Abstract
AGREE is a new approach to algebraic graph transformation. It provides sophisticated mechanisms for object cloning and non-local manipulations. It is for example possible to delete all edges in a graph by a single rule application. Due to these far-reaching effects of rule application, Church-Rosser results are difficult to obtain. Currently, there are only sufficient conditions for parallel independence, i.e. there are independent situations which are not classified independent by the existing conditions. In this paper, we characterise all independent situations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
For details compare Definition 2 below.
- 2.
This is mainly due to a pullback construction that is part of definition of rewrite in AGREE. For details compare Sect. 3.
- 3.
As in [2], we consider right-linear rules only.
- 4.
This is only half of the usually required adhesivity, compare [4].
- 5.
Compare [6, 7]. Facts F1 and F2 are consequences of the fact that all pushouts in a category with partial arrow classifiers are hereditary. A pushout \((f',g')\) of (g, f) as depicted in sub-diagram (1) of Fig. 2 is hereditary, if for all commutative situations as presented in Fig. 2 in which sub-diagrams (2) and (3) are pullbacks, and \(m,m_{P},\mathrm {\,and\,}m_{Q}\) are monomorphisms the following property is valid: the co-span \((f'_{m},g'_{m})\) is pushout of the span \((g_{m},f_{m})\), if and only if sub-diagrams (4) and (5) are pullbacks and \(m_{U}\) is monomorphism. Hereditariness implies stability of pushouts under pullbacks along monomorphisms. For some of the following results, however, we need requirement P3 which does not require monic \(m,m_{P},m_{Q},\mathrm {\,and\,}m_{U}\).
- 6.
Special case of F3 for \(d=\mathrm {id}_{B}\) and \(b=c\circ e\).
- 7.
Consequence of F3 for \(d=\mathrm {id}_{B}\).
- 8.
For the proof, see [11].
- 9.
Compare top part of Fig. 9.
- 10.
Compare left part of Fig. 9.
- 11.
Compare Fig. 9 where \((i\circ m')^{\bullet }\) is called \(m_{i}\)!.
- 12.
Equal up to isomorphism.
- 13.
These two lemmata demonstrate that gluing constructions in categories of abstract (right linear) spans possess the same composition and decomposition properties as simple pushouts in arbitrary categories.
- 14.
\((\pi _{2},\beta _{1})\) is pullback of \((\alpha _{1},t_{1})\), compare proof of Proposition 22.
References
Corradini, A., Duval, D., Echahed, R., Prost, F., Ribeiro, L.: AGREE – algebraic graph rewriting with controlled embedding. In: Parisi-Presicce, F., Westfechtel, B. (eds.) ICGT 2015. LNCS, vol. 9151, pp. 35–51. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21145-9_3
Corradini, A., Duval, D., Prost, F., Ribeiro, L.: Parallelism in AGREE transformations. In: Echahed, R., Minas, M. (eds.) ICGT 2016. LNCS, vol. 9761, pp. 37–53. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40530-8_3
Corradini, A., Heindel, T., Hermann, F., König, B.: Sesqui-pushout rewriting. In: Corradini, A., Ehrig, H., Montanari, U., Ribeiro, L., Rozenberg, G. (eds.) ICGT 2006. LNCS, vol. 4178, pp. 30–45. Springer, Heidelberg (2006). https://doi.org/10.1007/11841883_4
Ehrig, H., Ehrig, K., Prange, U., Taentzer, G.: Fundamentals of Algebraic Graph Transformation. MTCSAES. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-31188-2
Ehrig, H., Rensink, A., Rozenberg, G., Schürr, A. (eds.): Graph Transformations. LNCS, vol. 6372. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15928-2
Goldblatt, R.: Topoi. Dover Publications, Mineola (1984)
Heindel, T.: Hereditary pushouts reconsidered. In: Ehrig et al. [5], pp. 250–265 (2010)
Kennaway, R.: Graph rewriting in some categories of partial morphisms. In: Ehrig, H., Kreowski, H.-J., Rozenberg, G. (eds.) Graph Grammars 1990. LNCS, vol. 532, pp. 490–504. Springer, Heidelberg (1991). https://doi.org/10.1007/BFb0017408
Löwe, M.: Algebraic approach to single-pushout graph transformation. Theor. Comput. Sci. 109(1&2), 181–224 (1993)
Löwe, M.: Graph rewriting in span-categories. In: Ehrig et al. [5], pp. 218–233 (2010)
Löwe, M.: Characterisation of parallel independence in agree-rewriting. Technical report 2018/01, FHDW Hannover (2018)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Löwe, M. (2018). Characterisation of Parallel Independence in AGREE-Rewriting. In: Lambers, L., Weber, J. (eds) Graph Transformation. ICGT 2018. Lecture Notes in Computer Science(), vol 10887. Springer, Cham. https://doi.org/10.1007/978-3-319-92991-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-92991-0_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92990-3
Online ISBN: 978-3-319-92991-0
eBook Packages: Computer ScienceComputer Science (R0)