This paper discusses a dynamic argumentation framework (AF) that consists of odd-length cycles. When an AF has no stable extension, we investigate the position to which an argument is added so that it has a stable extension. We prove the unstability of an AF that consists of only odd-length cycles with some constraints, show the conditions for repairability using a reduced AF, and identify the positions. In contrast to existing studies on dynamic argumentation, we use a topological approach. In addition, we introduce a reduced form of an AF, which provides a proof that is easier to understand and reduces the computational burden.
There have been lots of studies of computational argumentation [
We often use argumentation to reach an agreement or to persuade others, and it is preferable to obtain a justified result. In this paper, we consider argumentation that becomes stuck and yields no outcome because of its cyclic structure, such as the structure in Figure 1, in which nodes represent arguments and edges represent counter-arguments. In this situation, an agent usually wants to persuade other agents to accept their claim by making one new counter-argument.
In Figure 1(a), if an agent wants their argument a to be accepted, they attack an argument c with a new argument, then c is defeated by the new argument, and as a result, a and the new argument are accepted. This is a unique solution (with respect to stable semantics), and it is easy to find such an attack position. In Figure 1(b), not only c but also d can be regarded as a position so that an argument a is accepted. Therefore, even if an agent does not find a new argument that attacks c, they can achieve their goal if they find a new argument that attacks d. However, it is difcfiult to rapidly identify the position when the AF is large. We identify such a position by considering the topology of an AF.
(a) case 1 (b) case 2
We consider a problem to resolve the stuck argumentation that consists of n-lemmas (i.e., odd-length cycles), through the addition of one new argument. We mainly discuss the position in which a new argument is added.
We discussed this problem in the case of a trilemma in our previous paper [
This paper is organized as follows. In Section 2, we describe the basic concepts. In Section 3, we introduce the concept of reduction and consider the stability of the reduced AF. In Section 4, we explore the positions to which a counter-argument is added to a given AF, based on the solution on its reduced AF. In Section 5, we compare our approach with related work. Finally, in Section 6, we present our conclusions and discuss future work.
The abstract AF, proposed by Dung [
Definition 1 (argumentation framework (AF)). An Argumentation Framework (AF) F is defined as a pair ⟨A , R⟩ where A is a set of arguments and R ⊆ A × A .
An AF can be represented as a graph in which each node corresponds to an argument, and each edge corresponds to an attack. In this paper, we consider a finite AF.
Definition 2 (path,cycle). For a pair of arguments A0 and An of an AF, if there exists a sequence of attacks (A0, A1), (A1, A2), . . . , (An− 1, An) where Ai ̸= A j (∀i, j; 0 ≤ i ̸= j ≤ n, n ̸= 0), then ⟨A0, . . . , An⟩ is called a path from A0 to An. If An = A0 (n ̸= 0), then it is called a cycle from A0 to A0.
Definition 3 (odd/even-length-far). For a path ⟨A0, . . . , An⟩, if i (0 < i ≤ n) is odd, Ai is called a node that is odd-length-far from A0 in the path; and if i is even, a node that is even-length-far from A0 in the path.
For an abstract AF, semantics is defined either by an extension or labeling, which has a
one-to-one relation [
A labeling is a total function from a set of arguments to a set of labels {in, out, undec}. Definition 4 (complete labeling). Let ⟨A , R⟩ be an AF. Labeling L is said to be complete if the following conditions are satisfied for any argument A ∈ A .
• L (A) = in iff ∀B ∈ A ; (B, A) ∈ R ⇒ L (B) = out.
• L (A) = out iff ∃B ∈ A ; L (B) = in ∧ (B, A) ∈ R.
Hereafter, the term “labeling” denotes a complete labeling unless otherwise indicated.
The set {A|A ∈ A , L (A) = in} is a set of accepted arguments corresponding to an extension in the extension-based semantics. The label of an argument is determined only by the labels of its direct attackers.
We use the following notations hereafter.
