Differential equation based on cost for recursive argumentation graph (DECRAG): Continuous framework for recursive bipolar argumentation with similarity

Abstract

This paper introduces an argumentation framework, differential equation based on cost for recursive argumentation graph (DECRAG), for natural language arguments. DECRAG’s argumentation graph is so flexible that attack and support relations, recursive relations, similarity relations between arguments, and default values can be represented. The base score and the necessity relation can be simulated by the framework. DECRAG maps an argumentation graph to a general system, and the solution of the system is used as the values of the nodes and edges of the graph. We illustrate interesting solutions such as multiple fixed points for the dilemma of even-length cycles and limit cycles for the paradox of odd-length cycles. DECRAG also satisfies many important properties. In this paper, focusing on the theoretical aspect of DECRAG, we show examples, properties, and evaluations of the framework.

Keywords

argumentation graph similarity cost model differential equation averaging

1. Introduction

In this paper, we propose an argumentation framework (AF). The formalism, examples, properties, and evaluations are shown in the following sections. Before this exposition, we briefly survey the field of argumentation and introduce the basic terminology in Section 1.1, detail our approach in Section 1.2, and explain the organization of this paper in Section 1.3.

1.1. Background

Argumentation is used in many fields. For example, design,¹ legal decision,² negotiation,³ online debate,⁴ fact check,⁵ and computational provenance⁶ are based on argumentation. The abstract structure used to represent this argumentation is modeled as an argumentation graph, consisting of a node representing a proposition (atomic argument) and an edge representing an attack relation from a source node to a destination node.

The argumentation graph has been extended to include more semantics found in the actual argumentation. The support relation⁷ is the relation of backing or deducing. The variations of the support relations are simplified into deductive support (the source implies the destination), necessity support (the destination implies the source), and evidential support (the sources collectively imply the destination).² The orientation of attack and support is called polarity, and the inclusion of attack and support is called bipolarity. We collectively call attacks and supports claim relations. Undercutting (attacking another attack) is expected in the real-world discussion. Gottifredi⁸ modified the definition of the argumentation graph so that the destination of an edge can be another edge. This type of attack is called recursive attack (higher-order attack⁹). Recursive support is also used. In this way, the destination of the claim relation can be a proposition or claim relation. To avoid repeating “a proposition or claim relation,” we collectively call propositions and claim relations proclaims in this paper.

The evaluation of an argumentation graph solves the conflict of interest using some reasoning model.⁷ The abstract AF¹⁰ determines the semantics of an argumentation graph as a set of accepted nodes called an extension. As desirable extensions, admissible, preferred, stable, and grounded extensions were proposed. A new extension type has been proposed to seek a more desirable set of nodes.

The extension is a two-way distinction between the accepted nodes and the other nodes. By labeling the nodes, the distinction was elaborated into three-way as accepted nodes labeled IN, rejected nodes (attacked from accepted nodes) labeled OUT, and undecided nodes (other nodes) labeled UNDEC.¹¹ The preorder between nodes¹² and (possibly compound¹³) numerical values on the nodes were introduced for a more precise comparison of the acceptability of the propositions. Depending on the applications, the values are also called strengths, scores, or ranks. In this paper, we call the determination of the values of nodes valuation. The aggregation approach^7,14,15 uses aggregation functions to produce the value of a node from the values of the nodes on which it depends. The equational approach^1,16–20 uses the values that satisfy the simultaneous equations for the nodes and their attacking nodes. The probabilistic approach^21,22 uses the probability that extensions have a node as the value of the node. The game-theoretical approach^23,24 uses the expected reward of the game theory as a value. The neural network approach²⁵ uses a value that minimizes the loss function reflecting the constraints of an argumentation graph. The differential equation approach²⁶ uses the fixed point solution of the differential equations constructed from an argumentation graph as a value. The differential equations are a continuous extension of repeated computation to find a fixed point satisfying simultaneous equations.^18,20

The similarity in argumentation was discussed in various contexts. Konat et al.²⁷ found that the rephrasing often occurred in some corpus from the corpus-based study. Walton^28–30 studied the analogical reasoning based on the similarity of the argument structure. Budán et al.³¹ proposed an analogy AF, which extends the abstract argumentation to cope with an analogy. Amgoud et al.³² proposed a method to reduce an argumentation graph based on the logical equivalence between logical arguments. Boltužić and Šnajder³³ and Reimers et al.³⁴ proposed a method to cluster similar arguments. Budán et al.³⁵ introduced a coalition-based analysis that utilizes the similarity of arguments, where the coalition is a maximal conflict-free set of support relations. Budán et al.³⁶ introduced a method to define the strength of attacks and supports using a custom-made similarity measure. Amgoud et al.³⁷ and Amgoud and David³⁸ introduced adjustment functions to discount the values of similar attackers.

1.2. Approach

We take a general model-based approach. We generate a general model from an argumentation graph and observe the model’s behavior. As a general model, we employ a dynamical system.²⁶ The dynamical system space is the higher-dimensional coordinate system for proclaims. A state at a time is a point in the space. Then, the entire valuation process is divided into the process to generate the dynamical system from an AF and the process to observe the behaviors of the dynamical system. We call the former a simulation phase and the latter an aggregation phrase. The advantage of this approach is that the dynamical system is so general that we can map various semantics onto it. Another advantage is that we can use well-established techniques of observation, such as averaging and sampling.

In the simulation phase, we generate a dynamical system, a map from a time and initial values to values at the time,³⁹ from an AF. The map is obtained from the solution (The solution is a function which satisfies the differential equations. This is not the solution of the minimization of the cost.) of differential equations generated from the AF. Unlike the previous approach using differential equations,²⁶ we generate differential equations from an AF through a cost model. The reason why we employ the cost model is that we need a common domain in which we can compare how each attack or support relation is satisfied, as explained in Sections 5.1.2 and 5.1.3. Each violation of the attack or support relation is specified by an expression of variables bearing values of proclaims. With this cost model placed between an argumentation graph and the dynamical system of differential equations, we have the model on which a sound interpretation of values is possible. Various semantics, such as similarity and negative support, as well as attack and support, can be described as constraints in the cost model. The differential equations are generated from the cost expression to reduce it like the gradient descent method. Unlike the standard gradient descent method, we fix some variables in the constraints of the claim relations to reflect their directionality. We call the cost model with this fixing trick a partially clasped cost model. The differential equations generated from the partially clasped cost try to minimize the cost under the restriction on variables that reflect the directionality of the claim relations. By fixing some variables, the cost may not be minimized by the generated differential equations. The appropriateness of the differential equations for our purpose is proved by the properties they satisfy, as shown in Section 4.

Once the dynamical system is created, we observe the behavior of the system. A point in the dynamical system is in a higher-dimensional coordinate system corresponding to the values of proclaims. From an initial point at time $0$ , the point moves according to the differential equations to form a trajectory. We can observe the trajectories from various initial points to understand the characteristics of an argumentation graph from which the dynamical system is generated.

We can summarize the observation into concise values by averaging over the time of trajectories and further averaging over the space of initial points. The advantage of employing averaging for summarization is that the averaging is linear. Equalities in variables are inherited by a linear combination of variables.

The constructed framework is named the differential equation based on cost for recursive argumentation graph (DECRAG).

1.3. Organization

This paper is organized as follows. Section 2 is devoted to the formalism of DECRAG. First, we introduce the simulation phase in three sections. Section 2.1 introduces an argumentation graph structure. Section 2.2 introduces a partially clasped cost model as the integration model. We derived differential equations to reduce the cost in Section 2.3. Next, we introduce the aggregation phase in three sections. Section 2.4 introduces the definition of time and space averages. Section 2.5 explains the averages for a single variable. Section 2.6 describes the method of sampling calculation of the averages.

Before going into theoretical properties, we show examples to illustrate what can happen in DECRAG in Section 3. The valuation of toy graphs is shown to convey the idea of how DECRAG works in Section 3.1. Section 3.2 shows the similarities and differences between extension-based semantics and DECRAG by examples. Section 3.3 shows the ones between order-based semantics and DECRAG by examples. As variants to the extension semantics, we show examples of the weighted AF (WAF) in Section 3.4, those of the AF with recursive attack (AFRA) in Section 3.5, and those of attack-support AF (ASAF) in Section 3.6. Section 3.7 introduces examples to show how complex the behaviors can be.

Section 4 shows the soundness of DECRAG. Section 4.1 explains the basic properties of DECRAG. The group properties in quantitative bipolar argumentation framework (QBAF) are discussed in Section 4.2. The properties of attack and support not covered by the group properties are discussed in Section 4.3, and the properties of attack only are examined in Section 4.4. The properties of an argumentation graph with acyclic dependency are discussed in Section 4.5, and the properties of similarity are discussed in Section 4.6. The proofs are given in Appendix A.

We evaluate DECRAG in Section 5. The advantages of DECRAG are shown in the use of the applications in Section 5.1 and in the satisfaction of the principles advocated previously in Section 5.2. The disadvantages of DECRAG are examined in Section 5.3. We discuss the possibility of employing the previous work in Section 5.4. Section 6 concludes this paper.

2. Formalism

This section introduces a formalism consisting of an argumentation graph, a partially clasped cost model, differential equations, averaging, and sampling calculation.

2.1. Argumentation graph

Definition 1 Argumentation graph

The argumentation graph of DECRAG is defined as a quintuple $G = (N, E, π, μ, w)$ . $N$ is a finite set of nodes representing propositions. $E$ is a finite set of edges representing the claim relations. $π$ produces the source, destination, and edge type for an edge defined as $π : E \to N \times (N \cup E) \times {\to, \Rightarrow, ↪}$ . $μ$ is a similarity weight between nodes defined as $μ : N \times N \to [0, \infty)$ with $μ (a, b) = μ (b, a)$ for $a, b \in N$ and $μ (a, a) = 0$ for $a \in N$ . $w$ is a claim weight on an edge defined as $w : E \to (0, \infty)$ .

For edge $e \in E$ such that $π (e) = (a, b, t)$ , $a$ is called a source node, $b$ is called a destination node or edge, and $t$ is an edge type of →, ⇒, or $↪$ . If $π (e) = (a, b, \to)$ , the claim relation $e$ means that $b$ should be false if $a$ is true. This claim relation is called an attack. We denote $e$ as a→b if such $e$ is unique. Note that an argumentation graph defined here can have multiple edges between the same combination of a source node, a destination node or edge, and an edge type. If $π (e) = (a, b, \Rightarrow)$ , the claim relation $e$ is called a support (inductive support), meaning that $b$ should be true if $a$ is true. We denote $e$ as a⇒b if such $e$ is unique. We introduce an additional claim “negative support” (Theoretically, there is one more claim relation from $a$ to $b$ meaning that $b$ should be true if $a$ is false. We omit this claim relation in this paper because we never used it.) so that the necessary support can be simulated with the claims of DECRAG, as exemplified in Example 8. If $π (e) = (a, b, ↪)$ , $e$ is called a negative support (In our previous papers,^40,41 we introduced backward support a⋅⇒b to mean that “ $a$ should be false if $b$ is false.” Because the direction of the arrow is confusing, we introduce the negative support in this paper. a⋅⇒b is b $↪$ a.), meaning that $b$ is false if $a$ is false. We denote $e$ as a $↪$ b if such $e$ is unique. For a→b, a⇒b, and a $↪$ b, $a$ is called an attacker, a supporter, and a negative supporter, respectively.

We collectively call a node and an edge a nodge in this paper because “a node or an edge” frequently appears as the destination of a claim. Using nodge, we can say that the destination of a claim relation is a nodge.

Unlike conventional graph theory, the destination of an edge can be an edge again. This edge is called a recursive edge. An attack by a recursive edge is called a recursive attack. We denote a recursive attack a→b for an attack $b = p \to q$ as a→( $p \to q$ ). A recursive support and recursive negative support are defined similarly.

The similarity weight $μ (a, b)$ can be $> 1$ . We denote $μ (a, b)$ as $μ_{a, b}$ for readability.

Definition 2 Isomorphism

An isomorphism of DECRAG from $G = (N, E, π, μ, w)$ to $G^{'} = (N^{'}, E^{'}, π^{'}, μ^{'}, w^{'})$ is the bijection $θ : N \cup E \to N^{'} \cup E^{'}$ such that $θ (N) = N^{'}$ , $θ (E) = E^{'}$ , $π^{'} (θ (e)) = (θ (a), θ (b), t)$ for $π (e) = (a, b, t)$ , $μ^{'} (θ (a), θ (b)) = μ (a, b)$ , and $w^{'} (θ (e)) = w (e)$ .

If there is an isomorphism from $G$ to $G^{'}$ , we say $G$ is isomorphic to $G^{'}$ .

2.2. Partially clasped cost model

We measure how the claim relation and similarity semantics are violated and call the measured violation cost. We make a variable for each proclaim. Because variables are in one-to-one and onto correspondence with nodges, we use them interchangeably and denote a variable with the same symbol for a nodge because whether a symbol represents a variable or a nodge can be understood from the context. If an edge is represented by a→b, a⇒b, or a $↪$ b, we denote it as $V_{a \to b}$ , $V_{a \Rightarrow b}$ , or $V_{a ↪ b}$ , respectively, for readability. In the logical context, a variable has a logical value of true or false. In the numerical context, it has a real value. True is represented by $1$ and false by $0$ by convention.

For attack a→b, we use the term $w | \bar{a} |^{p} | \bar{V_{a \to b}} |^{p} | b |^{p}$ for $p \geq 1$ . This term $w | \bar{a} |^{p} | \bar{V_{a \to b}} |^{p} | b |^{p}$ is the product of $w$ , $| \bar{a} |^{p}$ , $| \bar{V_{a \to b}} |^{p}$ , and $| b |^{p}$ , $| \bar{a} |^{p}$ is the $p$ -th power of $| \bar{a} |$ , and $| \bar{a} |$ is the absolute value of $\bar{a}$ . The coefficient $w$ is a claim weight in $(0, \infty)$ to control the relative strength of the term. We mainly use $p = 2$ , but we tentatively develop a framework for $p \geq 1$ for generality. $\bar{a}$ is a variable with the same value as $a$ and the meaning of the distinction is explained later. Suppose that $V_{a \to b} = 1$ . The term $w | \bar{a} |^{p} | \bar{V_{a \to b}} |^{p} | b |^{p}$ is $1$ if $a = 1$ and $b = 1$ , violating the attack relation, and $0$ if $a = 0$ or $b = 0$ , not violating the attack relation. Suppose that $V_{a \to b} = 0$ . Then, $w | \bar{a} |^{p} | \bar{V_{a \to b}} |^{p} | b |^{p}$ is $0$ , not violating the attack relation for any $a$ and $b$ because the attack a→b is not in effect if $V_{a \to b} = 0$ . Thus, $w | \bar{a} |^{p} | \bar{V_{a \to b}} |^{p} | b |^{p} = 1$ if and only if the constraint of a→b is violated.

Now, we explain the distinction of $\bar{a}$ and $a$ . Some constraints are directional. There is a preference for the variables to change to reduce each cost term. For example, a→b means that $b$ should be false if $a$ is true. The preference is realized by reducing the cost term $w | \bar{a} |^{p} | \bar{V_{a \to b}} |^{p} | b |^{p}$ by changing $b$ but not $a$ or $V_{a \to b}$ . To represent such a preference for variables to change, we introduce a fixed variable denoted by an overline on a variable such as $\bar{a}$ for variable $a$ . We call a variable that is not fixed a free variable. We call the cost model involving fixed variables a partially clasped cost model. The handling of fixed variables in the partially clasped cost is explained in Section 2.3. We use the cost term $w | \bar{a} |^{p} | \bar{V_{a \Rightarrow b}} |^{p} | 1 - b |^{p}$ for a⇒b because a⇒b means that $b$ should be $1$ if $a$ is $1$ . We use the cost term $w | 1 - \bar{a} |^{p} | \bar{V_{a ↪ b}} |^{p} | b |^{p}$ for a $↪$ b because a $↪$ b means $b$ should be $0$ if $a$ is $0$ . The cost term for a claim relation is called a claim term. Each term can have a different weight to adjust the strength of its constraint.

For the recursive attack c→(a→b), the claim term is $w | \bar{c} |^{p} | \bar{V_{c \to (a \to b)}} |^{p} | V_{a \to b} |^{p}$ for the claim weight $w$ . In this term, the variable $V_{a \to b}$ is used for the edge a→b that is the destination of the attack. Because this attack can exist only if the attack a→b exists, the claim term $w^{'} | \bar{a} |^{p} | \bar{V_{a \to b}} |^{p} | b |^{p}$ for the claim weight $w^{'}$ is also included in the cost. Note that $V_{a \to b}$ is a free variable in $w | \bar{c} |^{p} | \bar{V_{c \to (a \to b)}} |^{p} | V_{a \to b} |^{p}$ , and a fixed variable in $w^{'} | \bar{a} |^{p} | \bar{V_{a \to b}} |^{p} | b |^{p}$ .

