Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T08:41:52.750Z Has data issue: false hasContentIssue false
  • English
  • Français

Calculi of net structures and sets are similar

Published online by Cambridge University Press:  06 September 2007

Ludwik Czaja
Ludwik Czaja*
Affiliation:
Institute of Informatics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland; [email protected]
Get access

Abstract

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.

Keywords

Net structuressynchronised operationsdistributive latticeBoolean algebrarepresentation theorems
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 42 , Issue 2 , April 2008 , pp. 271 - 284
Copyright
© EDP Sciences, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
Best, E., Devillers, R., Koutny, M., The box algebra = Petri nets + process expressions. Inform. Comput. 178 (2002) 44–100. CrossRef
Czaja, L., 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. CrossRef
Czaja, L., Equations for message passing. Fund. Inform. 72 (2006) 81–93.
Czaja, L., Interpreted nets. Fund. Inform. 79 (2007) 283–293.
Degano, P., Meseguer, J. and Montanari, U., Axiomatising the algebra of net computations and processes. Acta Inform. 33 (1996) 641–667. CrossRef
Engelfriet, J., Branching processes of Petri nets. Acta Inform. 28 (1991) 575–591. CrossRef
R. Gorrieri, Refinement, atomicity and transactions for process description languages. Ph.D. Thesis. Dipartimento di Informatica, Universita di Pisa, TD - 2/91 (1991).
C.A.R. Hoare, Notes on Communicating Sequential Processes. Oxford University Computing Laboratory Technical Monograph PRG-33 (1983).
K. Kuratowski and A. Mostowski, Set Theory. North Holland, Amsterdam, PWN, Warsaw (1967).
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).
A. Mazurkiewicz, Introduction to Trace Theory, in The Book of Traces, edited by V. Diekert and G. Rozenberg, World Scientific (1995) 3–41.
J. Meseguer and U. Montanari. Petri nets are monoids. Inform. Comput. 88 (1990) 105–155.
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).
R. Milner, Communication and Concurrency. International Series in Computer Science, Prentice Hall (1989).
H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics. PWN, Warsaw (1968).
W. Reisig, Petri Nets, An Introduction. EATCS Monographs on Theoretical Computer Science, Springer Verlag (1985).
M.H. Stone, The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40 (1936) 37–111.
Wimmel, H. and Priese, L., Algebraic characterisation of Petri net pomset semantics, CONCUR'97: Concurrency Theory. Lect. Notes Comput. Sci. 1243 (1997) 403–420.
Winkowski, J., An algebraic description of system behaviours. Theoret. Comput. Sci. 21 (1982) 315–340. CrossRef
Winskel, G., Petri nets, algebras, morphisms and compositionality. Inform. Comput. 72 (1987) 197–238. CrossRef