Abstract
Sesqui-pushout rewriting is an algebraic graph transformation approach that provides mechanisms for vertex cloning. If a vertex gets cloned, the original and the copy obtain the same context, i.e. all incoming and outgoing edges of the original are copied as well. This behaviour is not satisfactory in practical examples which require more control over the context cloning process. In this paper, we provide such a control mechanism by allowing each transformation rule to refine the underlying type graph. We discuss the relation to the existing approaches to controlled sesqui-pushout vertex cloning, elaborate a basic theoretical framework, and demonstrate its applicability by a practical example.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For a proof of the if-part see [9].
- 2.
- 3.
Compare for example [11].
- 4.
Given arbitrary morphism \(a:P\rightarrow Q\) and monomorphism \(b:Q\rightarrowtail S\), the final pullback complement \((c:P\rightarrowtail R,d:R\rightarrow S)\) is constructed as follows, compare [2]: Vertices: \(R_{V}=P_{V}\uplus \left( S_{V}-b_{V}(Q_{V})\right) \), \(c_{V}=\mathrm {id}_{P_{V}}\), \(d_{V}=b_{V}(a_{V}(v))\) if \(v\in P_{V}\) and \(d_{V}=\mathrm {id}_{S_{V}}\) otherwise. Edges: \(R_{E}\) contains \(P_{E}\) and an edge “copy” \(\left( v,e,v'\right) \) for every edge \(e\in S_{E}-b_{E}(Q_{E})\) and pair of vertices \(v,v'\in R_{V}\) with \(s_{S}(e)=d_{V}(v)\mathrm {\, and\,}t_{S}(e)=d_{V}(v')\) with the following structure: \(s_{R}(v,e,v')=v\) and \(s_{R}(e)=s_{P}(e)\) if \(e\in P_{E}\), \(t_{R}(v,e,v')=v'\) and \(t_{R}(e)=t_{P}(e)\) if \(e\in P_{E}\), \(c_{E}=\mathrm {id}_{P_{E}}\), and \(d_{E}(v,e,v')=e\) and \(d_{E}(e)=b_{E}(a_{E}(e))\) if \(e\in P_{E}\).
- 5.
The typing is indicated by the graphical symbols.
- 6.
Note the edit-rule copies the node labelled “6, 7” from the left- to the right-hand side, and the publish-rule copies the node labelled “2, 3” from left to right.
- 7.
The objects of \(\mathcal {C}^{\rightarrow }\) are all morphisms of \(\mathcal {C}\) and a morphism i from \(f:A\rightarrow B\) to \(g:C\rightarrow D\) is a pair \((i_{A}:A\rightarrow C,i_{B}:B\rightarrow D)\) of \(\mathcal {C}\)-morphism such that \(i_{B}\circ f=g\circ i_{A}\).
- 8.
\(\mathcal {C}\!\downarrow \! T\) is the restriction of \(\mathcal {C}^{\rightarrow }\) to morphisms with co-domain T. Note that every category \(\mathcal {C}\) in our set-up is equivalent to \(\mathcal {C}\downarrow \mathrm {F}\), where \(\mathrm {F}\) is the final object in \(\mathcal {C}\).
- 9.
I.e. both components are epimorphisms.
- 10.
Morphism \(d_{X}\) is monic, since decomposition of pushouts provides that \((d_{X},r\left\langle m\right\rangle _{A})\) is pushout of \((r^{*},d)\), and pushouts preserve monomorphisms due to Fact F5.
- 11.
Note that the object \(G''\) cannot be typed in the refined type of the rule, since it contains unrefined parts, namely the parts outside m(L).
- 12.
Compare [10].
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, Heidelberg (2015)
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)
Duval, D., Echahed, R., Prost, F.: Graph transformation with focus on incident edges. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenberg, G. (eds.) ICGT 2012. LNCS, vol. 7562, pp. 156–171. Springer, Heidelberg (2012)
Ehrig, H., Ehrig, K., Prange, U., Taentzer, G.: Fundamentals of Algebraic Graph Transformation. Springer, Heidelberg (2006)
Ehrig, H., Pfender, M., Schneider, H.J., Graph-grammars: an algebraic approach. In: FOCS, pp. 167–180. IEEE (1973)
Kennaway, R.: Graph rewriting in some categories of partial morphisms. In: Ehrig, H., Kreowski, H.-J., Rozenberg, G. (eds.) Graph-Grammars and Their Application to Computer Science. LNCS, vol. 532, pp. 490–504. Springer, Heidelberg (1990)
Lack, S., Sobocinski, P.: Adhesive and quasiadhesive categories. ITA 39(3), 511–545 (2005)
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. Technical report 2010/02, FHDW-Hannover (2010)
Löwe, M.: A unifying framework for algebraic graph transformation. Technical report 2012/03, FHDW-Hannover (2012)
Löwe, M.: Polymorphic sesqui-pushout graph rewriting. Technical report 2014/02, FHDW-Hannover (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Löwe, M. (2016). Sesqui-Pushout Rewriting with Type Refinements. In: Echahed, R., Minas, M. (eds) Graph Transformation. ICGT 2016. Lecture Notes in Computer Science(), vol 9761. Springer, Cham. https://doi.org/10.1007/978-3-319-40530-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-40530-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-40529-2
Online ISBN: 978-3-319-40530-8
eBook Packages: Computer ScienceComputer Science (R0)