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On the Essence of Parallel Independence for the Double-Pushout and Sesqui-Pushout Approaches Read chapter Read first chapter

2018 | OriginalPaper | Chapter

Model Transformations as Free Constructions

Author : Michael Löwe

Published in: Graph Transformation, Specifications, and Nets

Publisher: Springer International Publishing

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Abstract

Hartmut Ehrig was an active researcher in Algebraic Specifications on the one hand and Graph and Model Transformations on the other hand. We demonstrate that these two research fields are closely connected, if we consider generating graph transformations only and use partial algebras instead of total algebras as the underlying category.

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previous chapter Towards a Navigational Logic for Graphical Structures
next chapter The Verigraph System for Graph Transformation
Footnotes
1
This transformation can serve as a prerequisite for the “compilation” of UML class diagrams to object-oriented programming languages like Java.
 
2
Up to isomorphism, if the semantics of transformations is using category theory.
 
3
For persistent free construction between abstract data types compare [5, 10, 11].
 
4
Like graphs, hypergraphs, graph structures [18], or general functor categories from a finite category to the category of sets and mappings.
 
5
Possibly without becoming total at last.
 
6
Alfred Horn, American mathematician, 1918 – 2011.
 
7
Compare [9], Example 3.6 on page 54.
 
8
This section uses material from [21].
 
9
The correspondence of the items in the left hand side of the rule to items in the right-hand side is always indicated by the visual correspondence in the layouts. In Fig. 1, we give an additional explicit definition of the mapping from left to right by index numbers. In most cases, this explicit indication is superfluous. Since mappings must be type-conform, i.e. classes can only be mapped to classes, attributes can only be mapped to attributes etc., and the mapping must respect the graphical structure, the mapping is uniquely determined for all rules that are depicted below. Thus, we do not use the index numbers in the following.
 
10
Standard key columns are unique and not null.
 
11
Standard foreign key columns are not null.
 
12
Merging of objects is expressed by non-injective rules.
 
13
Due to the merging, some of the generated junction tables for association may no longer accurately model the association’s semantics.
 
14
The rule cc2t is a simple modification of rule c2t in Fig. 1. Here, the class-to-table mapping is partial, since abstract classes are never mapped in CCT.
 
15
Rule cat2co also works for the own attributes of a concrete class due to the reflexivity rule \(\mathtt {t^{0}}\) in Fig. 4. This situation requires non-injective matches!
 
16
For the sake of simplicity of the presentation we do not generate a suitable check constraint in the relational schema.
 
17
A category has all small co-limits, if it has all co-limits for small diagram categories. A category is small if its collection of objects is a sets. As in [1], the family of morphisms in a category is a family of sets anyway.
 
18
Examples for such categories are all total or partial \(\varSigma \)-algebras for a given signature \(\varSigma \) or every epi-reflective sub-category of such categories of \(\varSigma \)-algebras, see below.
 
19
Neither rules nor matches are required to be monomorphisms. Rules and matches can be arbitrary morphisms.
 
20
Sub-diagram (1) in Fig. 7 denotes a direct derivation with rule \(r_{1}\) at match \(m_{1}\).
 
21
Object G being final wrt. system \(\mathbb {R}\) does not mean that there are no matches for rules in \(\mathbb {R}\) into G. But all these matches produce traces that are isomorphisms, i.e. do not have any effect.
 
22
The principle set-up follows [1], page 278 ff.
 
23
In this context, the objects L and R are called premise and conclusion respectively.
 
24
The morphism \(m^{r}\) is unique, if it exists, since r is epic.
 
25
\(A_{\mathbb {R}}\) is a small diagram, since the family of morphisms in a category is a family of sets, compare definition of categories in [1].
 
26
Given a family of sets \(A\!=\!\left( A_{s}\right) _{s\in S}\), \(k\ge 0\), and \(s_{1}\dots s_{k}\in S^{*}\), \(A^{s_{1}\dots s_{k}}=A_{s_{1}}\!\!\times \dots \times \!A_{s_{k}}\).
 
27
For a family of mappings \(f\!=\!\left( f_{s}\!:\!A_{s}\!\rightarrow \!B_{s}\right) _{s\in S}\), \(k\ge 0\), and \(w\!=\!s_{1}\dots s_{k}\in S^{*}\), \(f^{w}\!:\!A^{w}\!\rightarrow \!B^{w}\) is defined by \(f^{w}(x_{1},\dots ,x_{k})\!=\!\left( f_{s_{1}}\!(x_{1}),\dots ,f_{s_{k}}\!(x_{k})\right) \) for all \((x_{1},\dots x_{k})\!\in \!A^{w}\).
 
28
Compare [19].
 
29
Any two morphisms \(p,q:N\rightarrow M\) with \(p(k(x))=q(k(x))\) must coincide on all natural numbers due to the homomorphism condition (1).
 
30
It is easy to construct \(p,q:N\rightarrow M\) with \(p(k'(x))=q(k'(x))\) that differ at value 0.
 
31
Here, \(h(A)=\left( h_{s}(A_{s})\right) _{s\in S}\) and \(h_{s}(A_{s})=\{y\in B_{s}\mid y=h_{s}(x),x\in A_{s}\}\). For a proof of the proposition, compare [19].
 
32
An equivalence \(\equiv \) on a \(\varSigma \)-System A is a congruence, if \(f^{A}(x_{1},\dots x_{m})=(y_{1},\dots y_{n})\), \(f^{A}(x_{1}',\dots x_{m}')=(y_{1}',\dots y_{n}')\) and \(x_{i}\equiv x_{i}'\) for all \(1\le i\le m\) implies \(y_{j}\equiv y_{j}'\) for all \(1\le j\le n\) for all operations \(f\in O\).
 
33
Every variable set is a \(\varSigma \)-system with completely undefined operations!
 
34
Compare for example Theorem 99 (Homomorphism Theorem 1) in [19].
 
35
The notation indicates that we assume the same carrier for Basetype in CM and RS.
 
36
Again, Source and Target share sort Basetype with defined constant int.
 
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Metadata
Title
Model Transformations as Free Constructions
Author
Michael Löwe
Copyright Year
2018
Book
Graph Transformation, Specifications, and Nets

Print ISBN: 978-3-319-75395-9

Electronic ISBN: 978-3-319-75396-6

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-75396-6_8