Spider Graphs: a graph transformation system for spider diagrams

Abstract

The use of diagrammatic logic as a reasoning mechanism to produce inferences on subsets of some universe could provide a way to overcome the current limitations of visual modelling methods, which have to be integrated with textual languages to express complex constraints. On the other hand, graph transformations are becoming widespread as a way to express formal semantics for visual modelling languages, so that a mechanisation of diagrammatic logic based on graph transformation would facilitate language integration, based on a common underlying machinery. In this paper, we propose such a mechanisation for spider diagrams (SDs), an established language for reasoning with diagrams modelling relations between sets and constraints on their cardinalities. The concrete syntax of SDs extends that of Euler diagrams that use closed curves and the enclosed regions to represent sets and their intersections. The language is augmented with reasoning rules, i.e. syntactic transformation rules corresponding to logical inference rules. However, these rules are typically defined in procedural terms, so that a completely formal specification and an adequate mechanisation of them has not been achieved yet. We propose an abstract syntax for SDs in terms of typed graphs and define the corresponding language of Spider Graphs (SGs), expressing reasoning rules for SDs as graph transformation units. This enables a direct realisation of the reasoning system via graph transformation tools without resorting to ad hoc implementations, and we provide an implementation in AGG. Techniques for static analysis become available to reason on proof strategies and on possible optimisations.

Appendix: Presentation of rules 1–4,7

We complete the presentation of the TUs realising the whole reasoning system, and for each of them, we prove its correctness.

Rule 1   [Introduction of a strand] Figure 35 shows the rule for adding strands, realising the first alternative for Rule 1, where a NAC (not shown here) prevents the formation of the forbidden graph ($F1$) in Fig. 8. Figure 36 shows the rule for the second alternative, replacing a tie by a strand. In both cases, the rule receives as parameters the labels associated with the spiders to be connected and picks non-deterministically a zone inhabited by both spiders. The resulting TU is a choice between the two rules.

$$\begin{aligned}&IntroduceStrand (\mathtt{S1,S2}) =(\mathtt{addStrand(S1,S2)} \mid \\&\quad \mathtt{replaceTieWithStrand(S1,S2)}) \end{aligned}$$

Fig. 35

Rule addStrand(S1,S2)

Fig. 36

Rule replaceTie WithStrand(S1,S2)

Lemma 4

The TU IntroduceStrand realises Rule 1 of the SD system.

Proof

The SD rule corresponds to two separate $SD$ specifications:

The SG rules addStrand and replaceTieWithStr and clearly realise these specifications. Note that the condition $(\{s,t\},z)\not \in \tau \cup \upsilon$ for the first rule is guaranteed by its NAC. $\square$

Rule 2   [Extension of habitat] The spider to be extended is chosen by passing a parameter S to the TU, which will pass it on to the invoked rules. The zone in which to extend the chosen spider, and the spiders that are to have connections (of different types) with it added are chosen non-deterministically. The process is managed by a TU with control expression:

$$\begin{aligned}&\!\!\!\!\!ExtendHabitat(\mathtt{S}) =\\&\!\!\mathtt{extendSpiderToZone(S)};\\&\!\!\mathbf{asLongAsPossible}~~ \mathtt{markSpiderForConnections(S)}~~ \mathbf{end};\\&\!\!\mathbf{asLongAsPossible}~~ (\mathtt{addStrandForNewExtension(S)} \mid \\&\!\! \mathtt{addTieForNewExtension(S)} \mid \mathtt{noConnectionAddded})~\, \mathbf{end};\\&\!\! \mathtt{removeMarkFromZone}; \end{aligned}$$

The process starts with rule extendSpiderToZone in Fig. 37. A new inhabits edge is drawn from the chosen spider to a zone, which is marked to specialise the context for the rest of the TU.

Fig. 37

Rule extendSpiderToZone

The iteration on rule markSpiderForConnections in Fig. 38 marks each spider, other than the chosen one, inhabiting the marked zone as a candidate for connection. All marks are removed in the second iteration within the TU, where for each candidate spider, a choice is made of whether to add a strand from the chosen spider to the candidate one, to add a tie or do neither. We only present the rule for adding a tie in Fig. 39, the case for strand being analogous; the third alternative simply removes the mark from the candidate spider.

Fig. 38

Rule markSpiderForConnections

Fig. 39

Rule addTie ForNewExtension

Finally, the mark is removed from the zone (rule remove MarkFromZone, not shown here). In Lemma 5, in order to track the effects of marking in the SG, we introduce a new temporary set $M\subset \mathcal Z \cup \mathcal C \cup \mathcal S$, and we refer to this in the specification of the rules within the TU.

