2.1 DPO rewriting

2.1.1 Adhesive categories and (typed) hypergraphs

In order not to restrict ourselves to any concrete model of graphs, we work with adhesive categories (Lack and Sobociński Reference Lack and Sobociński2005). Adhesive categories are relevant because they have well-behaved pushouts along monomorphisms, and for this reason, they are convenient as ambient categories for DPO rewriting.

An important example is the category of finite directed hypergraphs ${\textbf{Hyp}_{}}$ . An object G of ${\textbf{Hyp}_{}}$ is a hypergraph with a finite set of nodes $G_\star$ and for each $k,l\in{{\mathbb{N}}}$ a finite set of hyperedges $G_{k,l}$ with k (ordered) sources and l (ordered) targets, that is, for each $0\leq i < k$ there is the ith source map $s_i: G_{k,l}\to G_\star$ and for each $0\leq j < l$ the jth target map $t_j : G_{k,l}\to G_{\star}$ . The arrows of ${\textbf{Hyp}_{}}$ are homomorphisms: functions $G_\star \to H_\star$ such that for each k,l, $G_{k,l}\to H_{k,l}$ they respect the source and target maps in the obvious way. The seasoned reader will recognise ${\textbf{Hyp}_{}}$ as a presheaf topos, and as such, it is adhesive (Lack and Sobociński Reference Lack and Sobociński2005).

We shall visualise hypergraphs as follows: $\bullet$ is a node and is a hyperedge, with ordered tentacles attached to the left boundary linking to sources and those on the right linking to targets

A signature $\Sigma$ consists of a set of generators $o: n \to m$ with arity n and coarity m where $m,n\in {{\mathbb{N}}}$ . Any signature $\Sigma$ can be considered as a hypergraph $G_{\Sigma}$ with a single node, in the obvious way. We can then express $\Sigma$ -labelled hypergraphs (briefly, $\Sigma$ -hypergraphs) as the objects of the slice category ${\textbf{Hyp}_{}} \downarrow G_{\Sigma}$ , denoted by ${\textbf{Hyp}_{\Sigma}}$ , which is adhesive, since adhesive categories are closed under slice (Lack and Sobociński Reference Lack and Sobociński2005). $\Sigma$ -hypergraphs are drawn by labelling hyperedges with generators in $\Sigma$

2.1.2 DPO rewriting

We recall the DPO approach (Ehrig and Kreowski Reference Ehrig and Kreowski1976) to rewriting in an adhesive category ${\mathbb{C}}$ . A DPO rule is a span $L \xleftarrow{} K \xrightarrow{} R$ in ${\mathbb{C}}$ . A DPO system $\mathcal{R}$ is a finite set of DPO rules. Given objects G and H in ${\mathbb{C}}$ , we say that G rewrites into H – notation $G {\rightsquigarrow_{\scriptscriptstyle{\mathcal{R}}}} H$ – if there exist $L \xleftarrow{} K \xrightarrow{} R$ in $\mathcal{R}$ , object C and morphisms such that the squares below are pushouts

A derivation from G into H is a sequence of such rewriting steps. The arrow $m : L \to G$ is called a match of L in G. A rule $L \xleftarrow{} K \xrightarrow{} R$ is said to be left-linear if the morphism $K\to L$ is mono. In this case, the matching m fully determines the graphs C and H; that is, for a fixed rule and a matching, there is a unique H such that $G {\rightsquigarrow_{\scriptscriptstyle{\mathcal{R}}}} H$ , if it exists. Here, by unique, we mean unique up-to isomorphism. More generally, the rewriting steps will always be up-to iso: in a step $G {\rightsquigarrow_{\scriptscriptstyle{\mathcal{R}}}} H$ , G and H should not be thought of as single graphs but rather as equivalence classes of isomorphic graphs.

2.1.2.1 Undecidability of confluence

In DPO rewriting, the confluence of terminating systems is not decidable, even if we restrict to left-linear rules.

Theorem 1 (Reference PlumpPlump 1993). Confluence of terminating DPO systems over ${\textbf{Hyp}_{\Sigma}}$ is undecidable.

Indeed, critical pair analysis for traditional DPO systems fails: for terminating DPO systems, joinability of critical pairs does not necessarily imply confluence.

Definition 2 (Pre-critical pair and joinability). Let $\mathcal{R}$ be a DPO system with rules $L_1 \xleftarrow{} K_1 \xrightarrow{} R_1$ and $L_2 \xleftarrow{} K_2 \xrightarrow{} R_2$ . Consider two derivations with common source S

We say that is a pre-critical pair if $[f_1,f_2] : L_1 + L_2 \to S$ is epi; it is joinable if there exists W such that .

We use the standard notation ${\rightsquigarrow_{\scriptscriptstyle{}}}^*$ for defining a possibly empty sequence of rewriting steps. Intuitively, in a pre-critical pair S should not be bigger than $L_1+L_2$ . In a critical pair, $L_1$ and $L_2$ must overlap in S, so that the two rewriting steps do not form a parallel pair.

Definition 3 (Parallel pair). Let $\mathcal{R}$ be a DPO system with rules $L_1 \xleftarrow{} K_1 \xrightarrow{} R_1$ and $L_2 \xleftarrow{} K_2 \xrightarrow{} R_2$ . Consider two derivations with common source S as in Definition 2. We say that is a parallel pair if there exist $g_1 : L_1 \to C_2$ and $g_2 : L_2 \to C_1$ making the diagram below commute

The key result for us is that parallel pairs are always joinable (see e.g., Corradini Reference Corradini2016), and a pre-critical pair is thus critical if it is not a parallel one. However, for the purposes of this paper, this restriction is immaterial, and we will mostly stick to pre-critical pairs in our results, as proofs are less tedious. For the sake of succinctness, most of the examples will instead display only the critical pairs. For a pre-critical pair which is also a parallel pair, see for instance the first picture of Section 6.1.