If a proposition is described in natural language, we encounter similar propositions, and it is natural to assume that similar propositions have similar values. If this adjustment is possible by claim relations, it is simpler and desirable. However, a naive solution of propagating the value of $a$ to $b$ and vice versa, such as a⇒b, a $↪$ b, b⇒a, and b $↪$ a, has the side effect of drawing $a$ and $b$ closer to $0$ or $1$ . This side effect is caused by the positive feedback from a⇒b and b⇒a. So, we need a distinct form and adopt the term $μ_{a, b} | a - b |^{p}$ in the cost, where $μ_{a, b}$ is the similarity weight between $a$ and $b$ . We call this a similarity term. A proper scheme to determine the similarity weight depends on the applications.

We introduce a proposition with the expectation that it is true. The principle that a proposition holds if it is not attacked is called the prima facie principle used in early argumentation.⁴² We use the term $λ_{a} | 1 - a |^{p}$ to represent this constraint and call this term a default true term. The term $λ_{a} | 1 - a |^{p}$ is the product of $λ_{a}$ and $| 1 - a |^{p}$ . The coefficient $λ_{a}$ is a default true weight to adjust the strength of the constraint. $| 1 - a |^{p}$ is the $p$ th power of the absolute value of $1 - a$ . We also introduce the term $ν_{a} | a |^{p}$ to draw $a$ closer to $0$ and call this term a default false term. $ν_{a}$ is a default false weight. We assume $ν_{a} \geq 0$ , $λ_{a} \geq 0$ , and $ν_{a} + λ_{a} > 0$ .

Definition 3 Cost

The cost $Φ_{a}$ for a proclaim $a$ with attacks b_i→a ( $1 \leq i \leq I$ ), supports d_k⇒a ( $1 \leq k \leq K$ ), negative supports e_l $↪$ a ( $1 \leq l \leq L$ ), and similarities to $g_{m}$ ( $1 \leq m \leq M$ ), is defined as $Φ_{a} = \sum_{i = 1}^{I} w_{i} | \bar{b_{i}} |^{p} | \bar{V_{b_{i} \to a}} |^{p} | a |^{p} + \sum_{k = 1}^{K} w_{k} | \bar{d_{k}} |^{p} | \bar{V_{d_{k} \Rightarrow a}} |^{p} | 1 - a |^{p} + \sum_{l = 1}^{L} w_{l} | 1 - \bar{e_{l}} |^{p} | \bar{V_{e_{l} ↪ a}} |^{p} | a |^{p} + \sum_{m = 1}^{M} μ_{g_{m}, a} | g_{m} - a |^{p} + λ_{a} | 1 - a |^{p} + ν_{a} | a |^{p}$ , where $w_{i}$ , $w_{k}$ , and $w_{l}$ are claim weights, $μ_{g_{m}, a}$ is positive similarity weights, and $λ_{a}$ and $ν_{a}$ are default weights. The cost is $Φ = \sum_{a} Φ_{a}$ .

In the following expression, we omit the suffix ranges $1 \leq i \leq I$ , $1 \leq k \leq K$ , $1 \leq l \leq L$ , and $1 \leq m \leq M$ , and the summations $\sum_{i}$ , $\sum_{k}$ , $\sum_{l}$ , and $\sum_{m}$ . The variables $V_{b_{i} \to a}$ , $V_{d_{k} \Rightarrow a}$ , and $V_{e_{l} ↪ a}$ are omitted (assumed to be $1$ ) if their completion is straightforward. The suffix $a$ of $ν_{a}$ and $λ_{a}$ is omitted if they do not depend on $a$ .

2.3. Differential equation

To handle the partially clasped cost, we take the derivatives of the cost with some variables fixed. Because each proclaim has its corresponding variable, the number of variables $D$ is the total number of proclaims. We represent all (not fixed) variables as a vector $x \in [0, 1]^{D}$ for descriptive conciseness. We call positions for the variables of the nodes in $x$ node indexes. The value of the variable $a$ in $x$ is denoted as $x (a)$ . $\bar{x}$ is a vector for the fixed variables. A partially clasped cost $Φ$ is an expression of the variables in $x$ and the fixed variables in $\bar{x}$ . From $Φ$ , the differential equations are derived as follows.

Definition 4 Differential equation

\begin{aligned} \frac{d x}{d t} = - (\nabla_{x} Φ) [\bar{x} := x] . \end{aligned}

In the calculation of $(\nabla_{x} Φ)$ , $\bar{x}$ is treated as a constant. The (textual) substitution $E [x := R]$ is the expression that is the same as $E$ , except that all the occurrences of $x$ are replaced by $(R)$ .⁴³ For example, $(\bar{b} a) [\bar{b} := b] = b a$ , and $({\bar{a}}^{p} a^{p}) [\bar{a} := a] = a^{2 p}$ . The notation “ $[\bar{x} := x]$ ” is meant to replace each fixed variable $\bar{a}$ with its original $a$ . So, $(\nabla_{x} Φ) [\bar{x} := x]$ contains no fixed variables.

Let the order of the variable be $a$ , $b$ , and $V_{b \to a}$ in the vector $x$ . The node indices of this order are $1$ and $2$ . For $Φ = | \bar{b} |^{p} | \bar{V_{b \to a}} |^{p} | a |^{p}, (d a / d t, d b / d t, d V_{a \to b} / d t) = - (\nabla_{x} Φ)$ $[\bar{x} := x] = - (\partial Φ / \partial a, \partial Φ / \partial b, \partial Φ / \partial V_{b \to a})$ $[\bar{a} = a, \bar{b} := b, \bar{V_{b \to a}} := V_{b \to a}] = - (p | \bar{b} |^{p} | \bar{V_{b \to a}} |^{p} sgn (a) | a |^{p - 1}, 0, 0) [\bar{a} = a, \bar{b} := b, \bar{V_{b \to a}} := V_{b \to a}] = (- p | b |^{p} | V_{b \to a} |^{p} sgn (a) | a |^{p - 1}, 0, 0),$ where

\begin{aligned} sgn (x) = {\begin{cases} 1, & x > 0, \\ 0, & x = 0, \\ - 1, & x < 0. \end{cases} \end{aligned}

Thus, we have a set of differential equations:

\begin{aligned} \frac{d a}{d t} = - p | b |^{p} | V_{b \to a} |^{p} sgn (a) | a |^{p - 1}, \frac{d b}{d t} = 0, \frac{d V_{a \to b}}{d t} = 0. \end{aligned}

Suppose that there are attacks b_i→a, supports d_k⇒a, negative supports e_l $↪$ a, and similarities $μ_{g_{m}, a}$ affecting $a$ . Then, $d a / d t = - p (w_{i} | b_{i} |^{p} sgn (a) | a |^{p - 1} - w_{k} | d_{k} |^{p} sgn (1 - a) | 1 - a |^{p - 1} + w_{l} | 1 - e_{l} |^{p} sgn (a) | a |^{p - 1} - μ_{g_{m}, a} sgn (g_{m} - a) | g_{m} - a |^{p - 1} + ν_{a} sgn (a) | a |^{p - 1} - λ_{a} sgn (1 - a) | 1 - a |^{p - 1})$ .

We denote the solution of this differential equation for an initial value $x_{0}$ and time $t$ as $ϕ_{t} (x_{0})$ . The trajectory is a function of $ϕ_{t} (x_{0})$ along $t$ .

For a more precise analysis of the properties of the solution, we fix $p = 2$ hereafter. Then, the differential equation becomes $d a / d t = - 2 (w_{i} b_{i}^{2} + w_{k} d_{k}^{2} + w_{l} (1 - e_{l})^{2} + μ_{g_{m}, a} + ν_{a} + λ_{a}) a + 2 (w_{k} d_{k}^{2} + μ_{g_{m}, a} g_{m} + λ_{a})$ for b_i→a, d_k⇒a, e_l $↪$ a, and $μ_{g_{m}, a}$ .

For an example of $p = 2$ , consider an argumentation graph of a→b. Its cost is $Φ = {\bar{a}}^{2} \bar{V_{a \to b}} b^{2} + ν_{a} a^{2} + ν_{b} b^{2} + λ_{a} (1 - a)^{2} + λ_{b} (1 - b)^{2}$ . Then, $d a / d t = - (\partial Φ / \partial a) [\bar{a} := a, \bar{b} := b, \bar{V_{a \to b}} := V_{a \to b}] = - 2 (ν_{a} a + λ_{a} (1 - a) (- 1))$ .

\begin{aligned} ϕ_{t} (x_{0}) (a) = \frac{λ_{a}}{ν_{a} + λ_{a}} + C e^{- 2 (ν_{a} + λ_{a}) t} \to \frac{λ_{a}}{ν_{a} + λ_{a}} \end{aligned}

t \to \infty

for some constant

C

depending on

x_{0}

. Similarly,

\begin{aligned} ϕ_{t} (x_{0}) (V_{a \to b}) \to \frac{λ_{a \to b}}{ν_{a \to b} + λ_{a \to b}} \end{aligned}

t \to \infty

. Then,

\begin{aligned} \frac{d b}{d t} = & - \frac{\partial Φ}{\partial b} [\bar{a} := a, \bar{b} := b, \bar{V_{a \to b}} := V_{a \to b}] \\ = & - 2 (a^{2} V_{a \to b}^{2} b + ν_{b} b + λ_{b} (1 - b) (- 1)) \\ = & - 2 (a^{2} V_{a \to b}^{2} + ν_{b} + λ_{b}) (b - \frac{λ_{b}}{a^{2} V_{a \to b}^{2} + ν_{b} + λ_{b}}) . \end{aligned}

t \to \infty

\begin{aligned} ϕ_{t} (x_{0}) (b) \to & \frac{λ_{b}}{ϕ_{t} (x_{0}) {(a)}^{2} ϕ_{t} (x_{0}) (V_{a \to b})^{2} + ν_{b} + λ_{b}} \\ \to & \frac{λ_{b}}{{(λ_{a} / (ν_{a} + λ_{a}))}^{2} {(λ_{a \to b} / (ν_{a \to b} + λ_{a \to b}))}^{2} + ν_{b} + λ_{b}} \end{aligned}

t \to \infty

. With

ν_{a} = ν_{b} = ν_{a \to b} = 0

and

λ_{a} = λ_{b} = λ_{a \to b} = 0.1

a \to 1

V_{a \to b} \to 1

, and

b \to 0.1 / 1.1 = 0.09 \dots

2.4. Averaging

The entire solution of the differential equation is a set of trajectories for every initial value. A trajectory for an initial value can converge to a point called a fixed point. The examples are shown in Sections 3.1–3.6 and 3.7.1. There is a possibility that a trajectory does not converge to a fixed point. In this case, if it reaches a periodic oscillation asymptotically, the final stable oscillation is called a limit cycle. The example is shown in Section 3.7.2. There still remains a possibility that a trajectory oscillates aperiodically. (There is no divergent case because a trajectory remains in $[0, 1]^{D}$ for DECRAG by Property 5.). This trajectory is called aperiodic oscillation. The example is shown in Section 3.7.3. For multiple fixed points, limit cycles, and aperiodic oscillations, we may need more compact information, for example, when comparing the values of variables. In such a case, we can use their averages. The time average is the generalization of the limit $t \to \infty$ and abstracts the time domain of trajectories. The time average smooths the limit cycle and the aperiodic oscillation. The time average is used in the properties of Sections 4.2 to 4.6. The space average abstracts the initial value dependency by averaging the time averages on the probability distribution of the initial values. This space average integrates values from multiple fixed points into a single value as shown in Figure 1 in Section 3.1, Figure 2 in Example 1, Figure 3 in Example 6, and Figure 4 in Example 8.

Time Average.
We take the time average of a trajectory. The distinction between a fixed point, a limit cycle, and aperiodic oscillation is lost. Because the entire solution is a set of trajectories, by taking the time average of the entire solution, we have a set of time-averaged values. Definition 5 Time average

The time average $ϕ (x_{0})$ of $ϕ_{t} (x_{0})$ for the initial value $x_{0}$ is $ϕ (x_{0}) = lim_{T \to \infty} (1 / T) \int_{0}^{T} ϕ_{t} (x_{0}) d t$ . If the solution is not fixed points or limit cycles, the time average is not guaranteed to exist.

Figure 1.

Results for the dilemma. $ν = 0$ and $λ = 0.1$ if not stated otherwise. (a) Argumentation graph and (b) histogram of time averages of $a$ for the 1000 randomly sampled initial values. The vertical dashed line shows the space average. (c) $a b$ -plane, $λ = 0.1$ . (d) $a b$ -plane, $λ = 0.25$ . (e) $a b$ -plane, $λ = 0.3$ . (f) Convergence of $a$ . The X-axis is time $T$ in the log scale. Upper and lower lines (their gap is small) indicate the estimated standard deviation above and below the average. (g) Convergence of the space average of $a$ . $T = 100, 000$ . Upper and lower lines indicate the estimated standard deviation above and below the average.

Figure 2.

Results for Figure 3.5 of Wakaki.⁴⁴ $ν = 0$ and $λ = 0.1$ . (a) Argumentation graph, (b) extension, and (c) value. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols after “:” is the same as in Figure 5(b).

Figure 3.

Results for the recursive attack example of Figure 4 of Baroni et al.⁴⁶ (a) argumentation graph, (b) extension, $X = {B, C, E, β, γ}$ , $Y = {B, C, E, F, β, γ, θ, ι}$ , and $Z = {A, B, C, E, β, γ, ζ}$ , and (c) value. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols after “:” is the same as in Figure 5(b). If there is no symbol after “:,” the value is distinct from others. $ν = 0$ and $λ = 0.1$ .

Figure 4.

Value for Figure 15. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols after “:” is the same as in Figure 5(b). If there is no symbol after “:,” the value is distinct from others. For $M = 1000$ , $\bar{σ} (F)$ , $\bar{σ} (J)$ , and $\bar{σ} (θ)$ can be the same. $ν = 0$ and $λ = 0.1$ .

We often omit the integral range of

\int_{0}^{T} d t

\int d t

Space Average.

We take the space average of the time averages (if they exist). The space average is a single value. Trajectories and their time averages depend on the initial values. We may accidentally pick up an exceptional trajectory if a predetermined initial value is used. For example, as to the dilemma of Example 3.1, if we use the initial value at $a = 0.5, b = 0.5$ , the trajectory is trapped at the unstable fixed point at $a = 0.39 \dots$ , $b = 0.39 \dots$ . So, we randomly select initial values according to some probability distribution $P_{0} (x_{0})$ to avoid the trap.

We define the space average $ϕ [P_{0}]$ for a time average $ϕ (x_{0})$ as follows. Definition 6 Space average

The space average $ϕ [P_{0}]$ for $ϕ (x_{0})$ with respect to the probability distribution $P_{0} (x_{0})$ is defined as $ϕ [P_{0}] = \int_{x_{0} \in [0, 1]^{D}} ϕ (x_{0}) P_{0} (x_{0}) d x_{0}$ .

We often omit the integral range of $\int_{x_{0} \in [0, 1]^{D}} d x_{0}$ as $\int d x_{0}$ .

We used the uniform distribution $P_{0} (x_{0})$ for $[0, 1]^{D}$ in the experiments.

\begin{aligned} P_{0} (x_{0}) = {\begin{cases} 1 & x_{0} \in [0, 1]^{D}, \\ 0 & otherwise . \end{cases} \end{aligned}

2.5. Average of variable

Here, we consider the average of a single variable $a$ . The projection of a trajectory to the variable $a$ is $ϕ_{t} (x_{0}) (a)$ . We let $a$ stand for $ϕ_{t} (x_{0}) (a)$ with implicit time $t$ and initial value $x_{0}$ for a concise description.

Time Average.
The time average $σ (a)$ for variable $a$ is defined as $σ (a) = lim_{T \to \infty} (1 / T) \int a d t = lim_{T \to \infty} (1 / T) \int ϕ_{t} (x_{0}) (a) d t = (lim_{T \to \infty} (1 / T) \int ϕ_{t} (x_{0}) d t) (a) = ϕ (x_{0}) (a)$ .

We extend the time average to an expression of variables. For example, $σ (a b) = lim_{T \to \infty} (1 / T) \int ϕ_{t} (x_{0}) (a) ϕ_{t} (x_{0}) (b) d t$ .
Space Average.
The space average $\bar{σ} (a)$ of $σ (a)$ with distribution $P (x_{0})$ is defined as $\bar{σ} (a) = \int σ (a) P_{0} (x_{0}) d x_{0} = \int ϕ (x_{0}) (a) P_{0} (x_{0}) d x_{0} = (\int ϕ (x_{0}) P_{0} (x_{0}) d x_{0}) (a) = ϕ [P_{0}] (a)$ .

The space average of a variable is extended to an expression of variables. For example, $\bar{σ} (a b) = \int σ (a b) P (x_{0}) d x_{0}$ .
2.6. Numerical and sampling computation

This section discusses a practical method to compute trajectories and their time and space averages. Obtaining all the trajectories is intractable because the time domain is continuous and infinite, and the space domain is continuous. We overcome the complexity problem by sampling from a trajectory and sampling from initial values. The sampling error provides the estimation of precision. The details are explained below.