Lemma 5

The TU ExtendHabitat realises Rule 2 of the SD system.

Proof

The SD rule has specification:

The rules extendSpiderToZone(18), markSpider ForConnections(19) and addStrandForNewExten sion addTieForNewExtension noConnectionAdd ed(20) in the TU have effects as follows:

$$\begin{aligned}&\!\!\!\!z\not \in {h(s)}\cup {M} \vdash {z}\in {h'(s)}\cap {M'}.\end{aligned}$$

(18)

$$\begin{aligned}&\!\!\! \!z\in {h(s)}\cap {M}\wedge \exists {t}\!\in \!\mathcal S [z\in {h(t)}\wedge {t}\notin {M}] \vdash {t}\!\in \!{M'}.\end{aligned}$$

(19)

$$\begin{aligned}&\begin{aligned}&\!\!\!z\!\in {h(s)}\cap {M}\wedge \exists {t}\in \mathcal S [z\in {h(t)}\wedge {t}\\&\quad \in {M}\wedge (\{s,t\},z)\notin \tau \cup \upsilon ] \vdash (\{s,t\},z)\\&\quad \in \tau '\wedge {t}\notin {M'}. \\&\!\!\!\! z\in {h(s)}\cap {M}\wedge \exists {t}\in \mathcal S [z\in {h(t)}\wedge {t}\\&\quad \in {M}\wedge (\{s,t\},z)\notin \tau \cup \upsilon ] \vdash \\&(\{s,t\},z)\in \upsilon '\wedge {t}\notin {M'}. \\&\!\!\! z\in {h(s)}\cap {M}\wedge \exists {t}\in \mathcal S [z\in {h(t)}\wedge {t}\\&\quad \in {M}\wedge (\{s,t\},z)\notin \tau \cup \upsilon ] \vdash \\&\quad t\notin {M'}. \end{aligned} \end{aligned}$$

(20)

To see that effect of the TU complies with the SD specification, suppose that we have $z\in \mathcal Z \wedge {z}\not \in {h(s)}$. Then, by entailment (18), the habitat of $s$ is extended, satisfying the outer post-condition, and the zone $z$ is marked, ensuring that subsequent rules apply only to this chosen zone. Entailment (19) marks any other spider that also inhabits the zone $z$. The alternative rules have effects that correspond to the three alternative nested pre–post-condition pairs in the SD specification [see entailment (20)]. The removal of the mark from the spider for each rule application ensures that each marked spider triggers exactly one of these alternative rules once only, and so the process terminates. All of the marks have been removed in the post-state: the marks of the spiders by the above process, while the removeMarkFromZone rule precisely ensures that $z\not \in M$ in the post-state. $\square$

Before we consider Rule 3, we need some notation.

Definition 7

Let $G(z), Con(s,z)$ and $Con_s(t,z)$ denote the strand–tie graph within zone $z$, the set of vertices in the connected component of $s$ in $G(z)$ and the set of vertices in the connected component of $t$ in $G(z){\setminus }\{s\}$, respectively.

Rule 3   [Erasure of a spider] The deletion of a spider $s$ is prepared by the deletion of all its connections to other spiders, reconstructing the connectedness of the ‘strand–tie graph’ in all the zones inhabited by $s$, when needed. To this end, we compute the separate connected components resulting from the removal of $s$, where each component has at least one spider that was directly connected to $s$ in the original graph and which is taken as the representative of the component. The process is started by selecting one representative (of one connected component of the strand–tie graph in the given zone after the deletion of $s$) via the rule startComputation(S) of Fig. 40, while the marking of the spiders in a connected component is achieved with rule computeConnectedComponent of Fig. 41. The two NACs prevent the original spider $s$, or an already marked spider, from being marked. Connections are then created between the representative of the computed component and another component (recognised by being directly connected to $s$ but not marked) to ensure that connectedness of the strand–tie graph is not affected by the deletion of $s$, through rule createStrandToOtherComponent of Fig. 42.