A notable feature of DPO rewriting is that, unlike in the case of term rewriting, joinability of all critical pairs is not enough to guarantee confluence, even for a terminating rewriting system.

Example 4 (Reference PlumpPlump 1993). Consider a DPO system $\mathcal{R}$ consisting of the following two rules, where we labelled nodes with numbers in order to make the graph morphisms explicit

Amongst the several pre-critical pairs, only the following two have non-trivial overlap

Both are obviously joinable. However, $\mathcal{R}$ is not confluent, as witnessed by the following

However, this “bug” in DPO rewriting can be fixed by considering graphs with interfaces, and DPO rules that respect the interface.

2.2 DPO rewriting with interfaces

Morphisms $G \xleftarrow{}J$ will play a special role in our exposition. When ${\mathbb{C}}$ is ${\textbf{Hyp}_{\Sigma}}$ , we will call them (hyper)graphs with interface. The intuition is that G is a hypergraph and J is an interface that allows G to be “glued” to a context. Note however that such morphisms are not necessarily mono, even if this will be the case in most of our examples.

Given $G \leftarrow J$ and $H \leftarrow J$ in ${\mathbb{C}}$ , G rewrites into H with interface J –notation $(G \xleftarrow{} J) {\rightsquigarrow_{\scriptscriptstyle{\mathcal{R}}}} (H \xleftarrow{} J)$ – if there exist rule $L \xleftarrow{} K \xrightarrow{} R$ in $\mathcal{R}$ , object C, and morphisms such that the diagram below commutes and the squares are pushouts

Hence, the interface J is preserved by individual rewriting steps.

When ${\mathbb{C}}$ has an initial object 0 (for instance, in ${\textbf{Hyp}_{\Sigma}}$ 0 is the empty hypergraph), ordinary DPO rewriting can be considered as a special case, by taking J to be 0.

Like for traditional DPO, rewriting steps are modulo isomorphism: $G_1 \leftarrow J: f_1$ and $G_2 \leftarrow J: f_2$ are isomorphic if there is an isomorphism $\varphi : G_1 \to G_2$ with $f_1{\,{;}\,} \varphi = f_2$ .

It can be rewritten in two different ways

(1)

which, unlike as in Example 4, results in two distinct hypergraphs with interface. Notably, the interface $\{0,1\}$ maintains the distinct identities of the two nodes initially connected to the hyperedge labelled by a, even after that hyperedge is removed. Notice that if (1) were considered as a critical pair, it would not be joinable. Hence, the counterexample of Plump (Reference Plump1993) (Example 1) would not work. This is the starting observation for our formulation of critical pair analysis: in Section 3 we will introduce pre-critical pairs for rewriting with interfaces and we will show that, as in term rewriting, joinability of pre-critical pairs entails confluence.

2.3 PROP rewriting

2.3.2 Rewriting in a PROP

Notation 7. Note that we write generic pairs and tuples using parentheses and reserve the notation $\langle l, r \rangle$ specifically for the case when the pair (l, r) forms a rewriting rule.

Definition 8. A rewriting system $\mathcal{R}$ in a PROP ${\mathbb{A}}$ consists of a set of rewriting rules, that is, pairs $\langle l, r \rangle$ of arrows $l, r : i \to j$ in ${\mathbb{A}}$ with the same arities and coarities. Given $a,b : m \to n$ in ${\mathbb{A}}$ , a rewrites into b via $\mathcal{R}$ , written $a {\Rightarrow_{\scriptscriptstyle {\mathcal{R}}}} b$ , if they are decomposable as follows, for some rule $\langle l, r \rangle \in \mathcal{R}$

(2)

In this situation, we say that a contains a redex for $\langle l, r \rangle$ .

The following well-known example illustrates the subtlety of critical pair analysis when rewriting in monoidal categories.

Example 9 (From Reference LafontLafont 2003, see also Reference MimramMimram 2014). Fix $\Sigma = \{ \gamma : 2 \to 2 \}$ and consider the rewriting system on ${\textbf{S}_{\Sigma}}$ consisting of the following rule

(3)

A critical pair analysis yields an infinite number of critical pairs. Indeed, as shown in Lafont (Reference Lafont2003); Mimram (Reference Mimram2014), any diagram $\phi:1+m \to 1+n$ that does not decompose non-trivially into $\phi=\mu + \nu$ for some $\mu,\nu$ yields a critical pair

in which clearly there are two embeddings of the left-hand side of (3) (depicted in blue and yellow, respectively, in a colour version of the paper) with an overlap (in green).

In Mimram (Reference Mimram2010) this problem was solved by adding duals to monoidal categories. In Section 4, we will show another solution based on Bonchi et al. (Reference Bonchi, Gadducci, Kissinger, Sobociński and Zanasi2020): a translation from PROPs to DPO rewriting with interfaces. The example below anticipates this encoding. It will be useful as a running example for the next section, which is devoted to critical pair analysis and confluence in DPO rewriting with interfaces.

Example 10. Treating the rewriting system of Example 4 as DPO system over ${\textbf{Hyp}_{\Sigma}}$ with $\gamma:2\to 2\in \Sigma$ yields the following DPO rule

The formal correspondence between PROPs and DPO rewriting with interfaces will be explained in Section 4. For the time being, the reader can observe the similarities between the left-hand side of (3) and the left-hand side of the above rule: each $\gamma$ in (3) corresponds to an hyperedge (labelled with $\gamma$ ) in the hypergraph above; moreover, each wires in (3) corresponds to a node above; finally, dangling wires in (3) are exactly the numbered nodes. A similar correspondence holds for the right-hand side, while the interface of the DPO rule, depicted in light blue, just collects all the numbered nodes.