2.6.1. Approximate time average calculation

We explain a method to compute an approximate time average for a variable. Because a trajectory projected to a variable may go to a fixed point while the trajectory projected to another variable goes to a limit cycle, the classification is on a per-variable basis.

We denote the numerically computed solution as $\tilde{ϕ_{t}} (x_{0})$ .

As an error bound, we use the variance of a value. Let $σ_{a}^{2}$ be the variance of the values of variable $a$ . Do not confuse the variance $σ_{a}^{2}$ with the time average $σ (a)$ .

In the following, $mean (S)$ is the average of samples in $S$ , and $var (S)$ is the variance of samples in $S$ . We underline the assumptions as they are required because the assumptions we need are dependent on contexts and difficult to state in one place.

First, we explain methods to calculate the time average for each case.

Fixed point.
The theoretical time average is $ϕ_{t} (x_{0}) (a)$ for $t \to \infty$ by Property 11 in Section 4. Numerically, we calculate ${\tilde{ϕ}}_{T} (x_{0}) (a)$ for large $T$ as the time average.
Limit cycle.
The theoretical time average is $(1 / C) \int_{0}^{C} O (t) d t$ for the limit cycle $O (t)$ and cyclic period $C$ by Property 11. Numerically, we calculate $mean ({{\tilde{ϕ}}_{ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}})$ for large $N_{n}$ as the time average. This $mean ({{\tilde{ϕ}}_{ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}})$ converges to $(1 / \tilde{C}) \int_{t = 0}^{\tilde{C}} \tilde{O} (t) d t$ unless $ξ / \tilde{C}$ is rational for the numerically calculated limit cycle $\tilde{O} (t)$ with cyclic period $\tilde{C}$ .
Aperiodic oscillation.
We assume the existence of the time average. For aperiodic oscillation, we need to consider the start time. Let $ζ_{j}$ be the $j$ th start time randomly sampled from $[0, ξ)$ . Calculate $mean ({y^{(j)} : 1 \leq j \leq N_{m}})$ for $y^{(j)} = mean ({{\tilde{ϕ}}_{ζ_{j} + ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}})$ . This is a sampling calculation of $lim_{T \to \infty} (1 / T) \int {\tilde{ϕ}}_{t} (x_{0}) (a) d t = lim_{N_{n} \to \infty} \frac{1}{ξ} \int_{ζ = 0}^{ξ} mean ({{\tilde{ϕ}}_{ζ + ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}}) d t$ for $ζ$ .

Next, we estimate the variance for each case.
Fixed point.
Let ε be the absolute tolerance, which is a parameter given to a differential equation solver. We assume that the variance of the difference between the calculated time average and the true time average is $ϵ^{2}$ .
Limit cycle.
The theoretical time average is $z = (1 / C) \int_{t = 0}^{C} O (t) d t$ by Property 11. The numerically calculated time average $lim_{T \to \infty} (1 / T) \int {\tilde{ϕ}}_{t} (x_{0}) (a) d t$ goes to $\tilde{z} = (1 / \tilde{C}) \int_{t = 0}^{\tilde{C}} \tilde{O} (t) d t$ . Let $y = mean ({{\tilde{ϕ}}_{ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}})$ . Let $Y$ , $Z$ , and $\tilde{Z}$ be the random variables corresponding to $y$ , $z$ , and $\tilde{z}$ , respectively. $Y - Z = (Y - \tilde{Z}) + (\tilde{Z} - Z)$ . The variance of the first term is estimated to be
$\begin{aligned} \frac{var ({{\tilde{ϕ}}_{ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}})}{N_{n}} \end{aligned}$
for large $N_{n}$ since the distribution of ${{\tilde{ϕ}}_{ξ i} (x_{0}) (a) : 1 \leq i < N_{n}}$ is close to that of ${\tilde{O} (t) (a) : 0 \leq t < \tilde{C}}$ if $N_{n}$ is sufficiently large. We assume the variance of the second term is $ϵ^{2}$ as in the fixed point case. We assume that the sampling error from the numerically calculated limit cycle and the error from the numerical calculation are independent. From these assumptions, the total variance is
$\begin{aligned} \frac{var ({{\tilde{ϕ}}_{ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}})}{N_{n}} + ϵ^{2} . \end{aligned}$

Aperiodic oscillation.
We assume the existence of the time average. For aperiodic oscillation, we need to consider the start time $ζ$ to get the time average $z$ as
$\begin{aligned} z = & lim_{T \to \infty} \frac{1}{T} \int {\tilde{ϕ}}_{t} (x_{0}) (a) d t = lim_{N_{n} \to \infty} \frac{1}{ξ} \int_{ζ = 0}^{ξ} mean ({{\tilde{ϕ}}_{ζ + ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}}) d t . \end{aligned}$
We use a sampling calculation on $ζ$ and let the $j$ th sample be $ζ_{j}$ for $1 \leq j \leq N_{m}$ . Let $z_{j} = lim_{N_{n} \to \infty} mean ({ϕ_{ζ_{j} + ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}})$ for the sample $ζ_{j}$ . Let $y^{(j)} = mean ({{\tilde{ϕ}}_{ζ_{j} + ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}})$ , and $\tilde{z_{j}} = lim_{N_{n} \to \infty} y^{(j)}$ . Let $Y^{(j)}$ , $\tilde{Z_{j}}$ , $Z_{j}$ , and $Z$ be the random variables corresponding to the actual values of $y^{(j)}$ , $\tilde{z_{j}}$ , $z_{j}$ , and $z$ , respectively.
$\begin{aligned} \frac{\sum_{1 \leq j \leq N_{m}} Y^{(j)}}{N_{m}} - Z = & (\sum_{1 \leq j \leq N_{m}} \frac{Y^{(j)} - \tilde{Z_{j}}}{N_{m}}) + (\sum_{1 \leq j \leq N_{m}} \frac{\tilde{Z_{j}} - Z_{j}}{N_{m}}) \\ + (\sum_{1 \leq j \leq N_{m}} \frac{Z_{j}}{N_{m}} - Z) . \end{aligned}$
Then, the variance of the first term is estimated as
$\begin{aligned} mean ({\frac{var ({{\tilde{ϕ}}_{ζ_{j} + ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}})}{N_{n}} : 1 \leq j \leq N_{m}}) . \end{aligned}$
We assume the variance of the second term as $ϵ^{2}$ . The variance of the third term is estimated to be
$\begin{aligned} \frac{var ({y^{(j)} : 1 \leq j \leq N_{m}})}{N_{m}} \end{aligned}$
because $y^{(j)}$ is close to $z_{j}$ . We assume that the sampling error of $Y^{(j)}$ , the numerical error of $\tilde{Z_{j}}$ , and the sampling error of $Z_{j}$ are independent. From these assumptions, the total variance is
$\begin{aligned} \frac{var ({y^{(j)} : 1 \leq j \leq N_{m}})}{N_{m}} + mean ({\frac{var ({{\tilde{ϕ}}_{ζ_{j} + ξ i} (x_{0}) (a) : 1 \leq i \leq N_{n}})}{N_{n}} : 1 \leq j \leq N_{m}}) + ϵ^{2} . \end{aligned}$

If we do not know whether the trajectory is a fixed point, a limit cycle, or an aperiodic oscillation, we can try the calculation for the aperiodic oscillation because additional repetition is harmless, except that it is slow.

The solutions with different start times and initial values are independent. So, the samples can be calculated in parallel.
2.6.2. Approximate space average calculation

To obtain an approximate space average, we have to sample initial values and take the average of the resulting time averages, as in Section 2.6.1. Let $N_{s}$ be the number of samples. Suppose that we have the time averages $y^{(s)}$ for the randomly sampled initial values $(x_{0})_{s}$ for $1 \leq s \leq N_{s}$ .

First, we explain methods to calculate the space average.

Fixed point, limit cycle, aperiodic oscillation.
The space average is $mean ({y^{(s)} : 1 \leq s \leq N_{s}})$ .

Next, we estimate the variance for each case.

Fixed point, limit cycle. Let $z_{s}$ be the true time average for the $s$ th initial value, and $z$ be the true space average. Let $Y^{(s)}$ , $Z_{s}$ , and $Z$ be the random variables for $y^{(s)}$ , $z_{s}$ , and $z$ , respectively.
$\begin{aligned} \frac{\sum_{1 \leq s \leq N_{s}} Y^{(s)}}{N_{s}} - Z = \frac{\sum_{1 \leq s \leq N_{s}} (Y^{(s)} - Z_{s})}{N_{s}} + (\frac{\sum_{1 \leq s \leq N_{s}} Z_{s}}{N_{s}} - Z) . \end{aligned}$
Let $σ_{s}^{2}$ be the variance of $Y^{(s)} - Z_{s}$ . Then, the variance of the first term is
$\begin{aligned} mean ({σ_{s}^{2} : 1 \leq s \leq N_{s}}) . \end{aligned}$
The variance of the second term is estimated to be
$\begin{aligned} \frac{var ({y^{(s)} : 1 \leq s \leq N_{s}})}{N_{s}} \end{aligned}$
because $y^{(s)}$ is close to $z_{s}$ . We assume that the sampling error of each time average and the sampling error of initial values are independent. From this assumption, the total variance is
$\begin{aligned} \frac{var ({y^{(s)} : 1 \leq s \leq N_{s}})}{N_{s}} + mean ({σ_{s}^{2} : 1 \leq s \leq N_{s}}) . \end{aligned}$

Aperiodic oscillation.
We assume the sampling errors of initial values and start positions are independent. Then, we can use the above procedure for the fixed point case with $y^{(s)} = mean ({{\tilde{ϕ}}_{ζ_{s} + ξ i} ((x_{0})_{s}) (a) : 1 \leq i \leq N_{n}})$ for the randomly sampled start time $ζ_{s}$ and the randomly sampled initial value $(x_{0})_{s}$ by combining the sampling of start times and initial values.
2.6.3. Decision on the equality of variables

Here, we discuss the decidability of the equality of values. Suppose that the value $a$ is of a random variable $A$ with an unknown mean and variance $σ_{a}^{2}$ , and $b$ is of a random variable $B$ with an unknown mean and variance $σ_{b}^{2}$ . If $A$ and $B$ share the same mean,

\begin{aligned} \frac{a - b}{\sqrt{σ_{a}^{2} + σ_{b}^{2}}} \end{aligned}

is a value of a random variable

Z

with mean

0

and variance

1

. We assume that

Z

obeys the normal distribution. Then, we consider that

a

and

b

can be the same and denote it as

a \approx_{θ} b

\begin{aligned} - θ < \frac{a - b}{\sqrt{σ_{a}^{2} + σ_{b}^{2}}} < θ \end{aligned}

for some threshold

θ

. In this paper, we use

\approx

to mean

\approx_{3}

. This threshold

3

corresponds to the confidence interval of 99.7%. For a constant

C

, the decision reduces to

\begin{aligned} - θ < \frac{a - C}{σ_{a}} < θ . \end{aligned}

Note that

\approx_{θ}

is not necessarily transitive.

Examples of the decision can be found in Section 3.

3. Examples

In this section, we show examples of how the formalism in Section 2 works. First, to get an impression of DECRAG, the valuation of the toy graphs is shown in Section 3.1. To show the relation between extension-based semantics and DECRAG, examples in their literature are illustrated in Sections 3.2. The relation between order semantics and DECRAG is shown in Section 3.3. As extensions of weights, support, recursive attack, and necessity support, we show examples of the WAF in Section 3.4, those of AFRA in Section 3.5, and those of ASAF in Section 3.6. Finally, as an example of complex behaviors, we show a limit cycle and aperiodic oscillation in Section 3.7.

3.1. Small example

First, we evaluate small graphs. If not stated otherwise, we use the prima facie principle ( $ν = 0$ and $λ > 0$ ).

Figure 5(b) shows the values for the attack in Figure 5(a). The number in parentheses is the estimated standard deviation (square root of the estimated variance) of the above value. $T$ , $ϵ$ , and $N_{s}$ were explained in Section 2.6.1. The variables $a$ and $V_{a \to b}$ have a fixed point at $a = 1, V_{a \to b} = 1$ . Variables having fixed points at $1$ are often omitted in the following table of values. $b$ has a fixed point at $0.090909$ and not straight $0$ because $ν = 0$ and $λ > 0$ . We show the six digits after the decimal point to match the absolute tolerance $10^{- 6}$ of the solution of the differential equation. The equality of values with respect to $\approx$ is shown by the symbols after the values delimited by “:” in Figure 5(b). Values followed by the same symbol are equal. For example, $σ (a) \approx \bar{σ} (a)$ , $σ (V_{a \to b}) \approx \bar{σ} (V_{a \to b})$ , $σ (a) \approx σ (V_{a \to b})$ , and $\bar{σ} (a) \approx \bar{σ} (V_{a \to b})$ . The numerical symbol denotes that the values are the same as the symbol. For example, $σ (a) \approx 1$ , and $σ (V_{a \to b}) \approx 1$ . The values followed by different symbols or no symbols are different. For example, $σ (a) ≉ σ (b)$ , $σ (b) ≉ σ (V_{a \to b})$ , and $σ (b) ≉ 1$ .

Figure 5.

Results for the attack. $ν = 0$ and $λ = 0.1$ . (a) Argumentation graph and (b) value. The number in parentheses is the estimated standard deviation of the above. The values followed by “:1” are the same with respect to $\approx$ . They are the same with $1$ with respect to $\approx$ also. The values followed by “:B” are the same with respect to $\approx$ .

The relation $\approx$ can be represented by symbols only if $\approx$ is transitive because $a$ and $c$ share the same symbol if $a \approx b$ and $b \approx c$ . We used this representation because $\approx$ is transitive in this experiment.

The values of the contradiction in Figure 6(a) are shown in Figure 6(b). $a$ is not low because $a$ ’s attacker ( $a$ itself) is not high.

Figure 6.

Results for the contradiction. $ν = 0$ and $λ = 0.1$ . (a) Argumentation graph and (b) value. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols after “:” is the same as in Figure 5(b).

Figure 7(b) shows the values for the recursive attack in Figure 7(a). a→b is attacked by $c$ and its value is $σ (V_{a \to b}) = 0.090909$ .

Figure 7.

Results for the recursive attack. $ν = 0$ and $λ = 0.1$ . (a) Argumentation graph and (b) value. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols after “:” is the same as in Figure 5(b).

Next, we consider the dilemma (mutual attacks (a→b, b→a)). This dilemma is a special case of Property 13 in Section 4. The argumentation graph is shown in Figure 1(a). Suppose that $ν = 0$ . At a fixed point,

\begin{aligned} σ (a) = \frac{λ}{σ {(b)}^{2} + λ} \end{aligned}

and

\begin{aligned} σ (b) = \frac{λ}{σ {(a)}^{2} + λ} . \end{aligned}

The solution of these simultaneous equations has the following two cases.

Case 1.

$σ (a) = σ (b)$ and $σ {(a)}^{3} + λ σ (a) - λ = 0$ . There is one solution between $0$ and $1$ . If $λ < 1 / 4$ , this fixed point is unstable. Otherwise, this fixed point is stable.

Case 2.

$σ (a) σ (b) = λ$ and $σ (a) + σ (b) = 1$ . There are two solutions when $λ < 1 / 4$ . These fixed points are stable.

The phrase planes (visualization of the directional derivatives³⁹) of $a$ and $b$ are shown in Figure 1(c) ( $λ = 0.1$ ), Figure 1(d) ( $λ = 0.25$ ), and Figure 1(e) ( $λ = 0.3$ ). From Property 17, for large $λ$ , the solution converges to a unique fixed point. In this case, the solution converges to the unique fixed point if $λ \geq 1 / 4$ . If $λ < 1 / 4$ , there are two stable fixed points $f_{1}$ and $f_{2}$ as shown in Figure 8. Figure 1(f) shows the convergence of $σ (a)$ over time. In this case, the solution becomes equal to the true value with respect to $\approx$ soon after time $100$ . The histogram of value $σ (a)$ for 1000 random samples is shown in Figure 1(b). In this experiment, $σ (a) = 0.112702$ occurred $504$ times and $σ (a) = 0.887298$ occurred $496$ times. Thus, the space average $\bar{σ} (a)$ is slightly lower than the theoretical mean of $0.5$ . Figure 1(g) shows the convergence of $\bar{σ} (a)$ along the number of samples. In this case, only a few samples are needed to make the solution equal to the true value with respect to $\approx$ .

Figure 8.

Value for Figure 1. Here, $ν = 0$ and $λ = 0.1$ . The number in parentheses is the estimated standard deviation of the above. The meaning of symbols after “:” is the same as in Figure 5(b).

Next, consider the relation between attack and similarity. The solution of attack a→b with various $μ_{a, b}$ is a fixed point. Figure 9 shows the values of $a$ and $b$ for a→b with different similarity weights. With larger similarity weights, values of $a$ and $b$ get closer. Note that the values of $a$ and $b$ converge to the value $c$ of contradiction c→c (Figure 6), where $c$ ’s default true weight is the sum of the ones of $a$ and $b$ . In this example, $λ_{a} = λ_{b} = 0.1$ and $λ_{c} = λ_{a} + λ_{b} = 0.2$ . This equality means that the strong similarity and the identity are smoothly connected in DECRAG.