For a label L, we define ¬L as follows: ¬L = ⎧ in ( if L = out ) ⎨ out ( if L = in ) ⎩ undec ( if L = undec )
L1 ∧ L2 = ⎨⎧ ionut (( iiff (L(1L=1=ouitn∧∧LL22==oouutt)) ∨ (L1 = out ∧ L2 = in) ∨ (L1 = in ∧ L2 = in)) ) ⎩ undec ( otherwise )
In this paper, we adapt semantics by labeling and regard stable semantics as the most natural
semantics for our purpose. An AF may not have a stable extension when it includes an
oddlength cycle, and some semantics were introduced to solve this problem. For example, CF2
semantics [
Definition 5 (stable labeling). Let ⟨A , R⟩ be an AF. For a complete labeling L , if {A|A ∈ A , L (A) = undec} = 0/, then it is called a stable labeling.
Definition 6 (stable AF). An AF with a stable labeling is called a stable AF, and one without it is called an unstable AF.
In addition to these concepts, we introduce several new concepts and terminology. Definition 7 (annihilator). For a new argument P and an attack I from P, a pair ⟨P, I⟩ is said to be an annihilator1 .
Definition 8 (entrance). For an AF F , the argument to which an annihilator is added is called an entrance of F .
Definition 9 (repair). For an unstable AF, the act of revision by adding a single annihilator to obtain a stable AF is called repair2, and the original AF is said to be repairable. An AF that cannot be repaired by taking any argument as an entrance is said to be unrepairable.
An entrance is always labeled out in the repaired AF, according to the definition of stable labeling.
We investigate an AF that consists of odd-length cycles.
Definition 10 (unit, n-lemma). An AF of the form ⟨{A1, A2, . . . , A2n− 1}, {(A1, A2), (A2, A3), . . . , (A2n− 2, A2n− 1), (A2n− 1, A1)}⟩ where n ≥ 1 is called a unit. For a specific number n, it is called an n-lemma.
An AF ⟨{A}, {(A, A)}⟩ is a 1-lemma, which contains a single node with a self-loop.
When an AF includes several units, some arguments are attacked by more than one argument. Such arguments play important roles in judging repairability.
Definition 11 (connector). Let F = ⟨A , R⟩ be an AF. For an argument C ∈ A , if there exists more than one argument A such that (A,C) ∈ R, then C is said to be a connector of F ; if there exists a unique argument A such that (A,C) ∈ R, then C is said to be a non-connector of F . Definition 12 (source/terminal/isolated connector). Let C be a connector of an AF F = ⟨A , R⟩. If there does not exist a connector C′(̸= C) such that (C′,C) ∈ R, and there exists a connector C′′(̸= C) such that (C,C′′) ∈ R, then C is said to be a source connector of F . If there exists a connector C′(̸= C) such that (C′,C) ∈ R, and there does not exist a connector C′′(̸= C) such that (C,C′′) ∈ R, then C is said to be a terminal connector of F . If there exists neither a connector C′(̸= C) such that (C′,C) ∈ R, nor a connector C′′(̸= C) such that (C,C′′) ∈ R, then C is said to be an isolated connector of F .
Example 1. In Figure 2, C is the source connector, D is the terminal connector, E is the
isolated connector, and the other nodes are non-connectors. Generally, there may be multiple
terminal/source/isolated connectors, although this example has one of each.
1In [
Generally, let F = ⟨A , R⟩ be an AF and L be its complete labeling. If C ∈ A is a connector that is attacked by k arguments, that is, (A1,C), (A2,C) . . . , (Ak,C) ∈ R, then L (C) = ¬( (. . . (L (A1) ∧ L (A2)) ∧ . . .) ∧ L (Ak) ). If D ∈ A is a non-connector that is attacked by an argument E, that is, (E, D) ∈ R, then L (D) = ¬L (E).
We clarify our target AF. In a practical stuck argumentation, interactions of cyclic arguments are observed but messy attack relations are uncommon. We set the target AF to reflect this situation.