Fig. 40

Rule startComputation

Fig. 41

Rule compute ConnectedComponent

Fig. 42

Rule createStrandToOtherComponent

The representative of the reached component is also marked, so that the process (of computing the connected component and then extending it by adding a connection) can be iterated, until all the spiders directly connected to the $s$ are now directly or indirectly connected with one another (i.e. lie in the same component).

$$\begin{aligned}&EraseSpider (\mathtt{S}) =\\&\quad \mathbf{asLongAsPossible}\\&\qquad \mathtt{startComputation(S)};\\&\qquad \mathbf{asLongAsPossible}\\&\qquad \qquad \mathbf{asLongAsPossible}\,\, \mathtt{computeConnected}\\&\qquad \qquad \qquad \mathtt{Component(S)}\,\, \mathbf{end};\\&\qquad \qquad \mathbf{asLongAsPossible}\,\, \mathtt{createStrandToOther}\\&\qquad \qquad \qquad \mathtt{Component(S)} \,\,\mathbf{end};\\&\qquad \mathtt{removeZoneMark}\,\, \mathbf end ;\\&\quad \mathbf{end};\\&\quad \mathbf{asLongAsPossible}\,\, \mathtt{removeConnection(S)}\,\, \mathbf{end};\\&\quad \mathbf{asLongAsPossible}\,\, \mathtt{removeSpiderMark} \,\,\mathbf{end};\\&\quad \mathtt{deleteSpider(S)} \end{aligned}$$

At the end of this iteration, all the connections associated with the spider can be removed by rule removeConnec tion in Fig. 43 (left), whose SPO application also removes the connects edge to any other spider, and the spider is finally deleted by rule deleteSpider in Fig. 43 (right). Note that the NAC would make this rule fail if the spider inhabited some shaded zones, making the whole TU fail. Such a check could also be performed by an additional rule at the beginning of the process.

Fig. 43

Rules removeConnection and deleteSpider

Lemma 6

The TU EraseSpider realises Rule 3 of the SD system.

Proof

The SD rule has specification:

The rules startComputation(21), computeConne ctedComponent(22), createStrandToOtherComp onent(23), removeConnection(24), delete- Spid er(25) in the TU have effects as follows:

$$\begin{aligned}&\begin{aligned}&z\in {h(s)}\cap {h(t_1)}\cap {h(t_2)}\\&\quad \wedge (\{s,t_1\},z),(\{s,t_2\},z)\in \tau \cup \upsilon \wedge \\&\quad (\{t_1,t_2\},z)\not \in \tau \cup \upsilon \vdash {z,t_1}\in M'. \end{aligned}\end{aligned}$$

(21)

$$\begin{aligned}&\begin{aligned}&z,t_1\in {M}\wedge {t_2}\not \in {M}\wedge {t_2}\ne {s}\wedge {z}\in {h(t_1)}\cap {h(t_2)}\wedge \\&\quad (\{t_1,t_2\},z)\in \tau \cup \upsilon \vdash {t_2}\in {M'}. \end{aligned}\end{aligned}$$

(22)

$$\begin{aligned}&\begin{aligned}&z\in {h(s)}\cap {h(t_1)}\cap {h(t_2)}\wedge {z},t_1\in {M}\wedge {t_2}\not \in {M}\\&\quad \wedge (\{s,t_2\},z)\in \tau \cup \upsilon \\&\quad \vdash (\{t_1,t_2\},z)\in \upsilon '\wedge {t_2}\in {M'}. \end{aligned}\end{aligned}$$

(23)

$$\begin{aligned}&z\in {h(s)}\wedge (\{s,t\},z){\in }\tau \cup \upsilon \vdash (\{s,t\},z)\not \in \tau '\cup \upsilon '.\end{aligned}$$

(24)

$$\begin{aligned}&h(s)\cap \mathcal Z ^*=\emptyset \vdash {s}\not \in \mathcal S '. \end{aligned}$$

(25)

We consider the effects of the nested part of the control condition of the TU, before considering the SD specification. The startComputation rule finds, and marks, an arbitrary (representative) spider $t_1$ that is connected to spider $s$ but is not connected to every other spider that $s$ is connected to [see entailment (21)]. If there are no such spiders, then the removal of spider $s$ cannot disconnect the strand–tie graph in $z$ since every pair of spiders connected to $s$ within $z$ are already connected to each other; note that in this case no match for this SG rule is found and the TU proceeds to removeConnection. Otherwise, iterating computeConnectedComponent (see entailment (22)) repeatedly marks every spider (except for $s$ itself) which is connected to a marked spider (by a connection), ending when all of $Con_s(t_1,z)$ is marked. If there is a spider $t_2$ such that $t_2\not \in {Con_s}(t_1,z)$ (hence $t_2$ is not marked) but there is a connection between $s$ and $t_2$ within $z$, then rule createStrandToOtherComponent [see entailment (23)] adds a strand between $t_2$ and $t_1$ within $z$. The iteration in the control condition marks all of the extended connected component (which is the union of the original connected components of $t_1$ and $t_2$ together with the new strand that joins them). This is repeated until there is only a single connected component. At this point, the deletion of the spider $s$ will not disconnect the strand–tie graph. The removeZoneMark rule just removes the marking on the zone. So, the total effect of this nested part of the control condition is to add enough strands to ensure that the deletion of spider $s$ does not disconnect any component of the strand–tie graph, satisfying the inner pre-/post-condition pair. Now, suppose we have a spider $s$ such that $h(s)\cap \mathcal Z ^*=\emptyset$, so the precondition of the eraseSpider SD rule is satisfied. Then, removeConnection [see entailment (24)] removes all connection nodes involving $s$, and deleteSpider [see entailment (25)] removes $s$, satisfying the outer post-condition. $\square$