Figure 9.

Effect of similarity. Similarity weight versus space average for a→b. $T = 100, 000, N_{m} = 100, ν = 0$ , and $λ = 0.1$ . The dotted line shows the space average of the contradiction with $λ = 0.2$ .

3.2. Classical extension-based semantics

This section shows the similarities and differences between the classical extension-based semantics and DECRAG by examples. First, we recall the definitions of popular extensions.¹⁰ Let $S \subseteq N$ and $S^{+}$ be the set of nodes that are attacked by some member of $S$ . Formally, $S^{+} = {x \in N : s \in S, s \to x}$ . $x \in N$ is acceptable with respect to $S$ if and only if $y \in S^{+}$ holds for every y→x. The set $S$ is conflict-free if and only if for each $x, y \in S$ , x→y does not hold. $S$ is admissible if and only if it is conflict-free, and every $x \in S$ is acceptable with respect to $S$ . The set $S$ is preferred if and only if it is a maximal (with respect to $\subseteq$ ) admissible set. $S$ is stable if and only if it is conflict-free, and $S^{+} = N - S$ . The set $S$ is complete if and only if it is an admissible set such that for each $x$ acceptable with respect to $S$ , $x \in S$ . The set $S$ is grounded if and only if it is the least fixed point of $F (T) = {x \in N : x is acceptable with respect to T}$ . We call the valuation of DECRAG consistent with extension $S$ if the minimum value of nodes in $S$ is higher than the maximum value of nodes in $N - S$ in this paper. The valuation of DECRAG for some initial values is consistent with a stable extension as shown in Property 19. A conflict arises if no stable extension exists, making variables have intermediate values.

Example 1 Extension with a cyclic structure

Figure 2(a) is the argumentation graph in Figure 3.5 of Wakaki,⁴⁴ Figure 2(b) is its extensions, and Figure 2(c) show the values by DECRAG. The fixed point $f_{1}$ is consistent with the extension ${a, d}$ , and $f_{2}$ is consistent with ${b, d}$ , which are admissible, complete, preferred, and stable.

In this example, DECRAG’s fixed points correspond to some extensions.

Next, we consider the effect of contradiction.

Example 2 Extension with contradiction

Figure 10(a) is the argumentation graph in Figure 3.6 of Wakaki,⁴⁴ Figure 10(b) is its extensions, and Figure 10(c) is the values by DECRAG. For the fixed point, $a$ ’s value is high, corresponding to the extension ${a}$ that is admissible, complete, preferred, and grounded. There is no stable extension because $c \in S$ or $c \notin S$ is impossible. In DECRAG, such a conflicting situation makes $c$ have an intermediate value. In addition, $d$ has an intermediate value because $d$ and $c$ have the same attacker $c$ .

Figure 10.

Results for Figure 3.6 of Wakaki.⁴⁴ $ν = 0$ and $λ = 0.1$ . (a) Argumentation graph, (b) extension, and (c) value. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols after “:” is the same as in Figure 5(b).

3.3. Order semantics

We show an example of order semantics.

Example 3 Order semantics example

Figure 11(a) is Figure 1 of Bonzon et al.¹² In DECRAG with $ν = 0$ and $λ = 0.1$ , we found a fixed point as shown in Figure 11(c). The five variables are distinct from the method in Section 2.6. Thus, the order of the space averages is $b > e > d > c > a$ ( $c$ is strictly higher than $a$ ). Figure 11(b) shows the orders by various semantics¹² and that of DECRAG. The result of DECRAG is the same as that Leite and Martins¹⁶ of support AF of SP $_{ϵ}$ with $ϵ = 0.1$ .

Figure 11.

Results for the order semantics example of Figure 1 of Bonzon et al.¹² $ν = 0$ and $λ = 0.1$ . (a) Argumentation graph, (b) orders¹² and that of the space average of (c) value. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols after “:” is the same as in Figure 5(b).

3.4. Weighted AF (WAF)

Dunne et al.⁴⁵ introduced WAF. WAF extends Dung’s AF by associating a weight (a positive real number) with each attack. In WAF, attacks having a total weight less than a given budget can be discarded when deciding on extensions. The counterparts of their weights in DECRAG are the claim term $w_{i} {\bar{b_{i}}}^{2} {\bar{V_{b_{i} \to a}}}^{2} a^{2}$ . In the following example, the valuation of DECRAG is consistent with some low-budget extensions.

Example 4 Weight in WAF

Figure 12(a) is the argumentation graph in Figure 2 of Dunne et al.⁴⁵ Figure 12(b) is $E_{ρ} (β)$ with type $ρ$ (=init, pr, or gr) from Table 1 of Dunne et al.⁴⁵ and Figure 12(f) is the values by DECRAG. We reflect a weight between the nodes of WAF as the weight on the corresponding claim term in DECRAG. Figures 12(c) to 12(e) are the $a_{4} a_{5}$ -phase planes of mutual attacks for various $λ$ s. Different weights for a₄→a₅ and a₅→a₄ skew the curves of $d a_{4} / d t = 0$ and $d a_{5} / d t = 0$ making there be only one fixed point at $λ = 0.1$ . $E_{ρ} (β)$ in the table is the set of extensions for the graphs without edges, the sum of whose weights is less than or equal to $β$ . For $λ = 0.1$ , there is only one fixed point consistent with ${a_{3}, a_{5}, a_{7}, a_{8}}$ found in $E_{g r} (β)$ for $β = 1$ .

Figure 12.

Results for various weights example of Figure 2 of Dunne et al.⁴⁵ $ν = 0$ and $λ = 0.1$ if not stated otherwise. (a) Argumentation graph, (b) extensions for various weights versus $β$ from Table 1 of Dunne et al.⁴⁵ $E_{ρ} (β)$ is the extensions in Figure 12(a) for $β$ and extension type $ρ$ . init is a set of unattacked nodes, pr is a preferred extension, and gr is a grounded extension. (c) $λ = 0.05$ . (d) $λ = 0.1$ . (e) $λ = 0.25$ . and (f) value. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols is the same as in Figure 5(b).

3.5. AF with recursive attack (AFRA)

AFRA⁴⁶ is a pair $(A, R),$ where $A$ is a set of propositions and $R$ is such a set that $R \subseteq A \times (A \cup R)$ .

Example 5 Recursive attack in AFRA

Figure 13(a) is the argumentation graph in Figure 1 of Baroni et al.⁴⁶ Figure 13(b) is the extension, and Figure 13(c) is the values by DECRAG. The values at the fixed point are consistent with the extension ${A, G, P, α, γ, ϵ}$ except for $δ$ , whose value is high in DECRAG because $δ$ is not attacked.

Figure 13.

Results for the recursive attack example of Figure 1 of Baroni et al.⁴⁶ $ν = 0$ and $λ = 0.1$ . (a) Argumentation graph, (b) extension, and (c) value. The number in parentheses is the estimated standard deviation of the above. The meaning of Symbols after “:” is the same as in Figure 5(b).

Example 6 Recursive attack 2 in AFRA

Figure 3(a) is the argumentation graph in Example 2 of Baroni et al.,⁴⁶ Figure 3(b) is the extensions, and Figure 3(c) shows the values by DECRAG. The fixed point $f_{1}$ is consistent with $Y$ except for $ζ$ , $η$ , and $κ$ , which are not attacked. The fixed point $f_{2}$ is consistent with $Z$ except for $η$ , $θ$ , $ι$ , and $κ$ , which are not attacked.

3.6. Attack-support AF (ASAF)

ASAF⁴⁷ is a tuple $(A, R, S),$ where $A$ is a set of nodes, $R \subseteq A \times (A \cup R \cup S)$ is a set of attack relations, and $S \subseteq A \times (A \cup R \cup S)$ is a set of necessity support relations. $S$ is assumed to be acyclic and $R \cap S = \emptyset$ . The recursive relation of ASAF is translated into AF with necessities (AFNs) with no recursive structure. AFN is a tuple $(A, R, N),$ where $A$ is a set of propositions, $R \subseteq A \times A$ is a set of attack relations, and $N \subseteq A \times A$ is a set of necessity support relations. For example, a support $α = (a \Rightarrow b)$ in ASAF is translated into $a \Rightarrow α^{+}$ , $α^{+} \to α^{-}$ , and $α^{-} \to b$ in AFN. AFN is again translated into AF. For example, $h \Rightarrow c$ , $c \to g$ , $g \Rightarrow b$ , $b \to d)$ , $d \to a$ , $e \Rightarrow a$ , and $a \to f$ in AFN is translated into c→g, c→b, b→d, d→a, and a→f in AF. The extension of ASAF is derived from that of the translated AF.

The values by DECRAG are consistent with some extensions except edges having low source nodes, as shown in the following examples.

Example 7 Recursive attack in ASAF

Figure 14(a) is the argumentation graph in Figure 2 of Nofal et al.,¹¹ Figure 14(b) is the extensions, and Figure 14(c) is the values by DECRAG. The fixed point is consistent with the extension ${b, d, e, h}$ except for $f$ . Due to the translation from ASAF to AF via AFN, $f$ is rejected in ASAF. In DECRAG, unattacked $f$ is high even though its source node $c$ is low.

Figure 14.

Results for the recursive attack example of Figure 2 of Nofal et al.¹¹ (a) Argumentation graph, (b) extension, and (c) value. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols after “:” is the same as in Figure 5(b). $ν = 0$ and $λ = 0.1$ .

Example 8 Necessity support in ASAF

Figure 15(a) is the argumentation graph in Figure 1 of Cohen et al.,⁴⁸ Figure 15(c) is the extensions, and Figure 4 is the values by DECRAG.

Figure 15.

Results for the necessity support example of Figure 1 of Cohen et al.⁴⁸ (a) Argumentation graph. $⟶$ is attack and $⟹$ is necessity support. (b) Translated argumentation graph. $⟶$ is attack, $⟹$ is support, and $⇢$ is negative support. Symbols’ suffixes are used to distinguish replicated symbols and have no specific meanings. (c) Extension.

In DECRAG, necessity support is translated into a combination of support and negative support. The rules of the translation are as follows.

Necessity support from $a$ to $b$ is translated into b⇒a and a $↪$ b. For example, necessity support from $B$ to $C$ becomes $C \Rightarrow B$ and $B ↪ C$ .

As one necessity support is translated into two claim relations, we must replicate a claim relation to the necessity support. For example, $δ$ attacking necessity support $β$ should attack both $β$ and $β_{Z}$ after the translation. Thus, we made $δ_{X}$ to attack $β_{Z}$ in addition to $δ$ . Suffixes “ $N$ ,” “ $X$ ,” “ $Y$ ,” and “ $Z$ ” are just used to distinguish replicated symbols and have no specific meanings.

If the destination of the necessity support is an edge, the edge is the source of the translated support. Because a source edge is not allowed in DECRAG, we have to make a new node having the same value as the edge. For example, the necessity support from $H$ to $δ$ is translated into δ⇒ℋ and ℋ $↪$ δ. Instead of δ⇒ℋ, we use N_δ⇒ℋ for $N_{θ}$ having the same value with $θ$ . To make $N_{θ}$ have the same value as $θ$ , we replicate claim relations with $θ$ as their destination to those with $N_{θ}$ as their destinations.

This translation is possible if there is no cyclic dependency involving necessary support. This is not a severe limitation of this translation of necessity support because necessity support relations $S$ are assumed to be acyclic in AFRA. The translated result of Figure 15(a) is Figure 15(b).

The fixed point $f_{1}$ was found in 85.0% and $f_{2}$ in 15.0% of the 1000 initial values. $f_{1}$ is consistent with $P_{2}$ except for $ε$ and $τ$ , which are not attacked. $f_{2}$ is consistent with $P_{1}$ except for $κ$ and $τ$ , which are not attacked.

To guarantee that a replicated nodge has the same values as the original, those variables should have the same values at time $t = 0$ . This restriction on the initial values does not affect the properties in Sections 4.2–4.6 because they hold for any initial value.

Practically speaking, for a necessity support $e$ from $a$ to $b$ , it is better to add claim terms $w (1 - a)^{2} {\bar{V_{e}}}^{2} {\bar{b}}^{2} + w (1 - \bar{a})^{2} \bar{V_{e}} b^{2} = w (1 - a)^{2} \bar{V_{e}} b^{2}$ for the claim weight $w$ to the cost because it can avoid the inflation of the number of nodges. Theoretically, it is still helpful to show that the argumentation graph having necessary support can be translated into the combination of existing claim relations because it can save the labor of proving the properties again for the additional claim term formulae.

3.7. Complex example

In this section, we show examples that exhibit interesting behaviors.

First, we see the general behaviors of even-length cycles. To identify nodes in a cycle, we use a cyclic index defined as an index considered modulo the cycle length, that is, $a_{n + 1} = a_{1}$ for the length $n$ of the cycle.

3.7.1. Even-length cycle (a_i→a_i+1 ( $1 \leq i \leq n$ ) even $n$ , cyclic index)

The dilemma in Section 3.1 is the simplest even-length cycle. In general, an even-length cycle (a_i→a_i+1 ( $1 \leq i \leq n$ ) even $n$ , cyclic index) has multiple fixed points at $f_{1} : a_{1} = 0.11 \dots$ , $a_{2} = 0.89 \dots$ , $a_{3} = 0.11 \dots$ , $\dots$ ; $f_{2} : a_{1} = 0.89 \dots$ , $a_{2} = 0.11 \dots$ , $a_{3} = 0.89 \dots$ , $\dots$ ; and $f_{3} : a_{1} = 0.39, \dots$ , $a_{2} = 0.39, \dots$ , $a_{3} = 0.39, \dots$ , $\dots$ for $ν = 0$ and $λ = 0.1$ from Property 13. Its space average is $a_{1} = 0.5$ , $a_{2} = 0.5$ , $a_{3} = 0.5$ , $\dots$ .

Next, we consider odd-length cycles.

3.7.2. Odd-length cycle (a_i→a_i+1 ( $1 \leq i \leq n$ ) odd $n$ , cyclic index)

For $n = 1$ , this is the contradiction. In general, this has one fixed point from Property 14. Let $ν = 0$ . The fixed point is unstable if $n \geq 7$ for $λ = 0.1$ and $n \geq 5$ for $λ = 0.01$ . From Property 6, we know $0.11 \dots \leq a_{i} \leq 0.89 \dots$ . Let us have a concrete example for $n = 17$ . A trajectory is traced from time $0$ to $100, 000$ , and an oscillation with cyclic period $\approx 62$ was identified. Figure 16(a) shows the oscillation of $a_{1}$ from time $99, 000$ to $100, 000$ . Figure 16(b) shows the limit cycle projected onto two variables. If $a_{1}$ is high, $a_{2}$ is low because $a_{1}$ attacks $a_{2}$ . Figure 16(e) shows the values by DECRAG. $N_{n}$ and $ξ$ were explained in Section 2.6.1. The time averages converge slowly as shown in Figure 16(c). The upper and lower lines indicate the estimated standard deviation above and below the average. Because $a_{1}$ oscillates, the standard deviation of ${ϕ_{ξ i} (x_{0}) (a_{1}) : 1 \leq i \leq N_{n}}$ is large. The space averages converge as shown in Figure 16(d). The upper and lower lines indicate the estimated standard deviation above and below the average. In this example, the variance of the space average does not decrease with an increase in the number of samples because the variance of each time average is much larger than the sampling error.

Figure 16.

Results for the odd-length cycle for $n = 17$ . Here, $ν = 0$ and $λ = 0.1$ . (a) Oscillation of $a_{1}$ at time 99,000–100,000. (b) $a_{1} a_{2}$ -plane. (c) Convergence of the time average of $a_{1}$ . At time $t$ , the time averages up to $t$ are shown. $N_{n} = 100, 000$ , $ξ = 1$ . Upper and lower lines indicate the estimated standard deviation above and below the average. (d) Convergence of the space averages of $a_{1}$ . $N_{n} = 100, 000$ , $ξ = 1$ , $N_{s} = 100$ . Upper and lower lines indicate the estimated standard deviation above and below the average. (e) Value. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols is the same as in Figure 5(b).

Experimentally, we found that the cyclic period becomes longer with larger $n$ and smaller $λ$ as shown in Figure 17.

Figure 17.

Odd-length cycle for $n = 17$ . Here, $ν = 0$ . (a) Cyclic period versus $n$ for $λ = 0.01$ and (b) cyclic period versus $λ$ for $n = 17$ .

As an extension of the periodic oscillation of an odd-length cycle, we found a case of aperiodic oscillation.

3.7.3. Aperiodic oscillation

We show an example of aperiodic oscillation based on the odd-length cycle of Example 3.7.2. Let $a_{i}$ and $b_{j}$ be the odd-length cycles of lengths $n$ and $m$ that have the cyclic periods $A$ and $B$ , respectively. Add a node $c$ and attacks a₁→c and b₁→c. If the ratio $B$ to $A$ is irrational, $c$ behaves as an aperiodic oscillation as Property 15. Figure 18(a) shows the oscillation of $c$ for $n = 31$ , $m = 17$ , $ν = 0$ , and $λ = 0.1$ . Here, we neglected the possibility that the ratio of the cyclic periods of $a_{i}$ and $b_{j}$ is rational because the possibility is $0$ . Figure 18(b) shows the convergence of the time average of a trajectory of $c$ . The upper and lower lines indicate the estimated standard deviation above and below the average. Over time, the estimated variance decreases. Figure 18(c) shows the average of the time averages for samples with different start times. In this example, the variances with different start times are quite small. Figure 19(a) shows the space average of the time averages. Each sample has a different start time. The average is almost constant, but has slight fluctuations for a small number of samples. In this experiment, the variances from different initial values are small compared to those of the time averages.

