Netherlands (1984). | MR 807209 | Zbl 0576.68044

[12] A. Mazurkiewicz, Introduction to Trace Theory, in The Book of Traces, edited by V. Diekert and G. Rozenberg, World Scientific (1995) 3-41. | MR 1478993

[13] J. Meseguer and U. Montanari. Petri nets are monoids. Inform. Comput. 88 (1990) 105-155. | MR 1070245 | Zbl 0711.68077

[14] J. Meseguer, U. Montanari and V. Sassone, On the Semantics of Place/Transition Petri Nets. Dipartimento di Informatica Universita di Pisa, TR - 27/92 (1992).

[15] R. Milner, Communication and Concurrency. International Series in Computer Science, Prentice Hall (1989). | Zbl 0683.68008

[16] H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics. PWN, Warsaw (1968). | Zbl 0122.24311

[17] W. Reisig, Petri Nets, An Introduction. EATCS Monographs on Theoretical Computer Science, Springer Verlag (1985). | MR 782303 | Zbl 0555.68033

[18] M.H. Stone, The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40 (1936) 37-111. | JFM 62.0033.04 | MR 1501865

[19] H. Wimmel and L. Priese, Algebraic characterisation of Petri net pomset semantics, CONCUR'97: Concurrency Theory. Lect. Notes Comput. Sci. 1243 (1997) 403-420.

[20] J. Winkowski, An algebraic description of system behaviours. Theoret. Comput. Sci. 21 (1982) 315-340. | MR 680920 | Zbl 0493.68049

[21] G. Winskel, Petri nets, algebras, morphisms and compositionality. Inform. Comput. 72 (1987) 197-238. | MR 878462 | Zbl 0622.68052