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Processes and unfoldings: concurrent computations in adhesive categories

Published online by Cambridge University Press:  26 June 2014

PAOLO BALDAN ,
ANDREA CORRADINI ,
TOBIAS HEINDEL ,
BARBARA KÖNIG  and
PAWEŁ SOBOCIŃSKI
PAOLO BALDAN
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, Italy Email: baldan@math.unipd.it
ANDREA CORRADINI
Affiliation:
Dipartimento di Informatica, Università di Pisa, Italy Email: andrea@di.unipi.it
TOBIAS HEINDEL
Affiliation:
School of Informatics, University of Edinburgh, Edinburgh, United Kingdom Email: tobias.heindel@googlemail.com
BARBARA KÖNIG
Affiliation:
Abteilung für Informatik und Angewandte Kognitionswissenschaft, Universität Duisburg-Essen, Germany Email: barbara_koenig@uni-due.de
PAWEŁ SOBOCIŃSKI
Affiliation:
ECS, University of Southampton, United Kingdom Email: ps@ecs.soton.ac.uk
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Abstract

We generalise both the notion of a non-sequential process and the unfolding construction (which was previously developed for concrete formalisms such as Petri nets and graph grammars) to the abstract setting of (single pushout) rewriting of objects in adhesive categories. The main results show that processes are in one-to-one correspondence with switch-equivalent classes of derivations, and that the unfolding construction can be characterised as a coreflection, that is, the unfolding functor arises as the right adjoint to the embedding of the category of occurrence grammars into the category of grammars.

As the unfolding represents potentially infinite computations, we need to work in adhesive categories with ‘well-behaved’ colimits of ω-chains of monos. Compared with previous work on the unfolding of Petri nets and graph grammars, our results apply to a wider class of systems, which is due to the use of a refined notion of grammar morphism.

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 24 , Issue 4: Structure Transformation , August 2014 , 240402
Copyright
Copyright © Cambridge University Press 2014 