Definition 13. An AF that satisfies the following [Cond] is called a target AF. [Cond] 1. It includes m-units (m ≥ 1). 2. Each node is included in at least one unit. 3. Each edge is included in at least one unit. 4. It includes no even-length cycle. 5. It is uncontroversial, that is, there exist no arguments A and B connected
by two different paths of even-length and odd-length.
6. No cycle consists only of connectors.
Note that a target AF does not always consist of the same n-lemmas. It may include, for example, 3-lemma and 5-lemma simultaneously. In addition, because of the third condition, we have only one stable labeling after repair.
We seldom encounter controversial argumentation in a practical argumentation, since it is unnatural that an argument A (in)directly attacks another argument B, and at the same time A (in)directly supports B. In such a case, we can divide B into several arguments considering that A attacks some aspect of B and supports another aspect of B.
The last condition excludes any AF that has a cycle containing only connectors, since both stability and repairability of such AFs depend on the configuration of the connectors. We do not discuss this issue here as we want to focus on repair of an unstable AF.
We have proven the following properties when a target AF consists of only trilemmas [
1. It is unstable. 2. It is repairable if there is only one connector or there is only one terminal connector. 3. Each connector is labeled out in the repaired AF. 4. When there is one connector, the connector and the node of an odd-length-far from it that does not attack itself are the entrances. 5. When there is more than one connector, the terminal connector and the node of an oddlength-far from the terminal connector that does not attack a connector are the entrances.
We investigate the properties corresponding to Theorem 1 in the general case of n-lemma.
In the case of a trilemma, the connectors are always connected; thus, there always exist a source connector and a terminal connector when there are multiple connectors. However, there may exist an isolated connector in the general case of an n-lemma.
It is difficult to understand the structure of a large and complicated AF, and it is computationally burdensome to explore its repairability directly. We introduce the reduced form of a target AF, while preserving the labels of connectors, because the label of the connector is the key to considering stability. Because two succeeding non-connectors in a path to a connector do not affect the label of the connector, we can reduce a sequence of non-connectors. Definition 14 (cpath). Let F = ⟨A , R⟩ be an AF and C, D be (possibly the same) connectors of F . Then a path ⟨C, A1, . . . , Ak, D⟩ where A1, . . . , Ak are non-connectors is said to be a cpath(k) from C to D, where k represents the number of nodes between C and D.
Definition 15 (reduction). Let F = ⟨A , R⟩ be an AF. For each pair of connectors Cp and Dp (possibly the same) in F , let p be an arbitrary cpath(k) between these connectors. We define three sets Sp, Tp, and Up: Sp = {A1, . . . , Ak}, Tp = {(Cp, A1), (A1, A2), . . . , (Ak, Dp)}, and Up = {(Cp, Dp)} if k is even, Up = {(Cp, Ep), (Ep, Dp)} where Ep is a new argument if k is odd.
Set S = ⋃︁p Sp, T = ⋃︁p Tp, and U = ⋃︁p Up; and A ′ = A \ S, R′ = (R \ T ) ∪ U . Then F ′ = ⟨A ′, R′⟩ is called a reduced form of F .
At least one connector in a target AF remains a connector in the reduced AF, but not all of them may remain as connectors in the reduced AF.
Example 2. Figure 3 shows target AFs and their reduced forms. In the figures, connectors are shown as red-colored nodes. Both connectors of F1 remain connectors of F ′1 (Figure 3(a)), while one connector of F2 becomes a non-connector of F ′2 (Figure 3(b)).
Proposition 1. Let F ′ = ⟨A , R⟩ be a reduced form of a target AF. Then F ′ is unstable. Proof. Let L ′ be a labeling of F ′.
If there is one connector C in F ′, then only odd-length cycles exist from C to C. Therefore, L ′(C) = undec. Thus, it is unstable.