Rule 4   [Erasure of shading] Shading can be erased from any shaded zone as shown in Fig. 44 for rule eraseShading.

Fig. 44

Rule eraseShading: erasing shading from a zone

Lemma 7

The rule eraseShading realises Rule 4 of the SD system.

Proof

The rule effect clearly satisfies the SD rule specification given by:

$\square$

Rule 7   [Equivalence of Venn and Euler forms] The final rule establishes the equivalence between SDs in Euler and Venn forms, meaning that they have underlying EDs or VDs (which have no missing zones), by showing how to transform one form into the other. We first present the TU for adding missing zones, in which missing zones can be progressively introduced as the twins of existing zones, as defined by the control condition:

$$\begin{aligned}&AddMissingZones() =\\&\quad \mathbf{asLongAsPossible}\quad \mathtt{addMissingTwin};\\&\qquad \mathbf{asLongAsPossible}\quad \mathtt{completeInsideRelations}\quad \mathbf{end};\\&\quad \mathtt{removeTwin}\\&\mathbf{end} \end{aligned}$$

Rule addMissingTwin in Fig. 45 creates a new zone $z_1$ and marks it as twin of some existing zone $z_2$, outside some curve $c$. The zone $z_1$ is inside the boundary curve and $c$, and $twin_c(z_1,z_2)$ will hold at the end of the TU.

Fig. 45

Rule addMissingTwin

The GAC in Fig. 46 checks that $z_2$ lacks a twin. The newly created $z_1$ must be inside all but only the curves that contain $z_2$, as guaranteed by rule completeInsideRelations in Fig. 47. A NAC, not shown here, ensures that only one inside edge is created between a zone and a curve. The rule removeTwin (not shown here) is then applied to remove all twin edges, before proceeding with identifying and constructing the next twin.

Fig. 46

A graphical representation of the GAC for identifying missing twins, to be checked on each match $m$ for the LHS of the rule addMissingTwin in Fig. 45. The associated formula reads: ‘for all matches $m_1:g_1\rightarrow {G}$ of the graph $g_1$ in the outer $\forall$-box extending $m$, it is not the case that there exists a match $m_2:g_2\rightarrow {G}$ of the graph $g_2$ in the outer $\wedge$-box extending $m_1$ such that, for each match $m_3:g_3\rightarrow {G}$ of the graph $g_3$ in the next $\forall$-box extending $m_2$, either a match extending $m_3$ exists for the graph in the $\vee$-box, or neither of the two graphs in the $\lnot$-boxes has a match extending $m_3$’

Fig. 47

Rule completeInsideRelations

We also consider the opposite transformation, from Venn to Euler, by which any shaded uninhabited zone can be erased (provided it is not the last zone inside any curve), realised by the TU whose control expression is:

$$\begin{aligned}&EraseShadedZone() =\\&\qquad \mathbf{asLongAsPossible} \quad \mathtt{markShadedZone}\quad \mathbf{end};\\&\qquad \quad \mathbf{asLongAsPossible} \quad (\mathtt{deleteMarkedZone} \mid \\&\qquad \qquad \qquad \quad \mathtt{removeMarkFromZone}) \quad \mathbf{end}; \end{aligned}$$

In order to simulate a non-deterministic choice of the zones to be deleted, we first iterate on rule markShaded Zone (see Fig. 48) to mark all shaded zones satisfying the following conditions: (1) the zone is not the only one inside a curve (checked in the LHS of the rule and in the GAC of Fig. 49, where the different ways in which a second zone can be inside a curve are presented) and (2) the zone is not inhabited by any spider (checked in the NAC). After that, an iteration is performed on the alternative between deleting a marked zone and simply removing the mark.

