Graph Transformations in Computer Science pp 234-247 | Cite as

# Single pushout transformations of equationally defined graph structures with applications to actor systems

## Abstract

This work has practically been motivated by an approach of modelling actor systems using algebraic graph grammars. It turned out that essential requirements on graph structures modelling computational states could nicely be expressed as conditional equations.

These and other examples lead then to a general investigation of single pushout transformations within categories of equationally defined graph structures i.e., certain algebras satisfying a set of given equations, and partial morphisms. Fundamentally we characterize pushouts in these equationally defined categories as the corresponding pushouts in the supercategory of graph structures without equations if and only if the pushout object already satisfies the given equations. For labeled graph structures this characterization can be inherited from the unlabeled case, but only for a restricted class of equations. For a special kind of so-called local equations in particular, interesting graph transformation results carry over to the new setting. The use and the effects of such equations are illustrated and discussed for corresponding graph grammar modellings of a client/server problem considered as an actor system.

## Keywords

Algebraic Graph Transformations Equationally Defined Graph Structures Actor Systems## Preview

Unable to display preview. Download preview PDF.

## References

- [Agh88]G.A. Agha,
*ACTORS: A Model of Concurrent Computation in Distributed Systems*, The MIT Press Cambridge, Massachusetts, London, England. 1988.Google Scholar - [BEHL87]P. Boehm, H. Ehrig, U. Hummert, and M. Löwe.
*Towards distributed graph grammars*, In Ehrig et al. [ENRR87].Google Scholar - [BFH87]P. Böhm, H.-R. Fonio, and A. Habel,
*Amalgamation of graph transformations: a synchronization mechanism*, Journal of Computer and System Science**34**(1987), 377–408.CrossRefGoogle Scholar - [Cli81]W.D. Clinger,
*Foundations of Actor Semantics*, Ph.D. thesis, MIT Artificial Intelligence Laboratory, AI-TR-633, May 1981.Google Scholar - [CR93]A. Corradini and F. Rossi,
*On the power of contrext-free jungle rewriting for term rewriting systems and logic programming*. Term Graph Rewriting: Theory and Practice, John Wiley & Sons Ltd, 1993.Google Scholar - [EH86]H. Ehrig and A. Habel,
*Graph grammars with application conditions*, The Book of L (G. Rozenberg and A. Salomaa, eds.), Springer. 1986, pp. 87–100.Google Scholar - [EHKP92]H. Ehrig, A. Habel, H.-J. Kreowski, and F. Parisi-Presicce,
*Parallelism and concurrency in High Level Replacement Systems*, Mathematical Structures in Comp. Sci.**1**(1992), 361–404.Google Scholar - [Ehr79]H. Ehrig,
*Introduction to the algebraic theory of graph grammars*, 1st Int. Workshop on Graph Grammars and their Application to Computer Science and Biology, Lecture Notes in Computer Science 73, Springer, 1979, pp. 1–69.Google Scholar - [EK80]H. Ehrig and H.-J. Kreowski,
*Application of Graph Grammar Theory to Consistency, Synchronization and Scheduling in Data Base Systems*, Inform. Systems**5**(1980).Google Scholar - [EKL91]H. Ehrig, M. Korff, and M. Löwe,
*Tutorial introduction to the algebraic approach of graph grammars based on double and single pushouts*, In Ehrig et al. [EKR91], pp. 24–37.Google Scholar - [EKMS+81]H. Ehrig, H.-J. Kreowski, A. Maggiolo-Schettini, B. K. Rosen, and J. Winkowski,
*Transformation of structures: an algebraic approach*, Mathematical Systems Theory**14**(1981), 305–334.Google Scholar - [EKR91]H. Ehrig, H.-J. Kreowski, and G. Rozenberg (eds.),
*4th Int. Workshop on Graph Grammars and Their Application to Computer Science, Lecture Notes in Computer Science 532*, Springer, 1991.Google Scholar - [EL93]H. Ehrig and M. Löwe,
*Categorical Principles, Techniques and Results for High-Level Replacement Systems in Computer Science*, Applied Categorical Structures (to appear 1993).Google Scholar - [EM85]H. Ehrig and B. Mahr,
*Fundamentals of algebraic specifications*, EACTS-Monographs in Theoretical Computer Science, vol. 6, Springer, Berlin, 1985.Google Scholar - [ENRR87]H. Ehrig, M. Nagl, G. Rozenberg, and A. Rosenfeld (eds.),
*3rd Int. Workshop on Graph Grammars and Their Application to Computer Science, Lecture Notes in Computer Science 291*, Springer, 1987.Google Scholar - [EPS73]H. Ehrig, M. Pfender, and H. J. Schneider,
*Graph grammars: an algebraic approach*, 14th Annual IEEE Symposium on Switching and Automata Theory, 1973, pp. 167–180.Google Scholar - [Hab92]A. Habel,
*Hyperedge replacement: Grammars and languages*, Springer LNCS 643, Berlin, 1992.Google Scholar - [HP91]B. Hoffmann and D. Plump,
*Implementing term rewriting by jungle evaluation*, Informatique théorique et Applications/Theoretical Informatics and Applications (Berlin), vol. 25 (5), 1991, pp. 445–472.Google Scholar - [JLR90]D. Jansens, M. Lens, and G. Rozenberg,
*Computation graphs for actor grammars*, Tech. report, University of Leiden, 1990.Google Scholar - [JR80]D. Janssens and G. Rozenberg,
*On the structure of node-label controlled graph grammars*, Information Science**20**(1980), 191–216.CrossRefGoogle Scholar - [JR89]D. Jansens and G. Rozenberg,
*Actor grammars*, Mathematical Systems Theory**22**(1989), 75–107.Google Scholar - [JR91]D. Jansens and G. Rozenberg,
*Structured transformations and computation graphs for actor grammars*, In Ehrig et al. [EKR91], pp. 446–460.Google Scholar - [Kor91]M. Korff,
*Application of graph grammars to rule-based systems*, In Ehrig et al. [EKR91], pp. 505–519.Google Scholar - [Kor92]M. Korff,
*Algebraic Transformations of Equationally Definied Graph Structures*, Tech. Report 92/32, Technical University of Berlin, 1992, CA. 120 pages.Google Scholar - [Kor93]M. Korff,
*An Algebraic Graph Grammar Model for Actor Systems*, Tech. Report 93/12, Technical University of Berlin, 1993.Google Scholar - [KW87]H.-J. Kreowski and A. Wilharm,
*Is parallelism already concurrency? part 2: non-sequential processes in graph grammars*, 3^{rd}Int. Workshop on Graph Grammars and Their Application to Computer Science, Lecture Notes in Computer Science 291, Springer, 1987, pp. 361–377.Google Scholar - [LE91]M. Löwe and H. Ehrig,
*Algebraic approach to graph transformation based on single pushout derivations*, 16th Int. Workshop on Graph Theoretic Concepts in Computer Science, Lecture Notes in Computer Science 484, Springer, 1991, pp. 338–353.Google Scholar - [Löw90]M. Löwe,
*Extended algebraic graph transformation*, Ph.D. thesis, Technical University of Berlin, Department of Computer Science, 1990.Google Scholar - [Pad82]P. Padawitz,
*Graph grammars and opewrational semantics*, Theoretical Computer Science**19**(1982), 37–58.Google Scholar - [Sch91]A. Schürr,
*Operationales Spezifizieren mit programmierten Graphersetzungssystemen*, Deutscher Universitätsverlag GmbH, Wiesbaden, 1991.Google Scholar