Figure 18.

Results for the aperiodic oscillation by two limit cycles with $n = 31$ and $n = 17$ . Here, $ν = 0$ and $λ = 0.1$ . (a) Aperiodic oscillation at 99,000 to 100,000 by odd-length cycles for $n = 31$ and for $n = 17$ . (b) Convergence of the time average of $c$ for one trajectory. $N_{n} = 100, 000$ , $ξ = 1$ . Upper and lower lines indicate the estimated standard deviation above and below the average, and (c) convergence of the time average of $c$ with multiple start points. $N_{n} = 100, 000$ , $ξ = 1$ , $N_{m} = 10$ . Upper and lower lines indicate the estimated standard deviation above and below the average.

Figure 19.

Results for the aperiodic oscillation by two limit cycles with $n = 31$ and $n = 17$ . Here, $ν = 0$ and $λ = 0.1$ . (cont.) (a) Converence of the space averages of $c$ . The time averages are for $N_{n} = 10, 000$ , $ξ = 1$ , and $N_{s} = 10$ . Upper and lower lines indicate the estimated standard deviation above and below the average. (b) Point plot of $c$ for increasing $a_{1}$ and $b_{1}$ on the trajectory of $N_{n} = 400, 000$ , $ξ = 1$ . Sampled from $200, 000$ to $400, 000$ . (c) Histogram of values of $c$ for $N_{n} = 400, 000$ , $ξ = 1$ . Sampled from $200, 000$ to $400, 000$ . and (d) value. The number in parentheses is the estimated standard deviation of the above. The meaning of symbols is the same as in Figure 5(b).

In the aperiodic case, the existence of a time average is not apparent. In this example, we can show the existence of the time average by Property 16. In the proof of the property, it was shown that the values of $c$ are uniquely determined from the increasing or decreasing values of $a_{1}$ and $b_{1}$ at $t \to \infty$ . Figure 19(b) shows the 3D plot of the values of $c$ for the increasing values of $a_{1}$ and $b_{1}$ . The other three plots of $c$ for different directions of $a_{1}$ and $b_{1}$ are omitted. Figure 19(c) shows the histogram of the values of $c$ as an approximation to $P (c)$ . Figure 19(d) shows the time and space averages.

4. Theoretical property of DECRAG

In this section, we explore the general properties of DECRAG. In Section 4.1, we explain the properties of DECRAG to show the soundness of DECRAG. These properties are summarized in Table 1. Then, the properties of the previous work are discussed in Sections 4.2 to 4.6 and are summarized in Table 2. The group properties in QBAF are discussed in Section 4.2. Other properties of attack and support are discussed in Section 4.3. The properties of attack are examined in Section 4.4. The properties of an argumentation graph with an acyclic dependency graph are discussed in Section 4.5. The properties of similarity are discussed in Section 4.6.

Table 1.
General property for DECRAG.

Property $p$ Ref.

Existence of solution $p \geq 1$ Property 1

Anonymity Property 2

Reconstructability $p > 1$ Property 3

Non-dilution $p \geq 1$ Property 4

Value range Property 5

Value range at $t \to \infty$ $p = 2$ Property 6

Existence of fixed point Property 7

Continuity Property 8

Constructibility Property 9

Set of numerical solutions Property 10

Time average of fixed point and limit cycle Property 11

Time average of claim relations and similarities Property 12

Even-length cycle solution Property 13

Odd-length cycle solution Property 14

Aperiodic oscillation by multiple odd-length cycles Property 15

Existence of time averages for aperiodic oscillation by multiple odd-length cycles Property 16

$a$ for $ν_{a} \to \infty$ or $λ_{a} \to \infty$ Property 17

Convergence for sufficiently large $λ$ Property 18

Stable extension Property 19

Property	$p$	Ref.
Existence of solution	$p \geq 1$	Property 1
Anonymity		Property 2
Reconstructability	$p > 1$	Property 3
Non-dilution	$p \geq 1$	Property 4
Value range		Property 5
Value range at $t \to \infty$	$p = 2$	Property 6
Existence of fixed point		Property 7
Continuity		Property 8
Constructibility		Property 9
Set of numerical solutions		Property 10
Time average of fixed point and limit cycle		Property 11
Time average of claim relations and similarities		Property 12
Even-length cycle solution		Property 13
Odd-length cycle solution		Property 14
Aperiodic oscillation by multiple odd-length cycles		Property 15
Existence of time averages for aperiodic oscillation by multiple odd-length cycles		Property 16
$a$ for $ν_{a} \to \infty$ or $λ_{a} \to \infty$		Property 17
Convergence for sufficiently large $λ$		Property 18
Stable extension		Property 19

Table 2.

Property of valuation. The results for DECRAG of $p = 2$ are listed for $σ (a)$ and $a$ at $t \to \infty$ or after some time. Some are shown under stronger conditions.⁴¹

Property	Pass/fail	Condition
Group property 1 (GP1)	Property 20	*1 (see Table 3)
Group property 2 (GP2)	Property 21	*1
Group property 3 (GP3)	Property 22	*1
Group property 4 (GP4)	Property 23	*1
Group property 5 (GP5)	Property 24	*1
Group property 6 (GP6)	Property 25	1, 2
Group property 7 (GP7)	Property 26	*1
Group property 8 (GP8)	Property 27	*1
Group property 9 (GP9)	Property 28	*1
Group property 10 (GP10)	Property 29	1, 2
Group property 11 (GP11)	Property 30	1, 2
Balanced	Property 31	1, 2, *3
Strictly balanced	–
Monotonic	Property 32	1, 2
Strictly monotonic	Property 33	1, 2
Compensation	–
Duality	Property 34	*4
Independence (In)	Property 35	*5
Void precedence (VP)	Property 36	*5
Self-contradiction (SC)	–
Cardinality precedence (CP)	–
Quality precedence (QP)	–
Counter-transitivity (CT)	Property 37	2, 5
Strict counter-transitivity (SCT)	Property 38	2, 5
Defense precedence (DP)	Property 39	*5
Distributed-defense precedence (DDP)	Property 40	5, 6
Strict addition of defense branch ( $\oplus$ DB)	–
Addition of defense branch (+DB)	–
Increase of defense branch ( $↑$ DB)	Property 41	5, 7
Addition of attack branch (+AB)	Property 42	5, 7
Increase of attack branch ( $↑$ AB)	Property 43	5, 8
Total (Tot)	Property 44
Nonattacked equivalence (NaE)	Property 45
Unique fixed point	Property 46	*9
Proportional	Property 47	*9
Attack versus full defense (AvsFD)	–
Extreme similarity	Property 48
Group properties 6–11 with similarity	Property 49	1, (2), *10
Effect of similarity to two variables	Property 50	*11

Table 3.

Conditions for Table 2.

*1 No negative support. No recursive relation. No similarity.

ι_{x} = ι

0 < τ (a) < 1

*2 The definitions of

=_{*}

\geq_{*}

, and

>_{*}

are modified.

λ w_{i} = ν w_{k}

for the corresponding

b_{i}

d_{k}

pair.

*4 No support. No negative support. No recursive relation. No similarity.

ι_{x} = ι

*5 No support. No negative support. No recursive relation. No similarity.

ι_{x} = ι

τ (a) = 1

R_{3} (a) = R_{3} (b) = \emptyset

*7 No odd-length path from

a

b

*8 No even-length path from

a

b

*9 Acyclic dependency graph.

*10

μ_{g_{m}, a} = μ_{g_{m}, b}

for any

g_{m} \in N - {a, b}

*11 No cyclic dependency of

a

a^{'}

a

a^{'}

except that from the similarity of

a

and

a^{'}

. Solutions are at a fixed point for any

μ_{a^{'}, a}

4.1. General property

In this section, we explain the general properties of DECRAG. Properties 1 to 5 are proved for $p \geq 1$ except $p > 1$ of Property 3 and Properties 6 to 19 are proved for $p = 2$ .

The differential equations should have a solution for the basic soundness of DECRAG. The following property guarantees it.

Property 1 Existence of solution⁴¹

$d x / d t = - (\nabla_{x} Φ) [\bar{x} := x]$ has a unique solution $ϕ_{t} (x_{0}) \in [0, 1]^{D}$ at time $0 \leq t < \infty$ for given initial value $x_{0} \in [0, 1]^{D}$ . The solution $ϕ_{t} (x_{0})$ is continuous on $t$ and $x_{0}$ .

We confirm that the isomorphic graphs have the same values. To state the property formally, we first introduce some terminology. Corresponding variables by the isomorphism $θ$ are such variables $a$ and $a^{'}$ that $θ (a) = a^{'}$ . Corresponding initial values by the isomorphism $θ$ are such $x_{0}$ and $x_{0}^{'}$ that $x_{0} (a) = x_{0}^{'} (θ (a))$ for any variable $a$ . A function $\bar{θ}$ is a map that relates the corresponding initial values such that $\bar{θ} (x_{0}) = x_{0}^{'}$ .

Property 2 Anonymity²⁰

If argumentation graphs are isomorphic, their corresponding variables have the same values at any time $t$ for the corresponding initial values by the isomorphism.

The anonymity property means that the isomorphic nodes have the same value. Conversely, is it true that node structures are isomorphic if their values are the same? The following property gives an affirmative answer to this question.

Property 3 Reconstructability ( $p > 1$ )

If $p > 1$ , the argumentation graph can be reconstructed from ${ϕ_{t} (x_{0}) : x_{0} \in [0, 1]^{D}, 0 \leq t < \infty}$ and the node indexes.

Reconstructability means that no information is lost during the simulation phase in the sense that all the trajectories and the node indices determine the argumentation graph structure. The node indexes are necessary because if the cost is ${\bar{a}}^{p} {\bar{b}}^{p} c^{p}$ , its original argumentation graph can be of a→c with $b = V_{a \to c}$ or b→c with $a = V_{b \to c}$ .

Next, for formally stating the nondilution property, we formalize the notion of dependency. If b→a is included in an argumentation graph, the value $a$ depends on $b$ and $V_{b \to a}$ . This dependency can be captured as a dependency graph $(V, A)$ for an argumentation graph such that edge $(X, Y) \in A$ means that the value of $X$ depends on that of $Y$ .

Definition 7 Dependency graph

The dependency graph $(V, A)$ of the argumentation graph $G$ is such a graph that $V$ is a set of variables for $G$ and $A$ is a set, each edge in which is from a free variable to other variables in the same cost term of $G$ . A fixed variable is changed to its original variable in $A$ . We neglect the term with a zero similarity weight for a similarity term.

For example, consider an argumentation graph $G$ with only attack b→a. Formally, $G = (N, E, π, μ, w),$ where $N = {a, b}$ , $E = {e}$ , $π (e) = (b, a, \to)$ , $μ_{a, a} = μ_{a, b} = μ_{b, a} = μ_{b, b} = 0$ , and $w (e) = 1$ . For $G$ , the claim term is $w (e) | \bar{b} |^{p} | \bar{V_{b \to a}} |^{p} | a |^{p}$ . Its free variables are only $a$ , and the other variables are $\bar{b}$ and $\bar{V_{b \to a}}$ . Their originals are $b$ and $V_{b \to a}$ . From $μ_{a, a} = μ_{a, b} = μ_{b, a} = μ_{b, b} = 0$ , the similarity terms do not contribute to $A$ . Because the default terms $ν_{a} | a |^{p}$ , $λ_{a} | 1 - a |^{p}$ , $ν_{b} | b |^{p}$ , and $λ_{b} | 1 - b |^{p}$ have no other variables, the default terms do not contribute to $A$ . In summary, $A = {(a, b), (a, V_{b \to a}))}$ . As another example, consider a→a. The cost term is $w (a \to a) | \bar{a} |^{p} | \bar{V_{a \to a}} |^{p} | a |^{p}$ . The free variables are only $a$ , and the other variables are $\bar{a}$ and $\bar{V_{a \to a}}$ . Their originals are $a$ and $V_{a \to a}$ . In summary, $A = {(a, a), (a, V_{a \to a})}$ . For $a$ and $b$ with similarity weight $μ_{a, b} > 0$ , the cost term is $μ_{a, b} | a - b |^{p}$ . The free variables are $a$ and $b$ , and the others are $b$ and $a$ , respectively. So, $(a, b), (b, a) \in A$ . Edges for other forms of the terms are $(a, b)$ and $(a, V_{b \Rightarrow a})$ for b⇒a, and $(a, b)$ and $(a, V_{b ↪ a})$ for b $↪$ a.

Property 4 Nondilution²⁰ and bivariate directionality¹⁵

If a nodge is added to an argumentation graph, the value of a variable that is not an antecedent of the nodge in the dependency graph after addition does not change.

From this property, for a→b, $b$ does not depend on whether a→c, a⇒d, or a $↪$ e exists if $c$ , $d$ , or $e$ does not affect $a$ and $V_{a \to b}$ . Therefore, the value of $b$ can be explained from that of $a$ regardless of what other $a$ attacks, supports, or negative supports. This nondilution means that we can calculate the value of a variable only from the variables on which the variable depends directly. Because a target argumentation graph is only a fraction of an unboundedly large graph, nondilution is a mandatory property a valuation system should have.

Formally, the domain of variables is $R$ . Let us confirm that the actual range of the variables is $[0, 1]$ .

Property 5 Value range⁴¹

For initial value $x_{0} \in [0, 1]^{D}$ , $ϕ_{t} (x_{0}) \in [0, 1]^{D}$ for $t$ such that $0 \leq t < \infty$ .

From now on, we assume $p = 2$ .

At $t \to \infty$ , even if the variables do not converge, the actual range may be narrower than $[0, 1]$ . To find the actual range at $t \to \infty$ , we introduce statements to describe the behavior at $t \to \infty$ . The statement “ $a = b$ at $t \to \infty$ ” means that for any $ϵ > 0$ , there exists $T_{0}$ such that $| a - b | < ϵ$ after $T_{0}$ . The statement “ $a \leq b$ at $t \to \infty$ ” means that for any $ϵ > 0$ , there exists $T_{0}$ such that $a \leq b + ϵ$ after $T_{0}$ .

Property 6 Value range at $t \to \infty$

If node $a$ has attacks $b_{i}$ $(1 \leq i \leq I)$ , supports $d_{k}$ $(1 \leq k \leq K)$ , negative supports $e_{l}$ $(1 \leq l \leq L)$ , and similarities $μ_{g_{m}, a}$ $(1 \leq m \leq M)$ , the value of $a$ is in the following range:

\begin{aligned} \frac{λ_{a}}{I + L + μ_{g_{m}, a} + ν_{a} + λ_{a}} \leq ϕ_{t} (x_{0}) (a) \leq \frac{K + μ_{g_{m}, a} + λ_{a}}{K + μ_{g_{m}, a} + ν_{a} + λ_{a}} \end{aligned}

t \to \infty

. If

a

’s attacker

b_{i}

is in

[{b_{i}}_{min}, {b_{i}}_{max}]

a

’s supporter

d_{k}

is in

[{d_{k}}_{min}, {d_{k}}_{max}]

a

’s negative supporter

e_{l}

is in

[{e_{l}}_{min}, {e_{l}}_{max}]

, and

g_{m}

is in

[{g_{m}}_{min}, {g_{m}}_{max}]

, this range can be improved to

\begin{aligned} \frac{{d_{k}}_{min}^{2} + μ_{g_{m}, a} {g_{m}}_{min} + λ_{a}}{{b_{i}}_{max}^{2} + {d_{k}}_{min}^{2} + (1 - {e_{l}}_{min})^{2} + μ_{g_{m}, a} + ν_{a} + λ_{a}} \leq ϕ_{t} (x_{0}) (a) \leq \frac{{d_{k}}_{max}^{2} + μ_{g_{m}, a} {g_{m}}_{max} + λ_{a}}{{b_{i}}_{min}^{2} + {d_{k}}_{max}^{2} + (1 - {e_{l}}_{max})^{2} + μ_{g_{m}, a} + ν_{a} + λ_{a}} \end{aligned}

t \to \infty

From this, if $λ_{a} > 0$ , then $a > ϵ$ for some $ϵ > 0$ after some time, and if $ν_{a} > 0$ , then $a < 1 - ϵ$ for some $ϵ > 0$ after some time.

At $t \to \infty$ , the variables may not converge as in the examples of Section 3.7. So, does there exist a fixed point at all? The following property gives an affirmative answer to this question.

Property 7 Existence of fixed point⁴¹

$d x / d t = - (\nabla_{x} Φ) [\bar{x} := x]$ has at least one fixed point.

