# Tile transition systems as structured coalgebras?

## Abstract

The aim of this paper is to investigate the relation between two models of concurrent systems: tile rewrite systems and coalgebras. Tiles are rewrite rules with side eects which are endowed with operations of parallel and sequential composition and synchronization. Their models can be described as monoidal double categories. Coalgebras can be considered, in a suitable mathematical setting, as dual to algebras. They can be used as models of dynamical systems with hidden states in order to study concepts of observational equivalence and bisimilarity in a more general setting.

In order to capture in the coalgebraic presentation the algebraic structure given by the composition operations on tiles, coalgebras have to be endowed with an algebraic structure as well. This leads to the concept of *structured coalgebras*, i.e., coalgebras for an endofunctor on a category of algebras.

However, structured coalgebras are more restrictive than tile models. Those models which can be presented as structured coalgebras are characterized by the so-called *horizontal decomposition property*, which, intuitively, requires that the behavior is compositional in the sense that all transitions from complex states can be derived by composing transitions out of component states.

## Keywords

Transition System Operational Semantic Monoidal Category Label Transition System Horizontal Arrow## Preview

Unable to display preview. Download preview PDF.

## References

- 1.R. Bruni, F. Gadducci and U. Montanari, Normal Forms for Partitions and Relations, Proc. 13th Workshop on Algebraic Development Techniques, Lisbon, April 2–4, 1998, Springer LNCS, 1999, to appear.Google Scholar
- 2.R. Bruni, J. Meseguer and U. Montanari, Executable Tile Specications for Process Calculi, in: Jean-Pierre Finance, Ed., FASE’99, Springer LNCS 1577, pp. 60–76.Google Scholar
- 3.R. Bruni, J. Meseguer and U. Montanari, Symmetric Monoidal and Cartesian Double Categories as a Semantic Framework for Tile Logic, to appear in MSCS.Google Scholar
- 4.R. Bruni and U. Montanari, Cartesian Closed Double Categories, their Lambda-Notation, and the Pi-Calculus, Proc. LICS’99, to appear.Google Scholar
- 5.B. Bloom, S. Istrail, and A.R. Meyer. Bisimulation can’t be traced.
*Journal of the ACM*, 42(1):232–268, 1995.zbMATHCrossRefMathSciNetGoogle Scholar - 6.A. Corradini and A. Asperti. A categorical model for logic programs: Indexed monoidal categories. In
*Proceedings REX Workshop, Beekbergen, The Netherlands, June 1992*, volume 666 of LNCS. Springer Verlag, 1993.Google Scholar - 7.A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors,
*Category Theory and Computer Science*, volume 1290 of LNCS, pages 87–105. Springer Verlag, 1997.CrossRefGoogle Scholar - 8.A. Corradini, M. Groe-Rhode, and R. Heckel. Structured transition systems as lax coalgebras. In B. Jacobs, L. Moss, H. Reichel, and J. Rutten, editors,
*Proc. Of First Workshop on Coalgebraic Methods in Computer Science (CMCS’98), Lisbon, Portugal*, volume 11 of Electronic Notes of TCS. Elsevier Science, 1998. http://www.elsevier.nl/locate/entcs. - 9.A. Corradini, M. Groe-Rhode, and R. Heckel. An algebra of graph derivations using nite (co-) limit double theories. In J.L. Fiadeiro, editor,
*Proc. 13th Works-hop on Algebraic Development Techniques (WADT’98)*, volume 1589 of LNCS. Springer Verlag, 1999.Google Scholar - 10.A. Corradini, R. Heckel, and U. Montanari. From SOS specications to structured coalgebras: How to make bisimulation a congruence. In B. Jacobs and J. Rutten, editors,
*Proc. of Second Workshop on Coalgebraic Methods in Computer Science (CMCS’99), Amsterdam*, volume 19 of Electronic Notes of TCS. Elsevier Science, 1999. http://www.elsevier.nl/locate/entcs. - 11.A. Corradini and U. Montanari. An algebraic semantics for structured transition systems and its application to logic programs.