Below, we give a DPO derivation with interface (in light blue), corresponding to a critical pair from the family identified in Example 9

4.1 From PROPs to hypergraphs with interfaces

In this subsection, we report a result from Bonchi et al. (Reference Bonchi, Gadducci, Kissinger, Sobociński and Zanasi2022) that is crucial for the encoding of PROP rewriting into DPO rewriting with interfaces in ${\textbf{Hyp}_{\Sigma}}$ (cf. Section 2.1).

First, we obtain our domain of interpretation by restricting the category ${\textsf{Csp}({{\textbf{Hyp}_{\Sigma}}})}$ whose objects are hypergraphs and arrows cospans, that is pairs $G_1 \xrightarrow{} G_2 \xleftarrow{}G_3$ of $\Sigma$ -hypergraphs morphisms, up-to isomorphism in the choice of $G_2$ .

Given a signature $\Sigma$ , we define a PROP morphism ${[\! [ {\cdot} ]\! ]} : \textbf{S}_\Sigma \to {{\textsf{Csp}_{D}({{\textbf{Hyp}_{{\scriptscriptstyle {\Sigma}}}}})}}$ . Since ${\textbf{S}_{\Sigma}}$ is the PROP freely generated by an SMT with no equations, it suffices to define ${[\! [ {\cdot} ]\! ]}$ on the generators: for each $o : n \to m$ in $\Sigma$ , we let ${[\! [ {o} ]\! ]}$ be the following cospan of type $n \to m$

Example 26. The two sides of the PROP rewriting rule (3) (Example 9) get interpreted as the following cospans in ${{\textsf{Csp}_{D}({{\textbf{Hyp}_{{\scriptscriptstyle {\Sigma}}}}})}}$

The encoding ${[\! [ {\cdot} ]\! ]}$ is an important part of Theorem 3 below. This is a pivotal result in our exposition, as it serves as a bridge between algebraic and combinatorial structures. Indeed, it provides a presentation, by means of generators and equations, for the PROP ${{\textsf{Csp}_{D}({{\textbf{Hyp}_{{\scriptscriptstyle {\Sigma}}}}})}}$ : the disjoint union of the SMTs of ${\textbf{S}_{\Sigma}}$ and ${{\textbf{Frob}}}$ .

The isomorphism ${\langle\! \langle {\cdot} \rangle \! \rangle}$ is given as the pairing $[{[\! [ {\cdot} ]\! ]}, {[ {\cdot} ]}] : {\textbf{S}_{\Sigma}} + {{\textbf{Frob}}} \to {{\textsf{Csp}_{D}({{\textbf{Hyp}_{{\scriptscriptstyle {\Sigma}}}}})}}$ , where ${[ {\cdot} ]} : {{\textbf{Frob}}} \to {{\textsf{Csp}_{D}({{\textbf{Hyp}_{{\scriptscriptstyle {\Sigma}}}}})}}$ is the PROP morphism mapping the generators of ${{\textbf{Frob}}}$ as follows

Here, ${{\textbf{Frob}}}$ is used to model those features of the graph domain that are not part of the syntactic domain, for example the ability of building a “feedback loop” around some $\alpha : 1 \to 1$ in $\Sigma$

4.2 Confluence for rewriting in ${\textbf{S}_{\Sigma}}+{{\textbf{Frob}}}$

We can use Theorem 3 to apply results for graphs with interfaces to ${\textbf{S}_{\Sigma}}+{{\textbf{Frob}}}$ . To this aim, first we need to interpret string diagrams as graphs with a single interface, instead of two as in their usual cospan interpretation. This can be easily achieved by applying the transformation ${\ulcorner {d} \urcorner}$ (introduced in Bonchi et al. Reference Bonchi, Gadducci, Kissinger, Sobociński and Zanasi2022), which “rewires” a syntactic term d of ${\textbf{S}_{\Sigma}}+{{\textbf{Frob}}}$ by turning all of the inputs into outputs

Syntactic rewriting with “rewired” graphs is equivalent to rewriting with the original ones, in the sense that $d {\Rightarrow_{\scriptscriptstyle {{{\left\langle {l},{r} \right\rangle}}}}} e$ if and only if ${\ulcorner {d} \urcorner} {\Rightarrow_{\scriptscriptstyle {{{\left\langle {{\ulcorner {l} \urcorner}},{{\ulcorner {r} \urcorner}} \right\rangle}}}}} {\ulcorner {e} \urcorner}$ . However, since the rewired rules have only one boundary, they are readily interpreted as hypergraphs with interfaces: if ${\langle\! \langle {d} \rangle \! \rangle} = i \rightarrow G \leftarrow j$ , then ${\langle\! \langle {{\ulcorner {d} \urcorner}} \rangle \! \rangle} = 0 \rightarrow G \leftarrow i + j$ , which we may simply write as the graph with interface $G \leftarrow i + j$ .

Observe that a rule in the rewrite system ${\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle}$ (defined as ${\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle} = \{ {{\left\langle {{\langle\! \langle {{\ulcorner {l} \urcorner}} \rangle \! \rangle}},{{\langle\! \langle {{\ulcorner {r} \urcorner}} \rangle \! \rangle}} \right\rangle}} \mid \langle l, r \rangle \in \mathcal{R} \}$ ) just consists of a pair of hypergraphs with a common interface; that is, it is a DPO rule of the form $L \leftarrow n+m \to R$ . Thus, PROP rewriting in ${{\textsf{Csp}_{D}({{\textbf{Hyp}_{{\scriptscriptstyle {\Sigma}}}}})}}$ coincides with DPOI rewriting: together with Theorem 3, this correspondence yields the following result.

