Normal Forms for Partitions and Relations

  • Roberto Bruni
  • Fabio Gadducci
  • Ugo Montanari
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1589)


Recently there has been a growing interest towards algebraic structures that are able to express formalisms different from the standard, tree-like presentation of terms. Many of these approaches reveal a specic interest towards their application in the “distributed and concurrent systems” field, but an exhaustive comparison between them is diffcult because their presentations can be quite dissimilar. This work is a first step towards a unified view, which is able to recast all those formalisms into a more general one, where they can be easily compared. We introduce a general schema for describing a characteristic normal form for many algebraic formalisms, and show that those normal forms can be thought of as arrows of suitable concrete monoidal categories.


Normal Form Partial Order Monoidal Category Sequential Composition Partition Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Abramsky, S. Gay, and R. Nagarajan. Interaction categories and the foundations of typed concurrent programming. In Deductive Program Design, Nato ASI Series, pages 403–442. Springer Verlag, 1994.Google Scholar
  2. 2.
    Z.M. Ariola and J.W. Klop. Equational term graph rewriting. Fundamenta Informaticae, 26:207–240, 1996.MathSciNetzbMATHGoogle Scholar
  3. 3.
    J.A. Bergstra and G. Stefanescu. Network algebra for synchronous and asynchronous dataflow. Technical Report Logic Group Preprint Series n. 122, Department of Philosophy, University of Utrecht, 1994.Google Scholar
  4. 4.
    R. Bruni, J. Meseguer, and U. Montanari. Process and term tile logic. Technical Report SRI-CSL-98-06, SRI International, 1998.Google Scholar
  5. 5.
    V.A. Cazanescu and G. Stefanescu. Classes of nite relations as initial abstract data types I. Discrete Mathematics, 90:233–265, 1991.CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    V.E. Cazanescu and G. Stefanescu. Towards a new algebraic foundation of flowchart scheme theory. Fundamenta Informaticae, 13:171–210, 1990.MathSciNetzbMATHGoogle Scholar
  7. 7.
    A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Category Theory and Computer Science, volume 1290 of LNCS. Springer Verlag, 1997.CrossRefGoogle Scholar
  8. 8.
    A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gsmonoidal categories. Applied Categorical Structures, 1998. To appear.Google Scholar
  9. 9.
    A. Corradini and F. Gadducci. Functorial semantics for multi-algebras. Presented at WADT’98, 1998.Google Scholar
  10. 10.
    P. Degano, J. Meseguer, and U. Montanari. Axiomatizing the algebra of net computations and processes. Acta Informatica, 33:641–647, 1996.CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    G. Ferrari and U. Montanari. A tile-based coordination view of the asynchronous π-calculus. In Mathematical Foundations of Computer Science, volume 1295 of LNCS, pages 52–70. Springer Verlag, 1997.Google Scholar
  12. 12.
    G. Ferrari and U. Montanari. Tiles for concurrent and located calculi. In C. Palamidessi and J. Parrow, editors, Expressiveness in Concurrency, volume 7 of Electronic Notes in Computer Science. Elsevier Sciences, 1997.Google Scholar
  13. 13.
    F. Gadducci and R. Heckel. An inductive view of graph transformation. In F. Parisi-Presicce, editor, Recent Trends in Algebraic Development Techniques, volume 1376 of LNCS, pages 219–233. Springer Verlag, 1998.CrossRefGoogle Scholar
  14. 14.
    F. Gadducci and U. Montanari. Axioms for contextual net processes. In Automata, Languages and Programming, volume 1443 of LNCS, pages 296–308. Springer Verlag, 1998.CrossRefGoogle Scholar
  15. 15.
