By Ulrike Golas

Graph and version differences play a principal position for visible modeling and model-driven software program improvement. in the final decade, a mathematical concept of algebraic graph and version modifications has been built for modeling, research, and to teach the correctness of adjustments. Ulrike Golas extends this conception for extra refined functions just like the specification of syntax, semantics, and version variations of advanced types. in keeping with M-adhesive transformation platforms, version alterations are effectively analyzed relating to syntactical correctness, completeness, useful habit, and semantical simulation and correctness. The built tools and effects are utilized to the non-trivial challenge of the specification of syntax and operational semantics for UML statecharts and a version transformation from statecharts to Petri nets maintaining the semantics.

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**Extra resources for Analysis and Correctness of Algebraic Graph and Model Transformations**

**Example text**

With e◦c∗ ◦a = c◦a = h ◦ c∗X ◦ a = h ◦ h ◦ b∗X = f ◦ a ◦ b∗X = f ◦ d ◦ e ◦ b∗X = e ◦ g ◦ e ◦ b∗X and e being a monomorphism it follows that c∗ ◦ a = g ◦ e ◦ b∗X . Pushout (7) implies that there is a unique i : H → F with c∗ = i ◦ c∗X and 3 M-Adhesive Transformation Systems 38 i ◦ h = g ◦ e . It further follows that e ◦ i = h using the pushout properties of H. By pushout decomposition, (8) is a pushout in C and the corresponding square in X\C is also a pushout. Therefore, (6) is an initial pushout over f in X\C.

Given the above commutative diagram, where (1) + (2) is a pushout, (2) is a pullback, w ∈ M, and (l ∈ M or k ∈ M), then (1) and (2) are pushouts and also pullbacks. 3. Cube pushout–pullback property. Given the above commutative cube (3), where all morphisms in the top and bottom faces are M-morphisms, the top face is a pullback, and the front faces are pushouts, then the following statement holds: the bottom face is a pullback if and only if the back faces of the cube are pushouts: 4. Uniqueness of pushout complements.

11 (Construction Theorem) If (C, M1 ), (D, M2 ), and (Cj , Mj ) for j ∈ J are M-adhesive categories, then also the following categories are M-adhesive categories: 1. 2 M-Adhesive Categories 27 pushouts along Mki -morphisms and Gi preserves pullbacks along M i morphisms, 2. any full subcategory (C , M1 |C ) of C, where pushouts and pullbacks along M1 are created and reﬂected by the inclusion functor, 3. the comma category (F, (M1 × M2 ) ∩ M orF ), with F = ComCat(F, G; I), where F : C → X preserves pushouts along M1 -morphisms and G : D → X preserves pullbacks along M2 -morphisms, 4.