Graph Rewriting and Relabeling with PBPO$^{+}$

Abstract

We extend the powerful Pullback-Pushout (PBPO) approach for graph rewriting with strong matching. Our approach, called PBPO$^{+}$, exerts more control over the embedding of the pattern in the host graph, which is important for a large class of graph rewrite systems. In addition, we show that PBPO$^{+}$ is well-suited for rewriting labeled graphs and certain classes of attributed graphs. For this purpose, we employ a lattice structure on the label set and use order-preserving graph morphisms. We argue that our approach is simpler and more general than related relabeling approaches in the literature.

Appendices

Appendix

A PBPO$^{+}$

We will need both directions of the well known pullback lemma.

Lemma 45

(Pullback Lemma). Consider the diagram on the right. Suppose the right square is a pullback square and the left square commutes. Then the outer square is a pullback square iff the left square is a pullback square.    $\square$

Lemma 10

(Top-Left Pullback). In the rewrite step diagram of Definition 9, there exists a morphism $u : K \rightarrow G_K$ such that $L \xleftarrow {l} K \xrightarrow {u} G_K$ is a pullback for $L \xrightarrow {m} G_L \xleftarrow {g_L} G_K$, $t_K = u' \circ u$, and u is monic.

Proof

In the following diagram, u satisfying $t_K = u' \circ u$ and $m \circ l = g_L \circ u$ is inferred by using that $G_K$ is a pullback and commutation of the outer square.

By direction $\Longrightarrow$ of the pullback lemma (Lemma 45), the created square is a pullback square, and so by stability of monos under pullbacks, u is monic.    $\square$

Lemma 11

(Uniqueness of u). In the rewrite step diagram of Definition 9 (and in any category), there is a unique $v : K \rightarrow G_K$ such that $t_K = u' \circ v$.

Proof

In the following diagram, the top-right pullback is obtained using Lemma 10, and the top-left pullback is a rotation of the match diagram:

By direction $\Longleftarrow$ of the pullback lemma, $L \xleftarrow {1_L \circ l} K \xrightarrow {u} G_K$ is a pullback for the topmost outer square.

Now suppose that for a morphism $v : K \rightarrow G_K$, $t_K = u' \circ v$. Then $\alpha \circ g_L \circ u = l' \circ u' \circ u = l' \circ t_K = l' \circ u' \circ v = \alpha \circ g_L \circ v$. Hence both v and u make the topmost outer square commute. Hence there exists a unique x such that (simplifying) $l \circ x = l$ and $u \circ x = v$. From known equalities and monicity of $t_K$ we then derive

$$ \begin{array}{rcl} u \circ x &{} = &{} v \\ u' \circ u \circ x &{} = &{} u' \circ v \\ t_K \circ x &{} = &{} t_K \\ t_K \circ x &{} = &{} t_K \circ 1_K \\ x &{} = &{} 1_K\text {.} \end{array} $$

Hence $u = v$.    $\square$

Lemma 12

(Bottom-Right Pushout). Let $K' \xrightarrow {r'} R' \xleftarrow {t_R} R$ be a pushout for cospan $R \xleftarrow {r} K \xrightarrow {t_K} K'$ of rule $\rho$ in Definition 9. Then in the rewrite step diagram, there exists a morphism $w' : G_R \rightarrow R'$ such that $t_R = w' \circ w$, and $K' \xrightarrow {r'} R' \xleftarrow {w'} G_R$ is a pushout for $K' \xleftarrow {u'} G_K \xrightarrow {g_R} G_R$.

Proof

The argument is similar to the proof of Lemma 10, but now uses the dual statement of the pullback lemma.    $\square$

B Expressiveness of PBPO$^{+}$

Lemma 27

Let $\rho$ be a canonical PBPO rule, $G_L$ an object, and $m' \circ e : L \rightarrow G_L$ a match morphism for a mono $m'$ and epi e. We have:

$$\begin{aligned} G_{L} \Rightarrow _{\rho }^{(m' \mathop {\circ } e),\alpha } G_{R} \quad \iff \quad G_{L} \Rightarrow _{\rho _e}^{m',\alpha } G_{R}\;\text {.} \end{aligned}$$

Proof

By using the following commuting diagram:

   $\square$

Theorem 33

In locally small, strongly amendable categories in which every morphism f can be factored into an epi e followed by a mono m, any PBPO rule $\rho$ can be modeled by a set of PBPO$^{+}$ rules.

Proof

From Corollary 28 we obtain a set of PBPO rules S that collectively model $\rho$ using monic matching, and by Lemma 32 each $\sigma \in S$ can be modeled by a monic rule $\tau _\sigma$ with a strong matching rewrite policy. By Proposition 24, the set $\{ \tau _\sigma \mid \sigma \in S \}$ corresponds to a set of PBPO$^{+}$ rules.    $\square$

C Category Graph $^{(\mathcal {L},\le )}$

Lemma 38

Graph $^{(\mathcal {L},\le )}$ is strongly amendable.

Proof

Let UGraph refer to the category of unlabeled graphs.

Given a $t_L : L \rightarrow L'$ in Graph $^{(\mathcal {L},\le )}$, momentarily forget about the labels and consider the unlabeled version (overloading names) in UGraph. Because UGraph is a topos, it is strongly amendable. Thus we can obtain a factorization $L {\mathop {\rightarrowtail }\limits ^{t_L'}} L'' {\mathop {\rightarrow }\limits ^{\beta }} L'$ on the level of UGraph that witnesses strong amendability, i.e., for any factorization of $L {\mathop {\rightarrowtail }\limits ^{m}} G {\mathop {\rightarrow }\limits ^{\alpha }} L'$ of $t_L$ in UGraph, there exists an $\alpha '$ such that

commutes and the left square is a pullback square.

The idea now is to lift the bottom unlabeled factorization into Graph $^{(\mathcal {L},\le )}$. As far as graph structure is concerned, we know that it is a suitable factorization candidate. Then all that needs to be verified are the order requirements $\le$ on the labels.

For the lifting of $L''$, choose the graph in which every element has the same label as its image under $\beta$. Then clearly the lifting of $\beta$ of UGraph into Graph $^{(\mathcal {L},\le )}$ is well-defined, and so is the lifting of $t_L'$ (using that $t_L$ is well-defined in Graph $^{(\mathcal {L},\le )}$).

Now given any factorization $L {\mathop {\rightarrowtail }\limits ^{m}} G {\mathop {\rightarrow }\limits ^{\alpha }} L'$ in Graph $^{(\mathcal {L},\le )}$, lift the $\alpha '$ that is obtained by considering the factorization on the level of UGraph. Then the lifting of $\alpha '$ is well-defined by $\alpha = {\beta \circ \alpha '}$ and well-definedness of $\alpha$ and $\beta$ in Graph $^{(\mathcal {L},\le )}$. All that remains to be checked is that the left square is a pullback as far as the labels are concerned, i.e., whether for every $x \in V_L \cup E_L$, $\ell _L(x) = {\ell _L(1_L(x)) \wedge \ell _G(m(x))}$. This follows using $\ell _L(x) \le \ell _G(m(x))$ and the complete lattice law $\forall a \ b . a \le b \implies a \wedge b = a$.    $\square$