Fig. 48

Rule markShadedZone

Fig. 49

A graphical representation of the GAC for checking that the zone is not the only one inside a curve, to be checked on each match $m$ for the LHS of the markShadedZone rule in Fig. 48. The associated formula reads: ‘for any match $m_1:g_1\rightarrow {G}$ of the graph $g_1$ in the $\forall$-box extending $m$, at least one of the two graphs inside the $\vee$-box has a match extending $m_1$’

Lemma 8

The TUs for AddMissingZones and EraseShadedZone realise Rule 7 of the SD system.

Proof

The addition of all of the missing zones is specified as:

The rule addMissingTwin in the TU (where the GAC is written in square brackets) followed by the complete InsideRelations rule has effect:

$$\begin{aligned} \begin{aligned} c_1\in \mathcal C \wedge {z_2}&=(X_2\cup \{u\},Y_2\cup \{c_1\})\in \mathcal Z \wedge \\ \forall {z_4}\in \mathcal Z [\lnot {z_4}&=(X_4\cup \{c\},Y_4)\wedge \forall {c_5}\\&\qquad \in \mathcal C [{z_2}=(X_5\cup \{c_5\},Y_5)\wedge \\ z_4&=(X_6\cup \{c_5\},Y_6)\vee (z_2\!=\!(X_5,Y_5\cup \{c_5\})\wedge \\ z_4&=(X_6,Y_6\cup \{c_5\}))]]\vdash {z_1}\\&=(X\cup \{c_1,u\},Y)\in \mathcal{Z ^*}'. \end{aligned} \end{aligned}$$

(26)

In the post-state of the SG after completing the iteration, $twins_c(z_1,z_2)$ holds. After the addMissingTwin rule, the Zone node for $z_1=(\{u,c\},Y)$ is not a correct twin for $z_2$ yet, but is related to $z_2$ via a twin indicator. Immediately afterwards, iterating the rule completeInsideRelations adds all of the inside edges from $z_1$ to the same set of curves as $z_2$, as required to ensure that it represents the required missing twin zone.

The above ensures that the addition of any missing twin zone can be achieved. It remains to show that one can just add missing twin zones to any SD in order to create the Venn form. Since any SD has an outermost zone, we can consider if there is any curve which does not have a zone which is inside only that curve (and the boundary curve). Such a zone would be a twin of the outermost zone w.r.t. that curve. The addition of any such missing zone ensures that every zone inside exactly the boundary curve and one other curve is present. Then we consider if there are any missing zones that are inside exactly two curves (plus the boundary curve), which are twins of the zones that are inside one curve (plus the boundary curve). Continuing this process ensures that all missing zones are added.

The removal of some of the shaded zones that are not inhabited by any spider (and do not contain the last zone within any curve) is specified as:

The markShadedZone rule is specified by entailment (27) (with the GAC indicated within the second pair of square brackets):

$$\begin{aligned} \begin{aligned} z_1&=(X_1\cup \{u,c\},Y_1)\in \mathcal Z ^*\wedge {z_2}=(X_2\cup \{u,c\},Y_2){\in }\mathcal Z \wedge \\&(\lnot \exists {s}\in \mathcal S \bullet {z_1}\in {h(s)})\wedge (\forall {c_k}\in \mathcal C {\setminus }\{u,c\}\\&\quad \bullet ({z_1}{=}(X_1\cup \{u,c,c_k\},Y_1))\wedge ((\exists {z_3}{=}(X_3\cup \{c_k\},Y_3))\!\!\! \\&\vee (\exists {z_2}=(X_2\cup \{u,c,c_k\},Y_2)))) \vdash {z_1}\in {M'}. \end{aligned} \end{aligned}$$

(27)

This marks a set of shaded zones. During the following iteration, at each step one of deleteMarkedZone or removeMarkFromZone is selected. The effect of the two rules is described in entailments (28) and (29), respectively.

$$\begin{aligned}&z\in \mathcal Z ^*\cap {M}\vdash {z}\not \in \mathcal Z '\wedge {z}\not \in {M'}.\end{aligned}$$

(28)

$$\begin{aligned}&z\in \mathcal Z ^*\cap {M}\vdash {z}\in \mathcal Z '\wedge {z}\not \in {M'}. \end{aligned}$$

(29)

After the iteration is performed, a (possibly empty) subset of $\mathcal Z ^*$ has been removed and no zone is marked. $\square$