Theorem 4 (Reference Bonchi, Gadducci, Kissinger, Sobociński and ZanasiBonchi et al. 2022). Let $\mathcal{R}$ be a rewriting system on ${\textbf{S}_{\Sigma}}+{{\textbf{Frob}}}$ . Then

\begin{equation*}d {\Rightarrow_{\scriptscriptstyle {\mathcal{R}}}} e \mbox{ iff } {\langle\! \langle {{\ulcorner {d} \urcorner}} \rangle \! \rangle} {\rightsquigarrow_{\scriptscriptstyle{\scriptscriptstyle{{\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle}}}}} {\langle\! \langle {{\ulcorner {e} \urcorner}} \rangle \! \rangle}.\end{equation*}

One can read Theorem 4 as: DPO rewriting with interfaces is sound and complete for any symmetric monoidal theory with a chosen special Frobenius structure, that is one of shape $(\Sigma + \Sigma_F, \mathcal{E} + \mathcal{E}_F)$ , with $(\Sigma_F, \mathcal{E}_F)$ the SMT of ${{\textbf{Frob}}}$ . There are various relevant such theories in the literature, such as the ZX-calculus (Coecke and Duncan Reference Coecke and Duncan2008), the calculus of signal flow graphs (Bonchi et al. Reference Bonchi, Sobociński and Zanasi2014), the calculus of stateless connectors (Bruni et al. Reference Bruni, Lanese and Montanari2006) and monoidal computer (Pavlovic Reference Pavlovic2013).

The combination of the result above with Theorem 2 is however not sufficient for ensuring the decidability of the confluence for a terminating rewriting system $\mathcal{R}$ on ${\textbf{S}_{\Sigma}}+{{\textbf{Frob}}}$ . Indeed, Theorem 2 and Theorem 4 ensure that if all the pre-critical pairs in ${\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle}$ are joinable, then the rewriting in $\mathcal{R}$ is confluent. However, for the decidability of confluence in $\mathcal{R}$ the reverse is also needed: if one pre-critical pair in ${\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle}$ is not joinable, then $\mathcal{R}$ should not be confluent. To conclude this fact, it is enough to check that all pre-critical pairs of ${\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle}$ lay in the image of ${\langle\! \langle {{\ulcorner {\cdot} \urcorner}} \rangle \! \rangle}$ , that is, that they all have discrete interfaces. The key observation is given by the lemma below.

Since in every rule $L \xleftarrow{} K \xrightarrow{} R$ in ${\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle}$ , K is discrete, from Lemma 29 and Theorem 2 we derive the following result.

Now, if $\mathcal{R}$ is terminating, then by Theorem 4 also ${\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle}$ is terminating. The latter is also computable and therefore joinability of pre-critical pairs of ${\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle}$ can easily be decided by following the steps in the second part of the proof of Corollary 24.

5.1 Monogamous acyclic hypergraphs

We first recall from Bonchi et al. (Reference Bonchi, Gadducci, Kissinger, Sobociński and Zanasi2020) a combinatorial characterisation of the image of ${[\! [ {\cdot} ]\! ]}$ . It is based on a few preliminary definitions. We call a sequence of hyperedges $e_1, e_2, \ldots, e_n$ a (directed) path if at least one target of $e_k$ is a source for $e_{k+1}$ and a (directed) cycle if additionally at least one target of $e_n$ is a source for $e_1$ . The in-degree of a node v in an hypergraph G is the number of pairs (h,i) where h is an hyperedge with v as its i-th target. Similarly, the out-degree of v is the number of pairs (h,j) where h is an hyperedge with v as its j-th source. We call input nodes those with in-degree 0, output nodes those with out-degree 0, and internal nodes the others. We write ${\textsf{in}({G})}$ for the set of inputs and ${\textsf{out}({G})}$ for the set of outputs.

1. (1) it contains no cycle (acyclicity) and

2. (2) every node has at most in- and out-degree 1 (monogamy).

A cospan $n \xrightarrow{f} G \xleftarrow{g} m$ in ${{\textsf{Csp}_{D}({{\textbf{Hyp}_{{\scriptscriptstyle {\Sigma}}}}})}}$ is monogamous acyclic (ma-cospan) when G is an ma-hypergraph, f is mono and its image is ${\textsf{in}({G})}$ , and g is mono and its image is ${\textsf{out}({G})}$ .

We call a hypergraph with interface $G \xleftarrow{} J$ monogamous acyclic (ma-hypergraph with interface) if it is of the form $G \xleftarrow{[i,o]} n + m$ for an ma-cospan $n \xrightarrow{i} G \xleftarrow{o} m$ , up to isomorphism between J and $n+m$ . Note that such an ma-cospan (and hence the ma-hypergraph with interface) may be uniquely fixed, so we can use the two representations interchangeably.

Finally, we say that a rule $L \xleftarrow{} i+j \xrightarrow{} R$ is an ma-rule if $i \xrightarrow{} L \xleftarrow{} j$ and $i \xrightarrow{} R \xleftarrow{} j$ are ma-cospans.