A fixed point can be unstable. The dilemma in Section 3.1 and the odd-length cycle in Section 3.7.2 have unstable fixed points.

Continuity is needed as explained in the smoothness principle of Section 5.2. It is guaranteed under some conditions.

Property 8 Continuity

At a fixed point, the value of a variable changes continuously according to the values of attackers, supporters, negative supporters, similar nodes, and weights if a small perturbation of the values is possible, and $a$ is at a fixed point at the perturbed values.

To perturb the values of an attacker $b$ , we can try adding a new attacker or supporter having a very low value to $b$ . If a small perturbation of the values is not possible, the property does not hold. For example, consider the unstable fixed point $σ (a) = σ (b)$ of the dilemma in Section 3.1. If $b$ is slightly perturbed, its value changes drastically to one of the other fixed points.

The actual domain of a variable is $[0, 1]$ . Is $[0, 1]$ covered by the values of some nodes? This holds as the following property states.

Property 9 Constructibility

If we can choose default weights, we can make a node $a$ with value $z$ by $ν_{a} = 1 - z$ and $λ_{a} = z$ at $t \to \infty$ .

We can create a node that has the value $z$ after some time for the specified default weights and tolerance $ϵ > 0$ .

Next, we consider an approximate solution by numerical calculation.

Property 10 Set of numerical solutions

For corresponding variables $a$ and $b$ in isomorphic argumentation graphs having numerical solutions $\tilde{ϕ_{t}}$ and ${\tilde{ϕ_{t}}}^{'}$ , ${\tilde{ϕ_{t}} (x_{0}) (a) : x_{0} \in [0, 1]^{D}} = {{\tilde{ϕ_{t}}}^{'} (x_{0}) (b) : x_{0} \in [0, 1]^{D}}$ at any $t$ .

The above property carries over to the time and space averages. For the time average, ${\tilde{ϕ} (x_{0}) (a) : x_{0} \in [0, 1]^{D}} = {{\tilde{ϕ}}^{'} (x_{0}) (b) : x_{0} \in [0, 1]^{D}}$ . For the space averages of them, $\tilde{ϕ} [P_{0}] (a) = {\tilde{ϕ}}^{'} [P_{0}] (b)$ .

The time average of a variable has a simple form if its behavior is a fixed point or a limit cycle.

Property 11 Time average of fixed point and limit cycle

If $ϕ_{t} (x_{0}) (a)$ converges to a fixed point $f$ , $ϕ (x_{0}) (a) = f$ . If $ϕ_{t} (x_{0}) (a)$ converges to a limit cycle $O (t)$ with a cyclic period $C$ , $ϕ (x_{0}) = (1 / C) \int_{0}^{C} O (t) d t$ .

The time average is a generalization of a simple limit in this sense.

The time average of claim relations and similarities is related, as shown in the next property.

Property 12 Time average of claim relations and similarities

Suppose that $a$ has b_i→a, d_k⇒a, e_l $↪$ a, and $μ_{g_{m}, a}$ . Then, $σ (a) = (1 / C) (λ_{a} - σ (b_{i}^{2} a) - σ (d_{k}^{2} a) - σ ((1 - e_{l})^{2} a) + σ (d_{k}^{2}) + μ_{g_{m}, a} σ (g_{m}))$ for $C = \sum_{m} μ_{g_{m}, a} + ν_{a} + λ_{a}$ .

We consider properties of even-length and odd-length cycles.

Property 13 Even-length cycle solution

An even-length cycle (a_i→a_i+1 ( $1 \leq i \leq n$ ) even $n$ , cyclic index) for $ν_{a_{i}} = ν$ and $λ_{a_{i}} = λ$ has solutions $σ (a_{i}) = a$ for the real solution of $a^{3} + (ν + λ) a - λ = 0$ , and $σ (a_{2 i}) = a$ and $σ (a_{2 i + 1}) = b$ for such $a, b$ that $a b = ν + λ$ and

\begin{aligned} a + b = \frac{λ}{ν + λ}, \end{aligned}

λ^{2} \geq 4 (ν + λ)^{3} .

Property 14 Odd-length cycle solution

An odd-length cycle (a_i→a_i+1 ( $1 \leq i \leq n$ ) odd $n$ , cyclic index) for $ν_{a_{i}} = ν$ and $λ_{a_{i}} = λ$ has only one solution $σ (a_{i}) = a$ for the real solution of $a^{3} + (ν + λ) a - λ = 0$ .

From these properties and Examples 3.7.1 and 3.7.2, the solutions of even-length cycles for small $λ$ are multiple fixed points, and those of odd-length cycles for large $n$ and small $λ$ are limit cycles.

Property 15 Aperiodic oscillation by multiple odd-length cycles

Suppose that $a_{i}$ and $b_{j}$ are odd-length cycles that have cyclic periods $A$ and $B$ , respectively, and the ratio $B$ to $A$ is irrational. Let a node $c$ be attacked by $a_{1}$ and $b_{1}$ , and $λ_{c} > 0$ . Then, $c$ behaves as an aperiodic oscillation.

Property 16 Existence of time averages for aperiodic oscillation by multiple odd-length cycles

Under the same assumptions as in Property 15, the time average of $c$ exists, and it can be calculated by the sampling calculation.

We can draw $a$ to $1$ by increasing $λ_{a}$ , and draw $a$ to $0$ by increasing $ν_{a}$ as the next property.

Property 17 $a$ for $ν_{a} \to \infty$ or $λ_{a} \to \infty$

If $ν_{a} < \infty$ and $λ_{a} \to \infty$ , $a \to 1$ at $t \to \infty$ , and if $ν_{a} \to \infty$ and $λ_{a} < \infty$ , $a \to 0$ at $t \to \infty$ .

By increasing $λ$ , we can make the trajectory converge.

Property 18 Convergence for sufficiently large $λ$

$ϕ_{t} (x_{0})$ converges to a unique fixed point for any $x_{0}$ for sufficiently large $λ$ .

Next, we consider the relation between the values by DECRAG and the extensions. For terminology, see Section 3.2.

Property 19 Stable extension

Suppose there is no support, negative support, recursive relation, or similarity. If there is a stable extension $S$ , there exists an initial value $x_{0}$ , $ν$ , and $λ$ such that the valuation at any time is consistent with $S$ .

4.2. Group property

In the following sections, if a property is said to hold, the property holds for any initial value $x_{0}$ . If minor additional conditions cannot make a property hold, we show a counterexample. Note that $p = 2$ is still assumed.

In this section, the group properties introduced in QBAF⁴⁹ are discussed. The group properties were designed to be so general that they covered many similar properties with a single property. In the following, we show the simplified definitions for our case. $N$ is the set of nodes. The set of attackers for $a$ is $R^{-} (a) = {b \in N : b \to a}$ and the set of supporters for $a$ is $R^{+} (a) = {b \in N : b \Rightarrow a}$ . $| A |$ is the cardinality of the set $A$ .

$A =_{*} B$ if and only if for $c \in B$ ( $σ (θ (c)) = σ (c)$ and $θ (c) = c$ at any time) for bijection $θ : B \to A$ .

$A \geq_{*} B$ if and only if for $c \in B$ ( $σ (θ (c)) \geq σ (c)$ and $θ (c) \geq c$ at any time) for injection $θ : B \to A$ .

$A >_{*} B$ if and only if $A \geq_{*} B$ and ( $| A | > | B |$ or for some $c \in B$ ( $σ (θ (c)) > σ (c)$ and $θ (c) > c$ at any time)) for injection $θ : B \to A$ .

Because “ $θ (c) = c$ at any time” implies $σ (θ (c)) = σ (c)$ , and “ $θ (c) \geq c$ at any time” implies $σ (θ (c)) \geq σ (c)$ , $σ (θ (c)) = σ (c)$ , and $σ (θ (c)) \geq σ (c)$ are redundant in the above definitions. However, “ $θ (c) > c$ at any time” does not imply $σ (θ (c)) > σ (c)$ , we leave the redundancy in the definitions. $A >_{*} B$ defined here is a bit stronger than $A \geq_{*} B$ and $B ≱_{*} A$ .

In the following, we use the base score. The base score^19,49 is a given proposition value if it does not have an attacker or a supporter. If $a$ has no attacker or supporter, $σ (a) = λ_{a} / (ν_{a} + λ_{a})$ from Property 12. To make $σ (a) = τ (a)$ , let $λ_{a} = τ (a) ι_{a}$ and $ν_{a} = (1 - τ (a)) ι_{a}$ for some $ι_{a} > 0$ .

We assume that there is no negative support, recursive relation, or similarity, $ι_{a} = ι$ , and $0 < τ (a) < 1$ (We can weaken this condition to allow $τ (a) = 0$ or $τ (a) = 1$ , but it considerably complicates the description of properties and proofs. So, we use this stronger condition in this paper.) for any nodge $a$ . From $0 < τ (a) < 1$ , we can guarantee that $a > ϵ$ and $a < 1 - ϵ$ for some $ϵ > 0$ after some time by Property 6. This property guarantees that every claim relation and its source node affect the value of destination nodge, and that no variable is saturated as $0$ or $1$ .

Property 20 GP1

If $R^{-} (a) = \emptyset$ and $R^{+} (a) = \emptyset$ , then $σ (a) = τ (a)$ and $a = τ (a)$ at $t \to \infty$ .

Property 21 GP2

If $R^{-} (a) \neq \emptyset$ and $R^{+} (a) = \emptyset$ , then $σ (a) < τ (a)$ and $a < τ (a)$ after some time.

For some initial value, $a < τ (a)$ does not hold. But eventually $a < τ (a)$ starts to hold. This is the reason why we need the condition “after some time” for $a < τ (a)$ . Formally, this condition is $\forall x_{0} \exists t_{0} > 0 \forall t > t_{0} (ϕ_{t} (x_{0}) (a) < τ (a))$ .

Property 22 GP3

If $R^{-} (a) = \emptyset$ and $R^{+} (a) \neq \emptyset$ , then $τ (a) < σ (a)$ and $τ (a) < a$ after some time.

Property 23 GP4

If $σ (a) < τ (a)$ , $R^{-} (a) \neq \emptyset$ .

Property 24 GP5

If $σ (a) > τ (a)$ , $R^{+} (a) \neq \emptyset$ .

Property 25 GP6

If $R^{-} (a) =_{*} R^{-} (b)$ , $R^{+} (a) =_{*} R^{+} (b)$ , and $τ (a) = τ (b)$ , then $σ (a) = σ (b)$ and $a = b$ at $t \to \infty$ .

Property 26 GP7

If $R^{-} (a) \subset R^{-} (b)$ , $R^{+} (a) = R^{+} (b)$ , and $τ (a) = τ (b)$ , then $σ (b) < σ (a)$ and $b < a$ after some time.

Property 27 GP8

If $R^{-} (a) = R^{-} (b)$ , $R^{+} (a) \subset R^{+} (b)$ , and $τ (a) = τ (b)$ , then $σ (a) < σ (b)$ and $a < b$ after some time.

Property 28 GP9

If $R^{-} (a) = R^{-} (b)$ , $R^{+} (a) = R^{+} (b)$ , and $τ (a) < τ (b)$ , then $σ (a) < σ (b)$ and $a < b$ after some time.

Property 29 GP10

If $R^{-} (a) <_{*} R^{-} (b)$ , $R^{+} (a) = R^{+} (b)$ , and $τ (a) = τ (b)$ , then $σ (b) < σ (a)$ and $b < a$ after some time.

Property 30 GP11

If $R^{-} (a) = R^{-} (b)$ , $R^{+} (a) <_{*} R^{+} (b)$ , and $τ (a) = τ (b)$ , then $σ (a) < σ (b)$ and $a < b$ after some time.

4.3. Property of attack and support

This section discusses the properties of attack and support⁴⁹ not covered by the group properties in Section 4.2. $N$ , $R^{-} (a)$ , $R^{+} (a)$ , $σ (a)$ , $τ (a)$ , $=_{*}$ , $\geq_{*}$ , and $>_{*}$ are the same as in Section 4.2. We use the assumptions in Section 4.2, including $p = 2$ .

Property 31 Balanced

The valuation is balanced if and only if the following holds for any graph satisfying $λ w_{i} = ν w_{k}$ for the corresponding $b_{i}, d_{k}$ pairs.

If $R^{-} (a) =_{*} R^{+} (a)$ , then $σ (a) = τ (a)$ and $a = τ$ at $t \to \infty$ .

If $R^{-} (a) >_{*} R^{+} (a)$ , then $σ (a) < τ (a)$ and $a < τ$ after some time.

If $R^{-} (a) <_{*} R^{+} (a)$ , then $σ (a) > τ (a)$ and $a > τ$ after some time.

Strictly balanced. The valuation is strictly balanced if and only if

If $σ (a) < τ (a)$ , $R^{-} (a) >_{*} R^{+} (a)$ .

If $σ (a) > τ (a)$ , $R^{-} (a) <_{*} R^{+} (a)$ .

Strictly balanced is not satisfied by DECRAG. For example, consider the case where $R^{-} (a)$ has two nodes with value $0.9$ and $R^{+} (a)$ has one node with value $1.0$ . Multiple weaker attacks can overcome a single stronger support in DECRAG.

Shaping triple. A shaping triple for node $a$ is defined as $S T (a) = (τ (a), R^{+} (a), R^{-} (a))$ . $≃_{*}$ , $⪯_{*}$ , and $≺_{*}$ among shaping triples are defined as follows.

$S T (a) ≃_{*} S T (b)$ if and only if $τ (a) = τ (b)$ , $R^{+} (a) =_{*} R^{+} (b)$ , and $R^{-} (a) =_{*} R^{-} (b)$ .

$S T (a) ⪯_{*} S T (b)$ if and only if $τ (a) \leq τ (b)$ , $R^{+} (a) \leq_{*} R^{+} (b)$ , and $R^{-} (a) \geq_{*} R^{-} (b)$ .

$S T (a) ≺_{*} S T (b)$ if and only if $S T (a) ⪯_{*} S T (b)$ and at least one of $τ (a) < τ (b)$ , $R^{+} (a) <_{*} R^{+} (b)$ , and $R^{-} (a) >_{*} R^{-} (b)$ holds.

Property 32 Monotonic

If $S T (a) ≃_{*} S T (b)$ , then $σ (a) = σ (b)$ and $a = b$ at $t \to \infty$ , and if $S T (a) ⪯_{*} S T (b)$ , then $σ (a) \leq σ (b)$ and $a \leq b$ at $t \to \infty$ .

Property 33 Strictly monotonic

If $S T (a) ≺_{*} S T (b)$ , then $σ (a) < σ (b)$ and $a < b$ after some time.

Compensation²⁰^,⁵⁰. If $τ (a) = τ (b)$ , $a$ has fewer supporters than $b$ does, all supporters of $a$ have the same value, all supporters of $b$ have the same value, and the value of supporters of $a$ is higher than that of supporters of $b$ , then $σ (a) = σ (b)$ .

Compensation does not hold for DECRAG. Whether the strength of supporters compensates for the smaller number of supporters depends on the supporters’ values.

Property 34 Duality⁵¹

For $a$ , $\tilde{a} = 1 - a$ at $t \to \infty$ if $\tilde{a}$ has exchanged attackers and supporters of $a$ , $τ (\tilde{a}) = 1 - τ (a)$ , and $ι_{a} = ι_{\tilde{a}}$ .

This property holds even if the trajectory is a limit cycle or aperiodic oscillation.

4.4. Property of attack

In this section, we discuss the properties of attack.¹² $N$ is a set of nodes. $R_{1} (a)$ is a set of direct attackers to $a$ defined as $R_{1} (a) = R^{-} (a)$ of Section 4.2. We assume that there is no support, negative support, recursive relation, or similarity, $p = 2$ , $ι_{a} = ι$ , and $τ (a) = 1$ . From these assumptions, for some $ϵ > 0$ , for any $a$ , $a > ϵ$ after some time, and $σ (a) > 0$ . “ $(a in G)$ ” is the value of $a$ calculated in graph $G$ . It is possible that $(a in G) \neq (a in G^{'})$ . For example, $(a in G) \neq (a in G^{'})$ if $G$ has only b→a and $G^{'}$ has both c→b and b→a. $G \cup G^{'}$ is the disjoint union of $G$ and $G^{'}$ .

Property 35 Independence (In)⁴

Suppose that $G$ and $G^{'}$ do not share a node. For node $a \in G$ , $σ (a in G) = σ (a in G \cup G^{'})$ and $(a in G) = (a in G \cup G^{'})$ at any time for the corresponding initial values by the isomorphism from $G \to$ $G$ ’ part of $G \cup G^{'}$ .

Property 36 Void precedence (VP)

If $R_{1} (a) = \emptyset$ and $R_{1} (b) \neq \emptyset$ , then $σ (a) > σ (b)$ and $a > b$ after some time.

Self-contradiction (SC). If a→a does not hold and b→b holds, $σ (a) > σ (b)$ .

SC does not hold for DECRAG. In DECRAG, the values of $a$ and $b$ depend only on the values of their attackers. Consider an argumentation graph of c→a and b→b. Because $c$ is not attacked and $b$ is attacked, $c > b$ after some time. So, $a < b$ after some time and $σ (a) \leq σ (b)$ .