If there is more than one connector in F ′, let C be any terminal connector. There exists a cycle ⟨C, E, D,C⟩ where D is another connector and E is a non-connector. If L ′(C) = out, then L ′(E) = in, L ′(D) = out, whereas if L ′(C) = in, then L ′(E) = out, L ′(D) = in. Both are contradictions, since C is a terminal connector that has an odd-length cycle from C to C. Therefore, L ′(C) = undec. Thus, F ′ is unstable.
F1 F ′1 (a) both connectors remain
F2 F ′2 (b) one of the connectors becomes a non-connector
Proposition 2. A target AF F is unstable.
Proof. Let F ′ be a reduced form of F , L and L ′ be labelings of F and F ′, respectively.
Let C,C1 and C2 be connectors in F , where C may coincide with C1 or with C2.
For simplicity, assume that there exist exactly two paths to C in F : an odd-length path p from C1 and an even-length path q from C2. Similar discussion is available if there are more than two paths, or only an odd/even-length path exists.
Let p be cpath(2k) from C1 to C and q be cpath(2h + 1) from C2 to C in F , respectively. Then, p is reduced to ⟨C1,C⟩ and q is reduced to ⟨C2, E,C⟩ in F ′, where E is a new node that is a non-connector in F ′.
Figure 4 illustrates a reduction and labeling. In this figure, pink nodes and blue nodes denote arguments labeled in and out, respectively.
(a) target AF: F (b) reduced form of AF: F ′
Then, L (C) = ¬(L (A2k) ∧ L (B2h+1)) = ¬(L (C1) ∧ ¬L (C2)).
On the other hand, L ′(C) = ¬(L ′(C1) ∧ L ′(E)) = ¬(L ′(C1) ∧ ¬L ′(C2)).
This shows that the relationship of the label of connector C and the labels of connectors C1 and C2 is the same in both F and F ′.
Since all connectors in F ′ are labeled undec by Proposition 1, all connectors in F are also labeled undec.
Proposition 3. Let F ′ = ⟨A ′, R′⟩ be a revised AF of the reduced form of a target AF by taking a connector as an entrance. F ′ is stable iff all its connectors are labeled out. Proof. Let L ′ be a labeling of F ′. (⇒) If there is only one connector in F ′, then it is labeled out. If there is more than one connector, then there exist connectors C and D, such that (C, D) ∈ R′ and (D, E), (E,C) ∈ R′ where E is a non-connector in F ′. Assume that there exists a connector labeled in. First, assume that L (C) = in, then L (D) = out, L (E) = in, that is a contradiction. Next, assume that L (D) = in, then L (E) = out, L (C) = out; in this case, there should exist a node labeled in that attacks C; however, there is no such node for a source connector, which is a contradiction. As a result, all connectors in F ′ are labeled out. (⇐) Let L ′ be a labeling of F ′. If L ′(C1) = L ′(C2) = out for all connectors C1 and C2, then L ′(C) = ¬(L ′(C1) ∧ ¬L ′(C2)) = out. Each connector has a path from another connector. Therefore, L ′ is stable.
Proposition 4. Let F = ⟨A , R⟩ be a target AF and F ′ = ⟨A ′, R′⟩ be its reduced form. When F ′ has one connector C, F is repairable by taking C as an entrance.
Proof. When F ′ has one connector C, C is also a connector in F . Assume that C is taken as an entrance of F , and let L be a labeling of the revised F . Then L (C) = out.
When F includes the unique connector C, for a cpath(2k) from C to C, we label in a manner such that L (Ai) = out if i (1 ≤ i ≤ 2k) is even and L (Ai) = in if i is odd. Then, L is stable.
When F includes a connector D other than C, D becomes a non-connector in the reduction process because there is more than one cpath(2k) from C to D. Let these paths be ⟨C, A1, . . . , A2k, D⟩ and ⟨C, B1, . . . , B2h, D⟩, respectively. We label in a manner such that L (Ai) = L (Bi) for each i (1 ≤ i ≤ min(2k, 2h)). Then, D is attacked from A2k and B2h, where L (A2k) = L (B2h) = L (C) = out. Therefore, L (D) = in. Since the length of a path from D to C is odd, L (C) = out. Then, L is stable.
