Incremental Consistency Checking of Heterogeneous Multimodels

Abstract

The local approaches to global consistency checking (GCC) of heterogeneous multimodels strive to reduce the model merging and matching workload within GCC. The paper’s contribution to such approaches is a framework allowing the user to do matching incrementally: to build the match required for checking the multimodel w.r.t. a new constraint, the user employs matches produced in previous GCC sessions.

This work is supported by the Automotive Partnership Canada via the Network on Engineering Complex Software Intensive Systems (NECSIS).

A Appendix. Some Operations Over Graphs and Models

Two operations over graphs and graph morphisms heavily employed in the paper are sketched below; a detailed specification can be found in, say, [3].

Coproducts. The coproduct $G_1 + G_2$ of two graphs $G_1, G_2$ is their disjoint union. Importantly, any coproduct is endowed with two canonic injections $i_k\!:\,G_k\,\hookrightarrow \,G_1+G_2$, $k=1,2$, which map each element to itself in the union.

Any pair of graph morphisms $f_{1,2}:G_{1,2}\rightarrow H$ gives rise to a unique morphism $[f_1,f_2]:G_1+G_2\rightarrow H$ compatible with injections: $i_{1,2}; [f_1,f_2] = f_{1,2}$. This property of coproducts is called universality and morphism specified above universal. It is easy to see that universality allows us to define the following operation over models (typed graphs): having typed graphs $A_1$, $A_2$ we define $A=A_1+A_2$ by setting $G_A=G_1+G_2$, $M_A=M_1+M_2$ and $\tau _A=[\tau _1;i_1,\tau _2;i_2]$ where $i_{1,2}\!:\,M_{1,2}\,\hookrightarrow \,M_A$ are coporduct injections.

Restriction/Retyping and Pullbacks. Given a model $A=(M,G,\tau )$ and a type graph map as shown in the inset diagram, we can define a new model $A'=(G',\tau ')$ over $M'$ by setting $G'=\left\{ {e=(t,x)}| \;{t\in M', x\in G, f(t)=\tau (x)}\right\}$ ⁵ with projection mappings $\tau '(e)=t$ and $f'(e)=x$. Further in the paper, we will often denote map $f'$ by $\tau ^*(f)$ and say that it is obtained by pulling f back along $\tau$, and similarly $\tau '=f^*(\tau )$ is obtained by pulling $\tau$ back along f; correspondingly, the entire operation of producing a span $(\tau ', f')$ from cospan $(\tau ,f)$ is called pull-back(PB) (of graphs).

If f is inclusion, then PB provides the (retyped) restriction of model A over the $M'$ part of the metamodel graph. Pullback operation can be seen as a generalization of model restriction for arbitrary mappings f, and we will often call it so. As any PB square is commutative, we can consider it as a special model morphism, which we will denote by a special arrow $f\!:A' \;{}^{\mathsf {pb}}\!\!\rightarrow A$.

Preservation properties. It is known that If f is inclusion, injective or surjective, then $f'$ is, resp., inclusion, injective or surjective as well.

Pullback composition and decomposition. Given $f\!:A \;{}^{\mathsf {pb}}\!\!\rightarrow B$ and $g\!:B \;{}^{\mathsf {pb}}\!\!\rightarrow C$, their composition is also PB, i.e., $f;g\!:A \;{}^{\mathsf {pb}}\!\!\rightarrow C$. Moreover, given that the second arrow and the composition are PBs, $f;g\!:A \;{}^{\mathsf {pb}}\!\!\rightarrow C$ and $g\!:B \;{}^{\mathsf {pb}}\!\!\rightarrow C$, it is possible to prove that the first arrow is also PB, $f\!:A \;{}^{\mathsf {pb}}\!\!\rightarrow B$.⁶

Coproducts and pullbacks (Extensivity). Given three typed graphs and morphism pairs , then if and only if $A_0 \cong A_1 + A_2$.