Cardinality precedence (CP). If $| R_{1} (a) | < | R_{1} (b) |$ , $σ (a) > σ (b)$ .

CP does not hold for DECRAG. If $a$ has an attacker with value $1.0$ and $b$ has two attackers with value $0.5$ , then $a < b$ after some time and $σ (a) \leq σ (b)$ .

Quality precedence (QP). If $\exists c \in R_{1} (b)$ $\forall d \in R_{1} (a)$ $(c > d)$ , then $σ (a) > σ (b)$ .

QP does not hold for DECRAG. If $b$ has an attacker with value $1.0$ and $a$ has two attackers with value $0.99$ , $σ (a) < σ (b)$ .

Property 37 Counter-transitivity (CT)

If $R_{1} (b) \geq_{*} R_{1} (a)$ , then $σ (a) \geq σ (b)$ and $a \geq b$ at $t \to \infty$ .

Property 38 Strict counter-transitivity (SCT)

If $R_{1} (b) >_{*} R_{1} (a)$ , then $σ (a) > σ (b)$ and $a > b$ after some time.

Next, we consider the effect of an indirect attack. Let $R_{i} (a)$ $(i > 0)$ be a multiset (bag) of $b_{i}$ of an attack sequence b_i→b_i−1, b_i−1→b_i−2, $\dots$ , b₁→b₀, $b_{0} = a$ .

Property 39 Defense precedence (DP)

If $| R_{1} (a) | = | R_{1} (b) |$ , $R_{2} (a) \neq \emptyset$ , and $R_{2} (b) = \emptyset$ , then $σ (a) > σ (b)$ and $a > b$ after some time.

For the following property, we need a more detailed structure of the attackers. The defense of $a$ is an attack on the attackers of $a$ . If each $d \in R_{2} (a)$ attacks exactly one $c \in R_{1} (a)$ , the defense of $a$ is called simple. If $| R_{1} (c) | = 1$ for every $c \in R_{1} (a)$ , the defense of $a$ is called distributed. For correspondence $θ$ from $c \in R_{1} (a)$ to its attacker $d \in R_{1} (c)$ , $θ$ is a function if the defense is distributed, and $θ$ is injective if the defense is simple. Because $θ$ is surjective from the definition of $R_{2} (a)$ , $θ$ is bijective if the defense is simple and distributed.

Property 40 Distributed-defense precedence (DDP)

If $| R_{1} (a) | = | R_{1} (b) |$ , $| R_{2} (a) | = | R_{2} (b) |$ , $R_{3} (a) = R_{3} (b) = \emptyset$ , the defense of $a$ is simple and distributed, and the defense of $b$ is simple but not distributed, then $σ (a) > σ (b)$ and $a > b$ after some time.

Next, we need a notion of even and odd branches. ${BR}_{n} (a) = {b \in R_{n} (a) : R_{1} (b) = \emptyset}$ , ${BR}^{+} (a) = ⋃_{even n} {BR}_{n} (a)$ , and ${BR}^{-} (a) = ⋃_{odd n} {BR}_{n} (a)$ . We also need attack sequence $P_{n} (a)$ to $a$ . $P_{n} (a) = (N, E, π, μ, w)$ where $N = {x_{n}, x_{n - 1}, \dots, x_{1}, x_{0} = a}$ , $E = {x_{i} \to x_{i - 1} : 1 \leq i \leq n}$ , $μ (x_{i}, x_{j}) = 0$ $(0 \leq i, j \leq n)$ , and $w ($ x_i→x_i−1 $) = 1$ $(1 \leq i \leq n)$ . We assume that $P_{n} (a)$ and other graphs share only node $a$ . The notation “ $G \oplus P_{2 n} (a)$ ” is the union of $G$ and $P_{2 n} (a)$ at node $a$ . A simple path has no repeating nodes. An odd-length path is a path (possibly not simple) with an odd number of edges. An even-length path is a path (possibly not simple) with an even number of edges.

Strict addition of defense branch ( $\oplus$ oplusDB). For $G$ and $a \in N$ of $G$ , $(a in G \oplus P_{2 n} (a)) > (a in G)$ .

Addition of defense branch (+DB). This property is a specialized version of $\oplus$ DB with the additional condition that $R_{1} (a) \neq \emptyset$ .

The contrary to $\oplus$ DB (and +DB) holds because the addition of an attacker reduces the value of a node, even if the attacker’s value is very low in DECRAG.

Property 41 Increase of defense branch ( $↑$ DB)

For $a, b \in N$ of $G$ , if $b \in {BR}^{+} (a)$ , $b \notin {BR}^{-} (a)$ , and there is no odd-length path from $a$ to $b$ in the dependency graph of $G$ , then $σ (a in G) > σ (a in G \oplus P_{2 n} (b))$ and $(a in G) > (a in G \oplus P_{2 n} (b))$ after some time.

Property 42 Addition of attack branch (+AB)

For $a, b \in N$ of $G$ , if $b \in {BR}^{+} (a)$ , $b \notin {BR}^{-} (a)$ , and there is no odd-length path from $a$ to $b$ in the dependency graph of $G$ , then $σ (a in G) > σ (a in G \oplus P_{2 n + 1} (b))$ and $(a in G) > (a in G \oplus P_{2 n + 1} (b))$ after some time.

Property 43 Increase of attack branch ( $↑$ AB)

For $a, b \in N$ of $G$ , if $b \in {BR}^{-} (a)$ , $b \notin {BR}^{+} (a)$ , and there is no even-length path from $a$ to $b$ in the dependency graph of $G$ , then $σ (a in G) < σ (a in G \oplus P_{2 n + 1} (b))$ and $(a in G) < (a in G \oplus P_{2 n + 1} (b))$ after some time.

Property 44 Total (Tot)

Values of $a$ and $b$ at time $t$ , $σ (a)$ and $σ (b)$ , and $\bar{σ} (a)$ and $\bar{σ} (b)$ can be compared.

Although this property seems simple, “compare” has some implications. First, the scales of the values should be the same. Consider the case where $a = 1$ and $b = 1.0001$ , and $a$ takes a value in $[0, 1]$ and $b$ takes a value in $[0, 2]$ . Numerically $b > a$ , but $b$ is not stronger than $a$ . In DECRAG, the scales of all nodes are in the same range $[0, 1]$ . Second, the values in different graphs should have the same meaning. Assume that the meanings are represented by values. Then, the nodes that have isomorphic dependency structures in each graph should have the same values. In DECRAG, this is guaranteed by the repeated application of Property 4. Third, the reliability of values should be decidable. If $σ (a) = 0.5$ and $σ (b) = 0.4999 \dots$ , we need their error bounds to conclude $σ (a) > σ (b)$ . In DECRAG, the variance estimation in Section 2.6 gives a principled way to decide on the equality of values.

Property 45 Nonattacked equivalence (NaE)

$σ (a) = 1$ and $a = 1$ at $t \to \infty$ for unattacked nodge $a$ .

4.5. Property of acyclic dependency graph

An argumentation graph with an acyclic dependency graph has some good properties. There is no similarity because the dependency graph must be acyclic. We assume that $p = 2$ and $0 \leq τ (a) \leq 1$ .

Property 46 Unique fixed point⁵²

A fixed point is unique, and the solution converges to it at $t \to \infty$ regardless of the initial values.

Property 47 Proportional

If an attack or a negative support is added to $a$ , the decrease of $σ (a)$ is proportional to $σ (a)$ . If a support is added, the increase in $σ (a)$ is proportional to $1 - σ (a)$ .

Attack versus full defense (AvsFD). ${BR}^{-} (a) = \emptyset$ , $| R_{1} (b) | = 1$ , and $R_{2} (b) = \emptyset$ , then $σ (a) > σ (b)$ .

AvsFD does not hold for DECRAG. Suppose that c_i→a $(1 \leq i \leq n)$ , d→c_i, d→b, $ν = 0$ , and $λ > 0$ . Then, $σ (a) = λ / [\sum_{i} σ (c_{i})^{2} + λ]$ , $σ (b) = λ / [σ {(d)}^{2} + λ]$ , $σ (c_{i}) = λ / [σ {(d)}^{2} + λ]$ , and $σ (d) = 1$ . For large $n$ , $\sum_{i} σ (c_{i})^{2} > σ {(d)}^{2}$ and $σ (a) < σ (b)$ .

4.6. Property of similarity

There is no restriction on an argumentation graph. We assume that $p = 2$ and $0 \leq τ (a) \leq 1$ .

Property 48 Extreme similarity⁴¹

If $μ_{a, b} \to 0$ , the result is the same as without the similarity term. For any $ϵ > 0$ and $T_{0} > 0$ , we can find $μ_{a, b}$ such that $| a - b | < ϵ$ after $T_{0}$ .

Note that even if the solution of an argumentation graph converges at $t \to \infty$ , additional similarity can make the solution of the graph not converge. For example, consider an odd-length cycle a_i→a_i+1 ( $1 \leq i \leq n$ , odd $n$ , cyclic index) having a limit cycle as in Example 3.7.2. We can make $d$ with a low value using Property 9. By adding a similarity between $d$ and $a_{1}$ , $a_{1}$ becomes lower, making the odd-length cycle converge.

Property 49 GP6–GP11 with similarity

Properties 25(GP6)–30(GP11) hold with similarity under the same assumptions of GP6–GP11 and $μ_{g_{m}, a} = μ_{g_{m}, b}$ for any $g_{m} \in N - {a, b}$ .

Property 50 Effect of similarity to two variables

Suppose that $a$ has b_i→a, d_k⇒a, e_l $↪$ a, $μ_{g_{m}, a}$ , and $a^{'}$ has b_i′→a′, d_k′⇒a′, e_l′ $↪$ a′, $μ_{g_{m}^{'}, a^{'}}$ , there is no cyclic dependency between $a$ and $a^{'}$ except for the similarity of $a$ and $a^{'}$ , and solutions are at a fixed point for any $μ_{a^{'}, a}$ . Then,

\begin{aligned} σ (a) = \frac{C C^{'} a_{0} + μ_{a^{'}, a} C a_{0} + μ_{a^{'}, a} C^{'} a_{0}^{'}}{C C^{'} + μ_{a^{'}, a} C + μ_{a^{'}, a} C^{'}}, \end{aligned}

where

C = σ (b_{i})^{2} + σ (d_{k})^{2} + (1 - σ (e_{l}))^{2} + ν_{a} + λ_{a}

C^{'} = σ ({b^{'}}_{i})^{2} + σ ({d^{'}}_{k})^{2} + (1 - σ ({e^{'}}_{l}))^{2} + ν_{a^{'}} + λ_{a^{'}}

, and

a_{0}

and

a_{0}^{'}

are

σ (a)

and

σ (a^{'})

when

μ_{a^{'}, a} = 0

, respectively.

Thus, $σ (a)$ is an interpolation of $a_{0}$ and $a_{0}^{'}$ . If $μ_{a^{'}, a} = 0$ , $σ (a) = a_{0}$ . As $μ_{a^{'}, a} \to \infty$ ,

\begin{aligned} σ (a) = \frac{C a_{0} + C^{'} a_{0}^{'}}{C + C^{'}} . \end{aligned}

This expression can be interpreted as that

a

is drawn to a more “discussed” proclaim having more attackers, supporters, and negative supporters.

5. Evaluation

In this section, we evaluate DECRAG from various aspects. First, we confirm the advantages of DECRAG in applications in Section 5.1 and in previous principles in Section 5.2. Then, we summarize the disadvantages of DECRAG in Section 5.3. We examine the possibility of using the previous work in Section 5.4.

5.1. Advantage in application

We analyzed the classroom discussion, the formal committee and informal social networking service (SNS) discussions, and the discussion on the ethical problem.^40,41,52 We applied a preliminary AF of DECRAG to the analysis, and the results are briefly explained in Sections 5.1.1–5.1.3. The advantages of DECRAG in these applications are summarized in Section 5.1.4.

5.1.1. Classroom discussion

We collected student opinions on user information regulation as argumentation graphs.^40,41 To encourage students to add propositions and claim relations, the outcome of the addition must be predictable. So, a proposition should come stronger if it gets an additional supporter, even if the value of the supporter is low. This predictability is partially guaranteed in Properties 26 and 27.

We collected 97 graphs with 1707 propositions in total from 68 participants. The graphs consisted of attacks and supports that were attacked and supported recursively. There was no cycle of attack or support. We merged propositions with the same text and reduced the number of unique propositions to 1153. If a→b is in one argumentation graph and a⇒b is in another, we were unable to merge them in one argumentation graph because our initial framework did not allow multiple edges between nodes. This limitation was removed in our current framework utilized in Section 5.1.3. We resolved this conflict by fixing edge types by manually inspecting the source and destination proposition texts and merging the argumentation graphs into a single graph. The solution of the merged argumentation graph had a single fixed point.

Some similar propositions had different values. Examples are the propositions “The management side can handle it quickly and easily” and “The management side can respond quickly to the problem.” They had $1.00$ and $0.02$ as their values without considering similarity. Using the similarity weights calculated from the document vectors of the propositions, we could obtain more reasonable values of $0.30$ and $0.17$ by DECRAG. Other examples were “The constitution is the highest law,” “Supreme law,” “It is difficult to regulate and manage,” and “Regulatory standards are difficult.”

Some textually similar propositions were not so similar. An example was “You should protect your right to know” and “You should also protect your right to know.” They were textually similar, but their points were slightly different. Another example was “damage caused by leakage of personal information” and “leading to leakage of personal information.” These propositions were textually similar, but the focus of the former was on the damage, while that of the latter was the leakage. These propositions had strong similarities, but the structure reflected the difference in focus, resulting in the difference in values. So, we tuned the balance between the similarities and claim relations using the claim and similarity weights.

5.1.2. Formal committee discussion and informal SNS discussion

We analyzed the formal discussion of the committee on the topic “lowering the legal age of majority” and the informal SNS discussion on the topic “second Tomei expressway construction.”⁵² We encoded these discussions into argumentation graphs. We are interested in the development of the discussions, and the graph development was recorded at each addition of propositions. Intuitively, we expected that the formal committee discussion would be convergent, while the informal SNS discussion would be divergent. To confirm this, we needed an objective measure of how convergent a discussion was. We considered that the claim terms of the argumentation graph, especially the last argumentation graph, of a convergent discussion have small values at a fixed point. For example, suppose that an argumentation graph is b→a and c⇒a conflicting at $a$ . With $ν = 0$ and $λ = 0.1$ , we have $a = 0.52$ , $b = 1.00$ , and $c = 1.00$ . The total of the claim terms is roughly ${1.00}^{2} {0.52}^{2} + {1.00}^{2} (1 - 0.52)^{2} = 0.50$ . The conflict is resolved if d→c is added to this graph. Now, the argumentation graph is b→a, c⇒a, and d→c. With $ν = 0$ and $λ = 0.1$ , the total of the claim terms is approximately ${1.00}^{2} {0.10}^{2} + {0.09}^{2} (1 - 0.10)^{2} + {1.00}^{2} {0.09}^{2} = 0.02$ , which is much smaller than $0.50$ . So, we used the total of the claim terms as the measure of convergence. We confirmed that the formal committee discussion was convergent and the informal SNS discussion divergent. For this comparison, we utilized the partially clasped cost model.

5.1.3. Discussion on ethical problem

We collected opinions⁵² on the ethical problem of

There are five workers ahead of a trolley that is out of control without brakes, and if we do nothing, they will surely die. If you push off the man on the bridge, you can save the five workers, but the person you push off will surely die. Should we push him off?⁵³

The propositions were given, and the participants submitted claim relations that included recursive attacks and supports. We merged the 45 argumentation graphs into a single argumentation graph. This merge was possible using the weights so that the weight of $1$ was given to an edge included in all the argumentation graphs and the weight of $1 / 45$ was given to an edge included in one argumentation graph. The solution of the merged argumentation graph had a fixed point. At the fixed point, the conclusion proposition “Should we push him off?” had a value of $0.7$ , which was lower than the value for the other variants of the ethical problem. Let the conclusion proposition be $a$ . Then, $d a / d t = - (b_{i}^{2} + d_{k}^{2} + λ) a + (d_{k}^{2} + λ) = 0$ at the fixed point. The largest positive term of this expression was the default weight $λ$ , different from the other variants of the problem. We could interpret this as saying that the participants were reluctant to support the conclusion proposition “Should we push him off?” For this analysis, we used the feature that the origins of the terms in differential equations were known.

5.1.4. Application summary

We could merge multiple argumentation graphs into one argumentation graph in DECRAG utilizing the weights of claim terms (Section 5.1.3).

Similarity was used in analyzing classroom discussions (Sections 5.1.1 and 5.1.3). The advantage of DECRAG in handling similarity is that it does not restrict the similarity measure. Another advantage is that the balance between similarities and claim relations can be controlled.

The total of claim terms at a fixed point could be used to assess the degree of convergence (Section 5.1.2). This use of the claim terms is the advantage of employing the partially clasped cost model between an argumentation graph and differential equations.