Thus, F is repairable by taking C as an entrance.
Proposition 5. Let F ′ = ⟨A ′, R′⟩ be a reduced AF and F ′ have one terminal connector C. For each connector D other than the terminal connector, if there exists a non-connector that attacks D, then F ′ is repairable by taking C as an entrance; otherwise, F ′ is unrepairable. Proof. Let L ′ be a labeling of the revised AF of F ′ by taking C as an entrance. Then, L ′(C) = out.
If (D,C) ∈ R′, then (C, E), (E, D) ∈ R′ where E is a non-connector; in this case, L ′(E) = in and L ′(D) = out. Otherwise, there exists another connector between D and C. Assume that there exists only one connector C′ in the path from D to C. Then L ′(C′) = out from the above discussion. And we can label all the connectors in the same way. As a result, all the connectors in F ′ are labeled out. Thus, L ′ is stable from Proposition 3.
On the other hand, assume that a connector D other than a terminal connector is not attacked by a non-connector. Then it is attacked only by the connectors labeled out. Then, L ′(D) = in. Thus, L ′ is unstable from Proposition 3. Proposition 6. Let F = ⟨A , R⟩ be a target AF, F ′ = ⟨A ′, R′⟩ be its reduced form, and F ′ have one terminal connector C. For each connector other than the terminal connector, if there exists a non-connector B such that (B,C) ∈ R′, then F is repairable by taking C as an entrance.
This proposition is proved in a similar manner to Proposition 4.
Accordingly, we can check the repairability and entrance taking a reduced form of a target AF. Theorem 2. Let F be a target AF.
1. It is unstable. 2. If F includes one connector, then F is repairable by taking that connector as an entrance. 3. If the reduced form of F includes only one terminal connector, and for each connector other than the terminal connector, there exists a non-connector that attacks it, then F is repairable by taking that terminal connector as an entrance.
We have found that the (terminal) connector in the reduced AF is an entrance of the original target AF. Generally, when a target AF is repairable, there may exist several entrances that do not appear in the reduced AF. These are the nodes that are deleted in the reduction procedure but exist in the original target AF. In this section, we discuss these possibilities depending on the number of connectors.
A target AF includes no connector when it includes a single unit. In this case, we do not consider its reduction.
Proposition 7. A target AF without a connector is unstable. It can be repaired and each node can be taken as an entrance.
Proof. It is trivially unstable and each node can be taken as an entrance.
In this case, the connector of a target AF is also a connector in the reduced AF. Several nodes that do not appear in F ′ can also be taken as entrances.
Proposition 8. Let F be a target AF and F ′ be its reduced form. Let L be a labeling of the AF obtained from F by taking a specified node as an entrance.
When F includes one connector C, let ⟨C, A1, . . . , At , At+1, . . . , Ak,C⟩ be an arbitrary cpath(k) from C to C in F , where A1, . . . , At are shared with all paths from C to C. (1) For 1 ≤ i ≤ k, if Ai is odd-length-far from C, then F is repaired by taking Ai as an entrance; in this case L (C) = out. (2) For 1 ≤ j ≤ t, if A j is even-length-far from C, then F is repaired by taking A j as an entrance; in this case L (C) = in. (3) If we take the other nodes as entrances, the resulting AF is not stable. Example 3. Figure 5 shows an AF F with one connector C. In the figures, pink and blue nodes show the arguments labeled in and out, respectively. There are two cycles ⟨C, a, b, d, e,C⟩ and ⟨C, a, b, d, f ,C⟩. Figure 5(a) shows a stable labeling in which a node a, odd-length-far from C, is taken as an entrance; in this case L (C) = out. Figure 5(b) shows a stable labeling in which a node b, even-length-far from C and shared with all paths from C to C, is taken as an entrance; in this case L (C) = in. Figure 5(c) shows an unstable labeling in which a node f , even-length-far from C but not shared with all paths from C to C, is taken as an entrance.
