Partial order semantics of Box expressions

  • Maciej Koutny
Full Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 815)


We develop a partial order semantics for the process expressions underlying the Petri Box Calculus. We aim at a semantics which would be equivalent to the standard partial order semantics of the Petri nets (Boxes) corresponding to such expressions. The solution we present is a variant of step sequence semantics in which actions are annotated with an additional information about the relative position of the parts of the expression from which they were derived, as first proposed by Degano, De Nicola and Montanari. This information is then used to capture all essential causal dependencies among actions, leading to the definition of a partial order of action occurrences. To represent Petri net markings within process expressions we employ an overbarring and underbarring technique which is related to that used in the event systems due to Boudol and Castellani. The partial order operational model turns out to be consistent with that defined in the Petri net theory. More precisely, if an expression can execute a partial order then the same holds for the corresponding Petri Box. The converse holds for all guarded expressions.


Causality partial order theory of concurrency net-based algebraic calculi structured operational semantics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.C.M.Baeten and W.P.Weijland: Process Algebra. Cambridge Tracts in Theoretical Computer Science (1990).Google Scholar
  2. 2.
    E.Best, R.Devillers and J.Esparza: General Refinement and Recursion Operators in the Box Calculus. Proc. of STACS-93, Springer-Verlag Lecture Notes in Computer Science Vol. 665, 130–140 (1993).Google Scholar
  3. 3.
    E.Best, R.Devillers and J.Hall: The Petri Box Calculus: a New Causal Algebra with Multilabel Communication. Advances in Petri Nets (ed. G.Rozenberg), Springer-Verlag Lecture Notes in Computer Science Vol.609, 21–69 (1992).Google Scholar
  4. 4.
    E.Best and R.P.Hopkins: B(PN) 2 — a Basic Petri Net Programming Notation. Proc. of PARLE-93, Springer-Verlag Lecture Notes in Computer Science Vol. 694, 379–390 (1993).Google Scholar
  5. 5.
    E.Best and H.G.Linde-Göers: Compositional Process Semantics of Petri Boxes. Proc. of MFPS (Mathematical Foundations of Programming Semantics), Springer-Verlag Lecture Notes in Computer Science (1993).Google Scholar
  6. 6.
    G.Boudol: Notes on Algebraic Calculi of Processes. In: Logics and Models of Concurrent Systems. K.R.Apt (ed.), 261–304 (1985).Google Scholar
  7. 7.
    G.Boudol and I.Castellani: Flow Models of Distributed Computations: Event Structures and Nets. Rapport de Recherche, INRIA, Sophia Antipolis (July 1991).Google Scholar
  8. 8. Cindio, G.De Michelis, L.Pomello and C.Simone. Milner's Communicating Systems and Petri Nets. In: Selected Papers of 3rd European Workshop on Applications and Theory of Petri Nets, IF 66 (Springer-Verlag, Heidelberg), 40–59 (1983).Google Scholar
  9. 9.
    P.Degano, R.De Nicola and U.Montanari: A Distributed Operational Semantics for CCS Based on C/E Systems. Acta Informatica 26 (1988).Google Scholar
  10. 10.
    P.Degano, R.De Nicola and U.Montanari: Partial Order Derivations for CCS. In: Proc. FCT, Lecture Notes in Computer Science Vol.199, Springer Verlag, 520–533 (1985).Google Scholar
  11. 11.
    R. de Simone: Higher-level Synchronising Devices in MEIJE-SCCS. Theoretical Computer Science Vol.37, 245–267 (1985).CrossRefGoogle Scholar
  12. 12.
    R.J. van Glabbeek and F.V.Vaandrager: Petri Net Models for Algebraic Theories of Concurrency. Proc. PARLE'87, Lecture Notes in Computer Science Vol.259, Springer Verlag, 224–242 (1987).Google Scholar
  13. 13.
    U.Goltz: On Representing CCS Programs by Finite Petri Nets. Arbeitspapiere der GMD Nr.290 (February 1988).Google Scholar
  14. 14.
    U.Goltz and A.Mycroft: On the Relationships of CCS and Petri Nets. In: J.Paredaens (ed.), Proc. 11th ICALP, Lecture Notes in Computer Science Vol.154, Springer Verlag, 196–208 (1984).Google Scholar
  15. 15.
    J.Hall, R.P.Hopkins and O.Botti: A Petri Box Semantics of occam. Advances in Petri Nets (ed. G.Rozenberg), Springer-Verlag Lecture Notes in Computer Science Vol.609, 179–214 (1992).Google Scholar
  16. 16.
    M.Koutny, J.Esparza and E.Best: Operational Semantics for the Petri Box Calculus. Hildesheimer Informatik-Berichte 13/93 (October 1993).Google Scholar
  17. 17.
    W.Li and P.E. Lauer: Using the Structural Operational Approach to Express True Concurrency. Technical Report 85-01, Departmant of Computer Science and Systems, McMaster University (1985).Google Scholar
  18. 18.
    H.G.Linde-Göers: Compositional Branching Processes of Petri Boxes. Ph.D. Thesis, Universität Hildesheim (October 1993).Google Scholar
  19. 19.
    D.May: occam. SIGPLAN Notices, Vol.l8(4), 69–79 (April).Google Scholar
  20. 20.
    R.Milner: Communication and Concurrency. Prentice Hall (1989).Google Scholar
  21. 21.
    M.Nielsen and P.S.Thiagarajan: Degrees of Nondeterminism and Concurrency. Proc. of 4th Conf. on Foundations of Software Technology and Theoretical Computer Science, Springer-Verlag Lecture Notes in Computer Science Vol.181 (eds. M.Joseph and R.Shyamasundar), 89–117 (1984).Google Scholar
  22. 22.
    E.R.Olderog: Operational Petri Net Semantics for CCSP. In: G. Rozenberg (ed.), Advances in Petri Nets 1987, Springer-Verlag Lecture Notes in Computer Science, Vol. 266, 196–223 (1987).Google Scholar
  23. 23.
    G.Plotkin: A Structural Approach to Operational Semantics. Report DAIMI FN-19, Århus University, Computer Science Department, Aarhus, Denmark (1981).Google Scholar
  24. 24.
    D.Taubner: Finite Representations of CCS and TCSP by Automata and Petri Nets. Lecture Notes in Computer Science, Vol. 369, Springer Verlag (1989).Google Scholar
  25. 25.
    G.Winskel: Petri Nets, Algebras, Morphisms and Compositionality. Info. Control 72, 197–238 (1987).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Maciej Koutny
    • 1
  1. 1.Department of Computing ScienceUniversity of NewcastleNewcastle upon TyneUK

Personalised recommendations