5.2 Convex rewriting and soundness

We are now in position to interpret PROP rewriting for ${\textbf{S}_{\Sigma}}$ in DPO rewriting for ma-hypergraphs with interfaces, via the mapping that takes string diagrams to ma-hypergraphs with interfaces. Unfortunately, as shown in Part II (Bonchi et al. Reference Bonchi, Gadducci, Kissinger, Sobociński and Zanasi2020), this interpretation is generally unsound. There are several things that can go wrong in the absence of Frobenius structure, as illustrated in the next two examples from Part II. These motivate our restrictions to PROP rewriting systems that make the interpretation sound, as presented in the sections below.

First, as noted in Part I (Bonchi et al. Reference Bonchi, Gadducci, Kissinger, Sobociński and Zanasi2022), a DPOI rule can have multiple pushout complements when it is not left-linear, only some of which make sense without Frobenius structure.

The rule is not left-linear, and therefore, pushout complements are not necessarily unique for the application of this rule. For example, the following pushout complement yields a rewritten graph that can be interpreted as an arrow in an SMC

On the other hand, if we choose a different pushout complement, we obtain a rewritten graph that does not look like an SMC morphism

The different outcome is due to the fact that f maps 0 to the leftmost and 1 to the rightmost node, whereas g swaps the assignments. Even though both rewriting steps could be mimicked at the syntactic level in ${\textbf{S}_{\Sigma}} + {{\textbf{Frob}}}$ , the second hypergraph rewrite yields a hypergraph that is illegal for $\mathcal{R}$ in ${\textbf{S}_{\Sigma}}$ . In particular, the rewritten graph in the second derivation is not monogamous: the outputs of $\alpha_1$ and $\alpha_3$ and the inputs of $\alpha_2$ and $\alpha_3$ have been glued together by the right pushout.

Next we can see that, even if a DPOI rewriting step yields a string diagram that can be expressed without Frobenius structure, it could be the case that equation itself cannot be proven in the SMT without introducing a feedback loop.

Example 33. Consider a $\Sigma = \{ e_1 : 1 \to 2, {e_2 : 2 \to 1}, {e_3 : 1 \to 1} , e_4 : 1 \to 1\}$ and the following rewriting rule in ${\textbf{S}_{\Sigma}}$

(12)

Left and right side are interpreted in ${{\textsf{Csp}_{D}({{\textbf{Hyp}_{{\scriptscriptstyle {\Sigma}}}}})}}$ as cospans

We introduce another diagram $c : 1\to 1$ in ${\textbf{S}_{\Sigma}}$ and its interpretation in ${{\textsf{Csp}_{D}({{\textbf{Hyp}_{{\scriptscriptstyle {\Sigma}}}}})}}$

Now, rule (12) cannot be applied to c, even modulo the SMC equations. However, their interpretation yields a DPO rewriting step in ${{\textsf{Csp}_{D}({{\textbf{Hyp}_{{\scriptscriptstyle {\Sigma}}}}})}}$ as below

Observe that the leftmost pushout above is a boundary complement: the input-output partition is correct. Still, the rewriting step cannot be mimicked at the syntactic level using rewriting modulo the SMC laws. That is because, in order to apply our rule, we need to deform the diagram such that $e_3$ occurs outside of the left-hand side. This requires moving $e_3$ either before or after the occurence of the left-hand side in the larger expression, but both of these possibilities require a feedback loop

Hence, if the category does not have at least a traced symmetric monoidal structure (Joyal et al. Reference Joyal, Street and Verity1996), there is no way to apply the rule.

The two examples motivate the definition of convex rewriting, as a restriction of DPOI rewriting that rules out the above counterexamples and ensures soundness. We briefly recall the relevant definitions from Part II (Bonchi et al. Reference Bonchi, Gadducci, Kissinger, Sobociński and Zanasi2020), referring to the discussion therein for more examples and properties of convex rewriting. As a preliminary step, we need to introduce a suitable restriction of the notion of pushout complement, called boundary complement.

Definition 34 (Boundary complement). For ma-cospans $i \xrightarrow{a_1} L \xleftarrow{a_2} j$ and $n \xrightarrow{b_1} G \xleftarrow{b_2} m$ and mono $f : L \to G$ , a pushout complement as depicted in $(\dagger)$ below

is called a boundary complement if $[c_1,c_2]$ is mono and there exist $d_1: n\to L^\perp$ and $d_2: m \to L^\perp$ making the above triangle commute and such that

(13) \begin{equation} j+n \xrightarrow{[c_2,d_1]} L^\perp \xleftarrow{[c_1,d_2]} m+i \end{equation}

is an ma-cospan.

Note that boundary complements are unique when they exist (Bonchi et al. Reference Bonchi, Gadducci, Kissinger, Sobociński and Zanasi2020) and that the pushout complement of Example 32 is not a boundary complement.

The next definition is a restriction on the possible matches, and rules out the other counterexample (Example 33).

We now have all the ingredients to recall the notion of convex rewriting step, which is essentially a DPOI rewriting step relying on a boundary complement and a convex match.

Definition 36. Given $n \rightarrow D \leftarrow m$ and $n \rightarrow E \leftarrow m$ ma-cospans, D rewrites convexely into E with interface $n+m$ –notation $(D \xleftarrow{} n+m) \Rrightarrow_{\mathcal{R}} (E \xleftarrow{} n+m)$ – if there exist ma-rule $L \xleftarrow{} i+j \xrightarrow{} R$ in $\mathcal{R}$ , object C, and morphisms such that the diagram below commutes and the squares are pushouts

(14)

and the following conditions hold

• $f : L \to D$ is a convex match, and

• $i+j \to C \to D$ is a boundary complement in the leftmost pushout.

Note that in the definition above we implicitly assume that $i \rightarrow L \leftarrow j $ is an ma-cospan.