(a) L (C) = out (b) L (C) = in (c) unstable
Proposition 9. Let F be a target AF and F ′ be its reduced form. Let L be a labeling of the AF obtained from F by taking a specified node as an entrance.
Assume that F includes two connectors C1 and C2, where C1 is the terminal connector in F ′. (1) When C2 is not a connector in F ′, the entrances of F are the same as entrances in the case of one connector. (2) When C2 is a connector in F ′. Let ⟨C, A1, . . . , Ak,C1⟩ be an arbitrary cpath(k) from C to C1 in F , where C is either C1 or C2. For 1 ≤ i ≤ k, if Ai is odd-length-far from C, then F is repaired by taking Ai as an entrance; in this case L (C) = out (Figure 6(a)(b)). (3) If we take the other nodes as entrances, the resulting AF is not stable (Figure 6(c)–(f)).
The proof is shown in the Appendix.
Proposition 10. Let F be a target AF and F ′ be its reduced form. If F ′ includes only one terminal connector C and for each connector C′ other than the terminal connector, there exists a non-connector that attacks C′ in F , then an odd-length-far node A j (1 ≤ j ≤ k) from C in an arbitrary cpath(k) from C to C in F , can be taken as an entrance in F .
Proof. Let L be a labeling of the revised AF. Then L (A j) = out and L (Ak) = in, because the length of the cycle from C to C is odd. Therefore, L (C) = out. The labeling of the other nodes can be done similarly to the case in which C is an entrance. Thus, L is stable. (a) (b) (a)(b) stable labeling (c) (d) (e) (c)–(f) unstable labeling (f)
Dynamic argumentation is an important and interesting topic in practical argumentation. In
general, the main consideration when changing an AF is an enforcing problem [
These studies were concerned mainly with the problem of finding a solution to realize a minimal change in an extension, which enables a specific argument to be credulously/skeptically accepted. In contrast, we focused on the position to which a new argument is added from a topological perspective.
Cyclic arguments have attracted substantial interest with regard to AFs. Gabbay discussed
a method of handling loops that appear in argumentation networks [
The target AFs in these related studies were more congested than ours and most allowed controversial AFs. In this paper, we ignore those topologies that are not frequently observed in practical argumentation.
We discussed repairing AF from a topological perspective, and investigated the position to which an argument is added so that it has a stable labeling.
We proved that an AF consisting of only odd-length cycles with some constraints is unstable, then showed the conditions for repairability using a reduced AF. We also showed the entrances when it is repairable.
When a stuck argumentation is repairable, the set of accepted arguments is obtained depending on the chosen entrance. If we want to make a specific argument accepted, we should take the entrance with which the obtained set includes it. If no such entrance exists, it is impossible that the argument is accepted by adding a single counter-argument. Our approach can help to achieve this goal.
In contrast to existing work regarding dynamic argumentation, we used a topological approach. In addition, we introduced a reduced form of an AF, which provides a proof that is easier to understand and that reduces the computational burden.
In future work, we will consider the following three issues. First, we will clarify the sufficient conditions for repairability in a case of more than two connectors. Second, we will consider the complexity of finding a position that a specific argument can be accepted. Third, we will extend the discussion to other structures of AFs especially an AF including even-length cycles, as well as other semantics.
This work was supported by JSPS KAKENHI Grant Number JP17H06103.
Proof. (1) There does not exist a cycle from C2 to C2, since C2 is not a connector in F ′. Therefore, we do not necessarily make L (C2) = out, and C2 can be considered as a non-connector in F .
(2-1) If Ai is an entrance, then L (Ai) = out. Then, L (Ak) = in, since the length of this cycle is odd. Therefore, L (C1) = out. Labeling of the other nodes proceeds as in the case when C1 is an entrance. Thus, L is stable (Figure 6(a)).