    F. Gadducci and U. Montanari. The tile model. In G. Plotkin, C. Stirling, and M. Tofte, editors, Proof, Language and Interaction: Essays in Honour of Robin Milner. MIT Press, 1999. to appear.Google Scholar
  16. 16.
    M. Hasegawa. Recursion from cyclic sharing: Traced monoidal categories and models of cyclic lambda-calculus. In Typed Lambda Calculi and Applications, volume 1210 of LNCS, pages 196–213. Springer Verlag, 1997.CrossRefGoogle Scholar
  17. 17.
    H.-J. Hoenke. On partial recursive de nitions and programs. In M. Karpinski, editor, Fundamentals of Computations Theory, volume 56 of LNCS, pages 260–274. Springer Verlag, 1977.CrossRefGoogle Scholar
  18. 18.
    P. Katis, N. Sabadini, and R.F.C. Walters. Bicategories of processes. Journal of Pure and Applied Algebra, 115:141–178, 1997.CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    P. Katis, N. Sabadini, and R.F.C. Walters. SPAN(Graph): A categorical algebra of transition systems. In M. Johnson, editor, Algebraic Methodology and Software Technology, volume 1349 of LNCS, pages 307–321. Springer Verlag, 1997.CrossRefGoogle Scholar
  20. 20.
    F.W. Lawvere. Functorial semantics of algebraic theories. Proc. National Academy of Science, 50:869–872, 1963.CrossRefzbMATHGoogle Scholar
  21. 21.
    S. Mac Lane. Categories for the working mathematician. Springer Verlag, 1971.Google Scholar
  22. 22.
    J. Meseguer. Membership algebra as a logical framework for equational specification. In F. Parisi-Presicce, editor, Recent Trends in Algebraic Development Techniques, volume 1376 of LNCS. Springer Verlag, 1998.Google Scholar
  23. 23.
    J. Meseguer and U. Montanari. Mapping tile logic into rewriting logic. In F. Parisi-Presicce, editor, Recent Trends in Algebraic Development Techniques, volume 1376 of LNCS. Springer Verlag, 1998.Google Scholar
  24. 24.
    R. Milner. Calculi for interaction. Acta Informatica, 33:707–737, 1996.CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    U. Montanari and F. Rossi. Contextual nets. Acta Informatica, 32:545–596, 1995.CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    U. Montanari and F. Rossi. Graph rewriting, constraint solving and tiles for coordinating distributed systems. Applied Categorical Structures, 1998. To appear.Google Scholar
  27. 27.
    M. Pfender. Universal algebra in s-monoidal categories. Technical Report 9522, University of Munich-Department of Mathematics, 1974.Google Scholar
  28. 28.
    J. Power and E. Robinson. Premonoidal categories and notions of computation. Mathematical Structures in Computer Science, 7:453–468, 1998.CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    E. Robinson and G. Rosolini. Categories of partial maps. Information and Computation, 79:95–130, 1988.CrossRefGoogle Scholar
  30. 30.
    V. Sassone. On the Semantics of Petri Nets: Processes, Unfolding and Infinite Computations. PhD thesis, University of Pisa-Department of Computer Science, 1994.Google Scholar
  31. 31.
    V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170:277–296, 1996.CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    M.R. Sleep, M.J. Plasmeijer, and M.C. van Eekelen, editors. Term Graph Rewriting: Theory and Practice. Wiley, London, 1993.zbMATHGoogle Scholar
  33. 33.
    G. Stefanescu. On flowchart theories: Part II. The nondeterministic case. Theoret. Comput. Sci., 52:307–340, 1987.CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    G. Stefanescu. Algebra of flownomials. Technical Report SFB-Bericht 342/16/94 A, Technical University of München, Institut für Informatik, 1994.Google Scholar

Copyright information

© IFIP International Federation for Information Processing 1999

Authors and Affiliations

  • Roberto Bruni
    • 1
  • Fabio Gadducci
    • 2
  • Ugo Montanari
    • 1
  1. 1.Dipartimento di InformaticaUniv. of PisaPisaItalia
  2. 2.TU Berlin, Fach. 13 InformatikBerlinGermany

Personalised recommendations