5.3 Failure of naive critical pair analysis for convex rewriting

The main issue with using the critical pair analysis technique delineated before is that convexity is not preserved by clipping. That is, we can have critical overlaps H which form non-convex subgraphs $H \subseteq G$ . Hence, it could be the case that a branching is joinable using convex rewriting starting from H, but the lifted branching will not be joinable.

This is what happens in the following example, taken from Part II (Bonchi et al. Reference Bonchi, Gadducci, Kissinger, Sobociński and Zanasi2020).

modulo the following equations

(15)

which can be represented as the following rewriting system

Suppose we start to consider the pre-critical pairs for this system. The following one is clearly joinable, since it involves parallel (and in fact totally disjoint) applications of ${\textbf{FS}_{3}}$ and ${\textbf{FS}_{4}}$

However, if we consider the middle graph in a larger context, for example

it no longer becomes joinable by convex rewriting. For example, suppose we apply ${\textbf{FS}_{3}}$ on the larger graph above

Then we are stuck: ${\textbf{FS}_{4}}$ no longer has a convex matching, because applying ${\textbf{FS}_{3}}$ introduced a new path from the output 5 to the input 0. Convexity guarantees we will not introduce cycles, and indeed in this case applying ${\textbf{FS}_{4}}$ would introduce a new path from 0 to 5, and hence a cycle.

We present two solutions to this problem. The first is to put a strong restriction, called left-connectedness, on the rewriting systems being considered. The second is to develop a more in-depth notion of critical pair analysis, which accounts for the context-sensitivity of convex rewriting using formal path extensions.

They both rely on a more refined notion of pre-critical pair. One cannot simply reuse Definition 11, as we want to enforce that the common source $S \xleftarrow{} J$ (cf. (4)) of the two derivations is an ma-hypergraph with interfaces, so that it is in the image of ${\langle\! \langle {{\ulcorner {\cdot} \urcorner}} \rangle \! \rangle}$ and we can reason about pre-critical pairs “syntactically” in ${\textbf{S}_{\Sigma}}$ . However, while Lemma 29 guarantees that this is always the case for rewriting systems on ${\textbf{S}_{\Sigma}}+{{\textbf{Frob}}}$ , with Definition 11, this is not guaranteed for ${\textbf{S}_{\Sigma}}$ , as shown by the example below.

Although $L_1 \xleftarrow{} K_1$ and $L_2 \xleftarrow{} K_2$ are left-hand sides of left-connected rules, S is not monogamous, thus this pre-critical pair does not correspond to anything syntactic in ${\textbf{S}_{\Sigma}}$ .

Recall from the end of Section 5.1 that an ma-hypergraph with interface is required to have an interface corresponding exactly to the inputs and outputs of the ma-hypergraph. Because of this, using the notion of pre-critical pair from Definition 11 may yield too many nodes in the interface.

Motivated by these two examples, we give the following definition.

Definition 40 (Ma-pre-critical pair). Let $\mathcal{R}$ be a rewrite system consisting of ma-rules $L_1 \xleftarrow{} K_1 \xrightarrow{} R_1$ and $L_2 \xleftarrow{} K_2 \xrightarrow{} R_2$ . Consider two derivations with source $S\xleftarrow{} J$

(16)

We say that is an ma-pre-critical pair if $[f_1,f_2] : L_1 + L_2 \to S$ is epi, $(\dagger)$ is a commuting diagram, and $S\xleftarrow{} J$ is an ma-hypergraph with interface; it is joinable if there exists an ma-hypergraph with interface $W\xleftarrow{}J$ such that .

Comparing it with Definition 11, we are dropping the requirement that $(\dagger)$ is a pulback. However, note that up to an isomorphic choice of J, there is at most one ma-hypergraph with interface $S\xleftarrow{} J$ . Indeed, all ma-pre-critical pairs are also pre-critical pairs, and if I is the pullback along $C_1 \rightarrow S \leftarrow C_2$ , the uniquely induced monomorphism $I\xleftarrow{} J$ just weeds out from I those items whose image is neither an input nor an output of S.

5.4 Confluence for left-connected rewriting in ${\textbf{S}_{\Sigma}}$

Intuitively, in Definition 41 strong connectedness prevents matches leaving “holes,” as in Example 33, whereas left-linearity guarantees uniqueness of the pushout complements, and prevents the problem in Example 32. We are then able to prove the following.

1. (1) if $d {\Rightarrow_{\scriptscriptstyle {\mathcal{R}}}} e$ then ${\langle\! \langle {{\ulcorner {d} \urcorner}} \rangle \! \rangle} \Rrightarrow_{{\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle}} {\langle\! \langle {{\ulcorner {e} \urcorner}} \rangle \! \rangle}$ ;

2. (2) if ${\langle\! \langle {{\ulcorner {d} \urcorner}} \rangle \! \rangle} \Rrightarrow_{{\langle\! \langle {{\ulcorner {\mathcal{R}} \urcorner}} \rangle \! \rangle}} {\langle\! \langle {{\ulcorner {e} \urcorner}} \rangle \! \rangle}$ then $d {\Rightarrow_{\scriptscriptstyle {\mathcal{R}}}} e$ .

The above theorem allows us to use DPOI rewriting as a mechanism for rewriting ${\textbf{S}_{\Sigma}}$ .

We could now recast in this setting the considerations on parallel and critical pairs, as well as on joinability, as given in Definition 14 and Proposition 16, respectively. We move instead directly to state the confluence theorem for left-connected systems.