(2-2) If Ai is an entrance, then L (Ai) = out. The length of a path from C2 to C1 in F is odd, since C1 is the terminal connector in F ′. Therefore, L (Ak) = in. Then, L (C1) = out.
The length of any path from C1 to C2 in F is even, since the length of any path from C2 to C1 in F is odd. Therefore, for any cpath(h) from C1 to C2 in F , L (Bh) = in. Then, L (C2) = out. Thus, L is stable (Figure 6(b)).
(3) Let L be a labeling of an AF obtained by adding an annihilator to the node that does not satisfy either condition shown in (1) or (2). We show that a contradiction arises in each case. (3-1) Let ⟨C1, A1, . . . , Ak,C1⟩ be an arbitrary cpath(k) from C1 to C1 in F .
If A j is an entrance that is even-length-far from C1, then L (Ak) = L (A j) = out.
For another cpath(h) from C1 to C1 without an annihilator, L (Bh) = out, since the length of the path is odd. This means that there is no node labeled in that attacks C1 in any path from C1 to C1. Therefore, there should exist a path from C2 to C1 that includes a node labeled in that attacks C1.
Here, we consider the sub-AF Fs obtained by removing all the paths from C1 to C1 from F . Since Fs is also a target AF, it is unstable. The terminal connector C1 in F ′ is not attacked by a node labeled in from the outside of Fs. Therefore, Fs is unstable. As a result, there is no node labeled in that attacks C1 in any path.
Thus, L is unstable (Figure 6(c)).
(3-2) When C2 is a connector in F ′, let ⟨C2, A1, . . . , Ak,C1⟩ be an arbitrary cpath(k) from C2 to C1 in F .
If A j is an entrance that is even-length-far from C1, then L (Ak) = L (A j) = out. Therefore, there should exist a path from C1 to C1 including a node labeled in that attacks C1; which is impossible (Figure 6(d)).
(3-3) When C2 is a connector in F ′, let ⟨C1, A1, . . . , Ak,C2⟩ be an arbitrary cpath(k) from C1 to C2 in F .
If A j is an entrance that is odd-length-far from C1 in a cpath(k) from C1 to C2 in F , then L (Ak) = L (A j) = out, since C2 is even-length-far from C1.
Assume that L (C2) = in. L (C1) = out, since C1 is odd-length-far from C2. As C2 is a connector, there exists either a path from C2 to C2 or another path from C1 to C2. As the length of the path from C2 to C2 is odd, the node in this path that attacks C2 is labeled in. As the length of another path from C1 to C2 is even, the node in this path that attacks C2 is labeled in. Therefore, there exists a node labeled in that attacks C2 in either case; which is a contradiction.
Assume that L (C2) = out. There should exist a node labeled in that attacks C1. As C1 is a connector, there exists either a path from C1 to C1, or another path from C2 to C1. As the length of the path from C1 to C1 is odd, the node in this path that attacks C1 is labeled in. As the length of another path from C2 to C1 is odd, the node in this path that attacks C1 is labeled out. Therefore, there does not exist a node labeled in that attacks C1 in either case; which is a contradiction (Figure 6(e)).
(3-4) When C2 is a connector in F ′, let ⟨C1, A1, . . . , Ak,C2⟩ be an arbitrary cpath(k) from C1 to C2 in F .
Assume that we add an annihilator to A j which is even-length-far from C1 in a cpath(k) from C1 to C2 in F . This is a contradiction by the same discussion with that in (3-3) (Figure 6(f)).
(3-5) When C2 is a connector in F ′, let ⟨C1, A1, . . . , Ak,C2⟩ be an arbitrary cpath(k) from C1 to C2 in F .
Assume that we add an annihilator to A j in a cpath(k) from C2 to C2 in F . This is a contradiction by the same discussion with that in (3-3).