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References

Baldan, P. (2000) Modelling Concurrent Computations: from Contextual Petri Nets to Graph Grammars, Ph.D. thesis, Dipartimento di Informatica, Università di Pisa.Google Scholar
Baldan, P., Busi, N., Corradini, A. and Pinna, G. M. (2004) Domain and event structure semantics for Petri nets with read and inhibitor arcs. Theoretical Computer Science 323 (1-3)129–189.CrossRefGoogle Scholar
Baldan, P., Chatain, T., Haar, S. and König, B. (2010) Unfolding-based diagnosis of systems with an evolving topology. Information and Computation 208 (10)1169–1192.Google Scholar
Baldan, P., Corradini, A., Heindel, T., König, B. and Sobociński, P. (2006) Processes for adhesive rewriting systems. In: Aceto, L. and Ingólfsdóttir, A. (eds.) FoSSaCS. Springer-Verlag Lecture Notes in Computer Science 3921 202–216.Google Scholar
Baldan, P., Corradini, A., Heindel, T., König, B. and Sobocinski, P. (2009) Unfolding grammars in adhesive categories. In: Kurz, A., Lenisa, M. and Tarlecki, A. (eds.) CALCO. Springer-Verlag Lecture Notes in Computer Science 5728 350–366.CrossRefGoogle Scholar
Baldan, P., Corradini, A. and König, B. (2008) A framework for the verification of infinite-state graph transformation systems. Information and Computation 206 869–907.Google Scholar
Baldan, P., Corradini, A. and Montanari, U. (2001) Contextual Petri nets, asymmetric event structures, and processes. Information and Computation 171 (1)1–49.Google Scholar
Baldan, P., Corradini, A., Montanari, U. and Ribeiro, L. (2007) Unfolding Semantics of Graph Transformation. Information and Computation 205 733–782.Google Scholar
Benveniste, A., Fabre, E., Haar, S. and Jard, C. (2003) Diagnosis of asynchronous discrete event systems, a net unfolding approach. IEEE Transactions on Automatic Control 48 (5)714–727.Google Scholar
Birkhoff, G. (1967) Lattice Theory, American Mathematical Society.Google Scholar
Braatz, B., Ehrig, H., Gabriel, K. and Golas, U. (2010) Finitary $\mathcal{M}$-adhesive categories. In: Ehrig, H., Rensink, A., Rozenberg, G. and Schürr, A. (eds.) Proceedings, Graph Transformations – 5th International Conference, ICGT 2010. Springer-Verlag Lecture Notes in Computer Science 6372 234–249.Google Scholar
Cockett, R. and Guo, X. (2007) Join restriction categories and the importance of being adhesive. Unpublished manuscript, slides from CT 07.Google Scholar
Corradini, A., Heindel, T., Hermann, F. and König, B. (2006) Sesqui-pushout rewriting. In: Corradini, A., Ehrig, H., Montanari, U., Ribeiro, L. and Rozenberg, G. (eds.) Proceedings, Graph Transformations – Third International Conference, ICGT 2006. Springer-Verlag Lecture Notes in Computer Science 4178 30–45.Google Scholar
Corradini, A., Hermann, F. and Sobociński, P. (2008) Subobject transformation systems. Applied Categorical Structures 16 (3)389–419.Google Scholar
Corradini, A., Montanari, U. and Rossi, F. (1996) Graph processes. Fundamenta Informaticae 26 241–265.CrossRefGoogle Scholar
Dyckhoff, R. and Tholen, W. (1987) Exponentiable morphisms, partial products and pullback complements. Journal of Pure and Applied Algebra 49 103–116.Google Scholar
Ehrig, H. (1979) Introduction to the algebraic theory of graph grammars. In Claus, V., Ehrig, H. and Rozenberg, G. (eds.) Graph-Grammars and Their Application to Computer Science and Biology. Springer-Verlag Lecture Notes in Computer Science 73 1–69.Google Scholar
Ehrig, H., Ehrig, K., Prange, U. and Taentzer, G. (2006) Fundamentals of Algebraic Graph Transformation, EATCS Monographs in Theoretical Computer Science, Springer-Verlag.Google Scholar
Ehrig, H., Golas, U. and Hermann, F. (2010) Categorical Frameworks for Graph Transformation and HLR Systems based on the DPO Approach. Bulletin of the EATCS 102 111–121.Google Scholar
Ehrig, H., Habel, A., Kreowski, H.-J. and Parisi-Presicce, F. (1991) Parallelism and Concurrency in High-Level Replacement Systems. Mathematical Structures in Computer Science 1 361–404.CrossRefGoogle Scholar
Ehrig, H., Pfender, M. and Schneider, H. (1973) Graph-grammars: an algebraic approach. In Proceedings of IEEE Conference on Automata and Switching Theory 167–180.CrossRefGoogle Scholar
Goltz, U. and Reisig, W. (1983) The Non-sequential Behaviour of Petri Nets. Information and Computation 57 125–147.Google Scholar
Habel, A., Müller, J. and Plump, D. (2001) Double-pushout graph transformation revisited. Mathematical Structures in Computer Science 11 (5)637–688.Google Scholar
Hayman, J. and Winskel, G. (2008) The unfolding of general Petri nets. In Proceedings of FSTTCS '08, Leibniz International Proceedings in Informatics 2.Google Scholar
Heindel, T. (2009) A Category Theoretical Approach to the Concurrent Semantics of Rewriting: Adhesive Categories and Related Concepts, Ph.D. thesis, Universität Duisburg-Essen.Google Scholar
Heindel, T. (2010) Hereditary Pushouts Reconsidered. In: Ehrig, H., Rensink, A., Rozenberg, G. and Schürr, A. (eds.) Proceedings, Graph Transformations – 5th International Conference, ICGT 2010. Springer-Verlag Lecture Notes in Computer Science 6372 250–265.Google Scholar
Heindel, T. and Sobociński, P. (2009) Van Kampen colimits as bicolimits in Span. In: Proceedings of CALCO '09. Springer-Verlag Lecture Notes in Computer Science 5728 335–349.Google Scholar
Hermann, F., Corradini, A., Ehrig, H. and König, B. (2010) Efficient analysis of permutation equivalence of graph derivations based on Petri nets. In: Proceedings of GT-VMT '10 (Workshop on Graph Transformation and Visual Modeling Techniques). Electronic Communications of the EASST 29.Google Scholar
Johnstone, P. (2002) Sketches of an Elephant volume 1, Oxford Science Publications.Google Scholar
Johnstone, P., Lack, S. and Sobociński, P. (2007) Quasitoposes, quasiadhesive categories and Artin glueing. In: Proceedings CALCO'07. Springer-Verlag Lecture Notes in Computer Science 4626 312–326.Google Scholar
Lack, S. and Sobociński, P. (2005) Adhesive and quasiadhesive categories. Theoretical Informatics and Applications 39 (2)511–546.Google Scholar
Lack, S. and Sobociński, P. (2006) Toposes are adhesive. In: Corradini, A., Ehrig, H., Montanari, U., Ribeiro, L. and Rozenberg, G. (eds.) Proceedings, Graph Transformations, Third International Conference, ICGT 2006. Springer-Verlag Lecture Notes in Computer Science 4178 184–198.Google Scholar
Löwe, M. (1993) Algebraic approach to single-pushout graph transformation. Theoretical Computer Science 109 181–224.CrossRefGoogle Scholar
Löwe, M. (2010) Graph rewriting in span-categories. In: Ehrig, H., Rensink, A., Rozenberg, G. and Schürr, A. (eds.) Proceedings, Graph Transformations – 5th International Conference, ICGT 2010. Springer-Verlag Lecture Notes in Computer Science 6372.Google Scholar
McMillan, K. (1993) Symbolic Model Checking, Kluwer.CrossRefGoogle Scholar
Meseguer, J., Montanari, U. and Sassone, V. (1997) On the semantics of Place/Transition Petri nets. Mathematical Structures in Computer Science 7 (4)359–397.CrossRefGoogle Scholar
Robinson, E. and Rosolini, G. (1988) Categories of partial maps. Information and Computation 79 (2)95–130.Google Scholar
Vogler, W., Semenov, A. and Yakovlev, A. (1998) Unfolding and finite prefix for nets with read arcs. In: Sangiorgi, D. and de Simone, R. (eds.) Proceedings of CONCUR '98. Springer-Verlag Lecture Notes in Computer Science 1466 501–516.Google Scholar
Winskel, G. (1987a) Event structures. In: Petri Nets: Applications and Relationships to Other Models of Concurrency. Springer-Verlag Lecture Notes in Computer Science 255 325–392.Google Scholar
Winskel, G. (1987b) Petri nets, algebras, morphisms, and compositionality. Information and Computation 72 (3)197–238.Google Scholar