Theorem 7. (Local confluence for left-connected systems). For a left-connected DPO system with interfaces, if all ma-pre-critical pairs are joinable then rewriting is locally confluent: given an ma-hypergraph with interface $G_0 \xleftarrow{} I$ and , there exists an ma-hypergraph with interface $W \xleftarrow{} I$ such that

The proof of Theorem 7 follows steps analogous to the one of Theorem 2. The essential difference is that ma-pre-critical pairs now have interfaces that are not necessarily pullbacks. The assumption of left-connectedness is nevertheless enough to ensure that the fundamental pieces, namely Constructions 19 and 12, can be reproduced.

We emphasise that the decision procedure relies on the fact that there are only finitely many pre-critical pairs to consider, the above one being the only one to feature a non-trivial overlap of rule applications. This is in contrast with a naive, “syntactic” analysis, which as we observe in Example 9 yields infinitely many pre-critical pairs for $\mathcal{R}$ .

5.5 Convex critical pair analysis via formal path extensions

It is natural to ask whether we can extend critical pair analysis for convex rewriting beyond left-connected systems. It turns out that this is true, but the usual checking of critical pairs does not suffice: they need to be checked in a variety of contexts to account for the possible existence of paths from an output of the critical pair to an input. However, while it might seem necessary to check infinitely many contexts to account for every way a critical pair can be embedded in a larger graph, we get around this problem by considering formal path contexts. These abstract over the particular graph in which a critical pair is embedded, and only capture whether certain paths exist.

Any morphism path-covers itself and path-covering is transitive, so $\lesssim$ is a pre-order (but not a partial order). We will write $m \sim m'$ for $m \lesssim m'$ and $m' \lesssim m$ .

We extend $\Sigma$ with 3 new formal path generators $\mathcal P =$ , and introduce a family of monos which can produce any path relation.

It follows by construction that $R = R_p$ . Note that there is more than one way to construct a path extension for a given R, but if $R = R_p = R_q$ then $p \sim q$ .

Lemma 48. Let $L \leftarrow K \rightarrow R$ be a left-linear ma-rule in ${\textbf{Hyp}_{\Sigma}}$ , $p: G' \to P$ a path extension, and $m : L \to P$ a convex match. Then, m factors as $L \xrightarrow{m'} G' \xrightarrow{p} P$ , and the convex rewrite of G’ at m’ extends to a convex rewriting step of P at m as follows

(17)

and furthermore q is a path extension.

The top pushouts in (17) are constructed as a convex DPO rewriting step. The bottom-left pushout is constructed as a pushout complement, which exists because m satisfies the gluing conditions with respect to K, so p satisfies the gluing conditions with respect to C’ (which contains K). The bottom-right square is a pushout. This corresponds to the original rewrite $P {\rightsquigarrow_{\scriptscriptstyle{}}} Q$ by uniqueness of the pushout complement D.

It only remains to show that q is a path extension. This follows from the fact that any $\mathcal P$ -hyperedges in the pushout yielding Q must come from D.

Lemma 49. Let $L \leftarrow K \rightarrow R$ , p, q, m’, and n’ be given as in Lemma 48. Then, for any mono $k : G' \to G$ such that $k \lesssim p$ the convex rewriting of G’ at m’ extends to a convex rewriting of G at $k \circ m'$ as follows, where $l \lesssim q$

(18)

It only remains to show that $l \lesssim q$ . Inspecting the bottom pushout squares of (17), we note that, because of monogamy of G’, it must be the case that any path from an output to an input of the image G’ in P must be in C’. Hence, the same path will be in the image of H’ in Q, so $R_p \subseteq R_q$ . By the symmetry of the DPO construction, it is also the case that $R_q \subseteq R_p$ so $R_p = R_q$ .

Applying the same argument to the bottom pushout squares of (18), we conclude that $R_l = R_k$ . Since $k \lesssim p$ , we have $R_l = R_k \subseteq R_p = R_q$ , so $l \lesssim q$ .

Lemma 50. For ma-rules $L_1 \leftarrow K_1 \rightarrow R_1$ and $L_2 \leftarrow K_2 \rightarrow R_2$ and an ma-pre-critical pair , let $k : G'_0 \to G_0$ be a mono such that the induced matches $L_1 \to G'_0 \to G_0$ and $L_2 \to G'_0 \to G_0$ are convex and $p : G'_0 \to P_0$ a path extension such that $k \lesssim p$ . Then, we can obtain the following rewrites by extending the critical pair along k and p, respectively

If the right branching is joinable by convex rewriting, then so is the left one.

Proof. We apply essentially the same technique as the proof of Theorem 2, except that we additionally need to show that, when we extend derivations from the critical pair $G'_0$ to the full graph $G_0$ following Construction 21

each of the matches $L_i \to G'_i \to G_i$ is convex.

If all the critical pairs in $\mathcal R$ are path-joinable, then there exists a path extension $p: G'_0 \to P_0$ that path-covers $G_0' \to G_0$ and is joinable. First, we can apply Lemma 48 to translate a convex rewrite $(P_0 \leftarrow I) {\rightsquigarrow_{\scriptscriptstyle{}}} (P_{1,i} \leftarrow I)$ to a convex rewrite $(G'_0 \leftarrow J) {\rightsquigarrow_{\scriptscriptstyle{}}} (G'_{1,i} \leftarrow J)$ . We can then apply Lemma 49 to extend this to a convex rewrite $(G_0 \leftarrow J) {\rightsquigarrow_{\scriptscriptstyle{}}} (G_{1,i} \leftarrow J)$ . This yields a path extension $q: G'_{1,i} \to P_{2,i}$ that path-covers $G'_{1,i} \to G_{1,i}$ , hence we can iterate this process to get a convex rewrite $(G_{1,i} \leftarrow J) {\rightsquigarrow_{\scriptscriptstyle{}}} (G_{2,i} \leftarrow J)$ , and so on.