*Theoret. Comput. Sci.*, 103:51–106, 1992.zbMATHCrossRefMathSciNetGoogle Scholar - 12.R. De Simone. Higher level synchronizing devices in MEIJE-SCCS.
*Theoret. Comput. Sci.*, 37:245–267, 1985.zbMATHCrossRefMathSciNetGoogle Scholar - 13.C. Ehresmann,
*Catègories Structurèes*: I and II, Ann. Ec. Norm. Sup. 80, Paris (1963), 349–426; III, Topo. et Geo. di. V, Paris (1963).Google Scholar - 14.G. Ferrari and U. Montanari, Tile Formats for Located and Mobile Systems, Information and Computation, to appear.Google Scholar
- 15.F. Gadducci and R. Heckel. A inductive view of graph transformation. In F. Parisi Presicce, editor,
*Recent Trends in Algebraic Development Techniques*, LNCS 1376, pages 223–237. Springer Verlag, 1998.Google Scholar - 16.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. An early version appeared as Tech. Rep. TR-96/27, Dipartimento di Informatica, University of Pisa, 1996. Paper available from http://www.di.unipi.it/ gadducci/papers/TR-96-27.ps.gz. - 17.J.F. Groote and F. Vandraager. Structured operational semantics and bisimulation as a congruence.
*Information and Computation*, 100:202–260, 1992.zbMATHCrossRefMathSciNetGoogle Scholar - 18.R. Heckel.
*Open Graph Transformation Systems: A New Approach to the Compositional Modelling of Concurrent and Reactive Systems*. PhD thesis, TU Berlin, 1998.Google Scholar - 19.G.M. Kelly and R.H. Street. Review of the elements of 2-categories. In G.M. Kelly, editor,
*Sydney Category Seminar*, volume 420 of Lecture Notes in Mathematics, pages 75–103. Springer Verlag, 1974.Google Scholar - 20.F.W. Lawvere. Some algebraic problems in the context of functorial semantics of algebraic theories. In
*Proc. Midwest Category Seminar II*, number 61 in Springer Lecture Notes in Mathematics, pages 41–61, 1968.Google Scholar - 21.K.G. Larsen, L. Xinxin,
*Compositionality Through an Operational Semantics of Contexts*, in Proc. ICALP’90, LNCS 443, 1990, pp. 526–539.Google Scholar - 22.J. Meseguer. Conditional rewriting logic as a unied model of concurrency.
*TCS*, 96:73–155, 1992.zbMATHCrossRefMathSciNetGoogle Scholar - 23.J. Meseguer,
*Rewriting Logic as a Semantic Framework for Concurrency: A Progress Report*, in: U. Montanari and V. Sassone, Eds.,*CONCUR’96: Concurrency Theory*, Springer LNCS 1119, 1996, 331–372.Google Scholar - 24.J. Meseguer. Membership algebra as logical framework for equational specication. In F. Parisi Presicce, editor,
*Recent Trends in Algebraic Development Techniques*, LNCS 1376, pages 18–61. Springer Verlag, 1998.Google Scholar - 25.J. Meseguer and U. Montanari. Petri nets are monoids.
*Information and Computation*, 88(2):105–155, 1990.zbMATHCrossRefMathSciNetGoogle Scholar - 26.J. Meseguer and U. Montanari. Mapping tile logic into rewriting logic. In Francesco Parisi-Presicce, editor,
*Recent Trends in Algebraic Development Techniques*, number 1376 in Spinger LNCS, pages 62–91, 1998.Google Scholar - 27.R. Milner.
*Communication and Concurrency*. Prentice-Hall, 1989.Google Scholar - 28.R. Milner, J. Parrow, and D. Walker. A calculus of mobile processes.
*Information and Computation*, 100:1–77, 1992.zbMATHCrossRefMathSciNetGoogle Scholar - 29.U. Montanari and F. Rossi, Graph Rewriting, Constraint Solving and Tiles for Coordinating Distributed Systems, to appear in Applied Category Theory.Google Scholar
- 30.G. Plotkin. A structural approach to operational semantics. Technical Report DAIMI FN-19, Aarhus University, Computer Science Deapartment, 1981.Google Scholar
- 31.J.J.M.M. Rutten. Universal coalgebra: a theory of systems. Technical Report CS-R9652, CWI, 1996. To appear in TCS.Google Scholar
- 32.D. Turi and G. Plotkin. Towards a mathematical operational semantics. In
*Proc. of LICS’97*, pages 280–305, 1997.Google Scholar