When we path-join the critical pair, we obtain $P_{m,1} \cong P_{n,2}$ . If we remove all of the $\mathcal P$ -hyperedges (and nodes connected only to $\mathcal P$ -hyperedges), this will restrict to an isomorphism $G'_{m,1} \cong G'_{n,2}$ , which in turn yields an isomorphism $G_{m,1} \cong G_{n,2}$ . Hence the branching is joinable by convex rewriting.

1. (1) a mono $m : G'_0 \to H$ exists for an ma-hypergraph H with $R_m = R$ ,

2. (2) the induced matchings $L_1 \to G'_0 \to H$ and $L_2 \to G'_0 \to H$ are convex, and

3. (3) no relation satisfying (1) and (2) is a proper superset of R.

The converse of this theorem is almost true, but with a small caveat that one needs to consider local confluence of ma-hypergraphs over the full signature $\Sigma + \mathcal P$ containing the formal path generators, rather than just $\Sigma$ .

The proof is immediate, since failing to join the path extension of a ma-pre-critical pair witnesses a failure of local confluence. There is no reason a priori that the above theorem would hold just for ma-hypergraphs. Hence, one can see the inclusion of the formal path generators as a sort of “stabilisation” of the theory that rules out certain degenerate cases of convex rewriting systems, such as those where certain paths never exist or can always be broken by rewriting.

Theorem 8 gives us an effective way to check local confluence. For an ma-pre-critical pair, we need to enumerate all the maximal path relations, and for each one, construct a path extension and check if it is joinable. While there could in principle be exponentially many of these for each critical pair, conditions (1)–(3) in Definition 51 rule many of them out. We will see this process in action in the case study in Section 6.2.

6.1 Left-connected and confluent: Non-commutative bimonoids

First case study is an application of the results on left-connected systems, showing confluence of the theory ${{\textbf{NB}_{}}}$ of non-commutative bimonoids (Example 6(c)). Below is the interpretation of the theory as a DPO system ${\langle\! \langle {{\ulcorner {\mathcal{R}_{\textbf{NBiM}}} \urcorner}} \rangle \! \rangle}$ , which was shown to be terminating in Bonchi et al. (Reference Bonchi, Gadducci, Kissinger, Sobociński and Zanasi2020)

Given that the system is terminating, it suffices to show local confluence. Observe that ${\langle\! \langle {{\ulcorner {\mathcal{R}_{\textbf{NBiM}}} \urcorner}} \rangle \! \rangle}$ is left-connected: monogamy is ensured by the fact that it is in the image of ${\langle\! \langle {{\ulcorner {\cdot} \urcorner}} \rangle \! \rangle}$ ; strong connectedness and left-linearity hold by inspection of the set of rules. We can thus use Theorem 7 and local confluence follows from joinability of the ma-pre-critical pairs. Amongst them, the pairs without overlap of rule applications pose no problem: they are trivially joinable. One example is given below, with the middle grey graph acting as the interface for all depicted derivation steps

Thus, we confine ourselves to analysing actual critical pairs, with overlapping rule applications. One such pair is given below, also involving rules ${{\textbf{NB}_{1}}}$ and ${{\textbf{NB}_{9}}}$ . Again, we show how it is joined, with the interface of each step drawn in the centre

Overall, there are 22 critical pairs to consider. For each of them, we only show the graph exhibiting the overlap. It is straightforward to check that the corresponding pairs are all joinable

We can thereby conclude that ${{\textbf{NB}_{}}}$ is a confluent rewriting system. Since it is also terminating, equivalence of terms in ${{\textbf{NB}_{}}}$ is decidable by means of rewriting. Note that, by virtue of Corollary 2(‘)@, the above pre-critical pair analysis can be automated.

6.2 A confluent, non-left-connected example

We now consider an example of a rewriting system that is not left-connected, and demonstrate a proof of confluence by means of path extensions. Let $\Sigma = \{ f : 2 \to 2, g : 1 \to 0, h : 0 \to 1 \}$ , satisfying one equation

which translates into the following ma-rule

The rule strictly decreases the number of f-labelled hyperedges in a graph, so NLC is clearly terminating. Hence, it suffices to check local confluence to prove that NLC is confluent.

The rule NLC has two types of ma-pre-critical pairs. The first type is a genuine critical pair

(19)

and the second is a parallel pair

(20)

All of the other ma-pre-critical pairs are variations of (19) and (20), obtained by joining some outputs to some inputs in such a way that the two matchings of ${{\textbf{NLC}_{1}}}$ remain convex.

The maximal path relations for (19) and (20) can be computed by exhaustive enumeration (see Appendix B). For (19), there is only one maximal path relation $R_p = \{ (7, 3) \}$ . Hence, we can form the extension by adjoining a $\mathcal P$ -hyperedge that creates a path from output 7 to input 3. This is then joinable

The ma-pre-critical pair (20) has three maximal path relations

\begin{align*} R_p & = \{ 6, 7, 8 \} \times \{ 3, 4, 5 \} \\ R_q & = \{ 9, 10, 11 \} \times \{ 0, 1, 2 \} \\ R_r & = \{ (11, 2), (8, 5) \} \\\end{align*}

We can construct path extensions for each of these as follows

Each of these is joinable by convex rewriting. The 24 variations of (19) and (20) are all basically identical to these two cases, with the only difference being that one or more paths in the path extension is replaced by an output connected directly to an input. These are also all joinable, and hence, the system NLC is confluent.