In differential equations, claim terms, similarity terms, and default terms are separated. From this separation, we could identify the decisive effects of these terms at a fixed point (Section 5.1.3).

5.2. Advantage in principle

This section shows that DECRAG satisfies the principles advocated in the previous works.

Similarity.
Rephrasing often occurred in some corpus from the corpus-based study.²⁷ DECRAG draws the values of rephrased propositions closer with the help of the large similarity weight of rephrased propositions.

Two text units, despite being semantically similar, could serve different argumentative functions.²⁷ Because the reliability of the similarities and claim relations varies depending on the application, the balance between them should be controlled. DECRAG controls the balance using the claim and similarity weights.

If a degree of similarity was lowest, the result should be the same as without the similarity.³⁸ In DECRAG, if the degree of similarity is lowest $0$ , the result is the same as that without the similarity by Property 48.
Flexibility.
The requirements for an AF vary from one application to another.⁴ To meet the requirements, the framework should be adjustable to various requirements. In DECRAG, weights can be used to adjust the framework to applications, and the framework is open to further extension by adding cost terms.

An unrestricted recursive notion of attack to attack was proposed.⁴⁶ DECRAG allows unrestricted recursion of any combination of attack, support, and negative support.

In some previous work, a necessity support was used. The necessity support was elaborated as (1) “If $b$ is accepted, then $a$ is accepted.” (2) “If $c$ attacks $a$ and $a$ is necessary for $b$ , then $c$ attacks $b$ .” (3) “If $a$ attacks $c$ and $a$ is necessary for $b$ , then $b$ attacks $c$ .”⁵⁴ Suppose that $c$ is true. Because $a$ is attacked by $c$ , $a$ is false. From (2), if $a$ is necessary for $b$ , then $b$ is false because $b$ is attacked by $c$ . In total, $b$ is false if $a$ is false. This relation is the negative support. DECRAG is flexible enough to add the negative support.
Separability.
The separability⁵⁵ requires attackers and supporters to be independent if no explicit relations among them exist. Using the fixed variable of DECRAG, for a→b, $b$ does not affect $a$ and for c⇒b, $b$ does not affect $c$ . In this way, attackers and supporters are separated. Also, an attacker $a$ and attack a→b are separated in such a way that they have different variables $a$ and $V_{a \to b}$
Non-dilution.
Non-dilution²⁰ (bivariate directionality¹⁵) requires that if a proclaim is added to an argumentation graph, values of the proclaims that do not depend on it do not change. In DECRAG, non-dilution is guaranteed by Property 4.
Bipolarity-bias.
The three types of bipolarity-bias¹⁵ are (1) attacks are as important as supports, (2) attacks are more important than supports, and (3) supports are more important than attacks. To reflect the bipolarity-bias, an attack and a support can have different weights in DECRAG.

To guarantee that the bipolarity-bias control is fair, we can require that attacks and supports work symmetrically by exchanging attacks and supports if other conditions are symmetrical. In DECRAG, the symmetry on attack and support is guaranteed by Property 34.
Paradox and dilemma.
Even-length cycles could be seen as representing dilemmas, and odd-length cycles could be seen as representing paradoxes, and that, in many contexts, it was desirable to be able to represent and distinguish paradoxes and dilemmas.⁵⁶ In DECRAG, even-length cycles have three fixed points for small $λ$ by Property 13, and their solutions converge to two fixed points in Example 3.1. Odd-length cycles have one fixed point by Property 14, and their solutions are limit cycles for large $n$ and small $λ$ in Example 3.7.2. These properties agree with the view that even-length cycles represent dilemmas and odd-length cycles represent paradoxes. In DECRAG, the combination of paradoxes can be more complex than in Example 3.7.3.
Open-mind.
The open-minded semantics⁵¹ should be able to move the values of variables arbitrarily close to the extreme values $0$ or $1$ if sufficient evidence against or for the proposition is given. In DECRAG, it is possible to make a proposition take any value between $0$ and $1$ as guaranteed by Property 9.
Total.
All values should be comparable.¹² In DECRAG, this is guaranteed by Property 44.

Order semantics¹² abstract away the exact values and focus on the relative order of values. The order has meaning if most values are distinct. In DECRAG, any value in $[0, 1]$ can be constructible by Property 9, and the values of proclaims with non-isomorphic claim structures are distinct by Property 3.
Smoothness.
The discontinuity is the abrupt increase of a proposition’s value by adding a weak attack or a weak support, and it should be avoided.¹ In DECRAG, the continuity is guaranteed by Property 8.
Stable extension.
A question about whether the score function can assign a final score of $1$ to the members of the unique stable extension¹⁰ and $0$ to every other proposition was posted.¹⁹ In DECRAG, the nodes in the stable extension have higher values than the other nodes as guaranteed by Property 19.
Base score.
Base score^19,49 (weight,⁵¹ basic weight,⁴ and basic score¹) is a given proposition value if it has no attacker or supporter. In DECRAG, the base score can be simulated by the default weights, and each proclaim can have a different base score by Property 12.

The prima facie principle⁴² was used in many previous works. In DECRAG, the prima facie principle can be simulated by $τ (a) = 1$ with appropriate $ι_{a}$ for each proclaim $a$ . The parameter $ι_{a}$ determines the balance between the a priori confidence and argument structures.
5.3. Disadvantage

We review the failed properties in Section 4 to examine the reasons for failures in Section 5.3.1. We discuss maximality and minimality in Section 5.3.2. Section 5.3.3 describes the computational complexity issue.

5.3.1. Failed properties

The following properties are not valid for DECRAG.

SC (Section 4.4): As shown in Figure 9, the attack a→b smoothly converges to SC as the similarity weight between $a$ and $b$ increases to $\infty$ in DECRAG.

CP (Section 4.4), QP (Section 4.4), compensation (Section 4.3): Both cardinality and quality are considered in DECRAG. Whether the strength of supporters compensates for the smaller number of supporters depends on the supporters’ values in DECRAG.

Compensation was also defined as a semantics that violates both CP and QP (Section 4.4).⁴ DECRAG satisfies this compensation.

$\oplus$ DB (Section 4.4), attack versus full defense (Section 4.5): Adding an attacker reduces the value of a node even if the attacker’s value is very low in DECRAG.

Strictly balanced (Section 4.3): DECRAG modified the definition of $=_{*}$ , $\geq_{*}$ , and $>_{*}$ to incorporate the time dependency of variables. As a result, they are slightly stronger.

5.3.2. Maximality and minimality

Maximality⁵⁷ is the property that the value of node $a$ is $1$ if $a$ has no attacker, and minimality²⁰ is the property that the value of node $a$ is $0$ if $a$ has no supporter and $τ (a) = 0$ . In DECRAG with $p = 2$ , maximality is satisfied if $τ (a) = 1$ and minimality is satisfied. No maximality ( $0 < τ (a) < 1$ ) was chosen to satisfy Properties 26, 27, 31, 32, and 33 (Sections 4.2 and 4.3). Maximality ( $τ (a) = 1$ ) was chosen to satisfy Properties 36 and 38 (Section 4.4). Maximality could be chosen for Properties 46 to 50 (Sections 4.5 and 4.6).

Next, we consider the combination of maximal and minimal values. As seen in Figure 5, for an argumentation graph of a→b, $σ (a) = 1$ , $σ (V_{a \to b}) = 1$ , and $σ (b) = 0.0909 \dots$ for $ν = 0$ and $λ = 0.1$ . In this result, $σ (a)$ and $σ (V_{a \to b})$ are the maximal $1$ , but $σ (b)$ is not the minimal $0$ in DECRAG. This may seem unintuitive, but the intuition that $σ (b)$ should be the minimal 0 is not as self-evident as it first appears in the continuous framework of DECRAG. For example, suppose that the claim term of a→b has weight $w$ . If we follow the intuition, $σ (b) = 0$ for $w > 0$ , and $σ (b) = 1$ for $w = 0$ as the claim term of a→b disappears when $w = 0$ . This discontinuity conflicts with the smoothness principle. For another example, suppose that there is an attack a→b and $σ (b) = 0$ . Then, the additional attack c→b cannot further decrease $σ (b)$ . To avoid such a saturating effect, we employed $τ (b) < 1$ in Sections 4.2 and 4.3.

5.3.3. Computational complexity

To get the time average, we have to get the solution of the differential equations for a sufficiently long time. To get the space average, we have to get the time averages for multiple initial values. So, the computational complexity is potentially high for DECRAG.

In our applications (Section 5.1), the solutions quickly converged to fixed points, even if some argumentation graphs had a cyclic part. So, we consider that most of the solutions in actual applications are (multiple) fixed points, and complex cases such as limit cycles and aperiodic oscillations are rare.

If this is not the case, there are a few methods to control the complexity.

The solution becomes simpler for larger $λ$ as shown in Property 18. The properties in Sections 4.2 to 4.6 are valid for any $λ$ .

We can trade the shorter calculation time for the finer discrimination of the values (Section 2.6).

We can reduce the elapsed time by parallel processing⁴⁰ because the calculations of time averages are completely independent of each other.

5.4. Previous work

In this section, we examine whether previous work could be used in our applications of Section 5.1 satisfying the principles of Section 5.2.

The extension-based semantics decides the extension of propositions, and the distinction is inherently binary: member (IN) or nonmember (OUT). The labeling semantics extends the distinction to IN, OUT, and UNDEC or more, but still only to a countable number of labels. The preorder semantics decides the preorder among propositions, but not their values. None of them satisfies the smoothness principle.

It is not straightforward to incorporate similarity into the aggregation and equational approaches. For example, in a local gradual valuation,¹⁴ the value of proposition $a$ is defined as $σ (a) = g (h_{s u p} (σ (c_{1}), \dots, σ (c_{p})), h_{d e f} (σ (b_{1}), \dots, σ (b_{n})))$ , where b_i→a, c_k⇒a, $h_{a t t} (\dots)$ and $h_{s u p} (\dots)$ evaluate the qualities of the attacks and supports, and $g (\dots)$ combines the evaluated attacks and supports. It is not obvious how to incorporate similarity into this model.

The probabilistic approach is the method to extend the extension-based semantics by probability. A natural way to incorporate similarity is to consider similarity as the probability that the propositions under similarity are identical. Suppose that $E$ is a set of extensions. The probability of proposition $a$ for $E$ is defined as $| {E : a \in E, E \in E} | / | E |,$ where $| A |$ is the cardinality of $A$ . However, for an odd-length cycle (including a contradiction), this probability becomes indefinite (zero divided by zero) because no extension exists. This indefiniteness is undesirable because we want to analyze an odd-length cycle to meet the “paradox and dilemma” principle.

In the game-theoretical approach, the value of a proposition is based on the degree of acceptability $ϕ (P, O)$ of $P$ with respect to $O$ , where $ϕ (P, O) = (1 / 2) (1 + f (| O^{\leftarrow P} |) - f (| P^{\leftarrow O} |))$ for $B^{\leftarrow A} = {(a, b) \in A \times B : a \to b}$ and the saturating function $f (n) = n / (n + 1)$ . Extending this to support and similarity is not a trivial task, and it is not easy to satisfy the similarity principle.

The neural network approach uses many parameters to produce a value. To tune the parameters, we need an enormous volume of data. We did not have such a volume of data in our applications.

The differential equation approach cannot satisfy the principles of flexibility and separability. For example, the quadratic energy model²⁶ uses the energy $\sum supporters' values - \sum attackers' values$ to decide the derivative of the attacked and supported proposition. No claim relations’ values are used, violating the separability principle. It is not straightforward to incorporate similarity, which means that this approach does not satisfy the similarity principle. In addition, this approach lacks the cost model, which is utilized in our applications (Section 5.1.4).

Next, we examine previous work on similarity for the argumentation. Analogy AF³¹ is a tuple $⟨ A, Δ, α_{μ}, E_{μ} ⟩,$ where $A$ is a set of propositions, $Δ$ is a context (not used here), $α_{μ}$ is a similarity function, and $E_{μ}$ is an attack relation such that $(X, Y) \in E_{μ}$ for $X, Y \in A$ if $α_{μ} (X, Y) < θ$ for threshold $θ$ . This discretization of similarity at some threshold cannot satisfy the smoothness principle.

The similarity-based bipolar AF³⁶ is defined based on the similarity ${Sim}_{C} (A, B)$ between $A$ and $B$ using the Jaccard similarity

\begin{aligned} {Coef}_{d} (A, B) = w_{d} \cdot \frac{| ν_{d}^{A} \cap ν_{d}^{B} |}{| ν_{d}^{A} \cup ν_{d}^{B} |} \end{aligned}

for a set of semantic values

ν_{d}^{A}

A

with weight

w_{d}

for the descriptor

d

. This approach uses a specific form of similarity. Because in our application, we had to use the similarity determined by the natural language processing, this does not satisfy the similarity principle.

To cope with sockpuppet behavior (multiple attacks from similar texts),⁵⁸ a method to readjust the values assigned by other valuation methods was proposed.^37,38 In our applications, we did not observe the sockpuppet behavior. The problem we faced was that the values of similar propositions have very different values. So, their method does not satisfy the similarity principle.

6. Conclusion

In this paper, we proposed an AF DECRAG. Taking the approach in Section 1.2, we developed the formalism in Section 2. Based on the examples in Section 3 and the theoretical properties in Section 4, the framework was shown to be useful in the evaluation of Section 5.

We took the general model-based approach to put an argumentation into a general system to observe the behavior by simulation, and to summarize the observation by averages. Due to the generality of the system, we can observe interesting behaviors, such as multiple fixed points for dilemmas and limit cycles for paradoxes. Even though it seems that the generality of the system makes it difficult to handle the system, we succeeded in proving many useful properties.

The formalism combines the argumentation graph, partially clasped cost model, differential equation, time and space averaging, and sampling calculation. Each has its part, as seen in Section 5. The key to this combination is the fixed variable, which makes the solutions of the cost model show richer behaviors than the simple minimization model.

We concentrated on the fundamental theoretical aspect of DECRAG, and there are still many research topics. We want to study the theoretical aspect of the framework further: the relationship between the structure of argumentation graphs and the behaviors of their differential equations, the meaning of the partially clasped cost and the differential equations generated from it, the condition of the existence of the time average, and better estimations of the variances. We studied the relationship between existing models and DECRAG by examples in Section 3 and found Property 19. We would like to study the relationship with more existing models theoretically and empirically. We applied a preliminary version of DECRAG to the limited cases.^40,41,52 We would like to extend these experiments to more practical situations to clarify DECRAG’s applicability. We also want to study artificial argumentation graphs by DECRAG, extending the experiment⁴⁰ on argumentation graphs in International Competition on Computational Models of Argumentation (ICCMA) (https://argumentationcompetition.org/).

Footnotes

Acknowledgment

We would like to thank anonymous reviewers for their constructive and excellent comments. This work was partially supported by JSPS KAKENHI Grant Number JP24K15093.

ORCID iDs

Kazunori Yamaguchi

Yoshitatsu Matsuda

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix A Proof

First, we clarify the notation and typical expressions. In the proofs, we consider some property in variable $a$ , and there are attacks b_i→a ( $1 \leq i \leq I$ ), supports d_k⇒a ( $1 \leq k \leq K$ ), negative supports e_l $↪$ a ( $1 \leq l \leq L$ ), and similarities $μ_{g_{m}, a}$ to $g_{m}$ ( $1 \leq m \leq M$ ). $I$ is the number of attacks, $K$ is that of supports, $L$ is that of negative supports, $M$ is that of similar nodes. The claim, similarity, and default terms for them are $w_{i} | \bar{b_{i}} |^{p} | a |^{p} | \bar{V_{b_{i} \to a}} |^{p}$ , $w_{k} | \bar{d_{k}} |^{p} | 1 - a |^{p} | \bar{V_{d_{k} \Rightarrow a}} |^{p}$ , $w_{l} | 1 - \bar{e_{l}} |^{p} | a |^{p} | \bar{V_{e_{l} ↪ a}} |^{p}$ , $μ_{g_{m}, a} | g_{m} - a |^{p}$ , $ν_{a} | a |^{p}$ , and $λ_{a} | 1 - a |^{p}$ . Note that $b_{i}$ , $d_{k}$ , $e_{l}$ , $V_{b_{i} \to a}$ , $V_{d_{k} \Rightarrow a}$ , and $V_{e_{l} ↪ a}$ can be $a$ . The cost affecting $a$ is $Φ_{a} = w_{i} | \bar{b_{i}} |^{p} | a |^{p} | \bar{V_{b_{i} \to a}} |^{p} + w_{k} | \bar{d_{k}} |^{p} | 1 - a |^{p} | \bar{V_{d_{k} \Rightarrow a}} |^{p} + w_{l} | 1 - \bar{e_{l}} |^{p} | a |^{p} | \bar{V_{e_{l} ↪ a}} |^{p} + μ_{g_{m}, a} | g_{m} - a |^{p} + λ_{a} | 1 - a |^{p} + ν_{a} | a |^{p}$ . The differential equation for $a$ is $d a / d t = - p (w_{i} | b_{i} |^{p} | V_{b_{i} \to a} |^{p} sgn (a) | a |^{p - 1} - w_{k} | d_{k} |^{p} | V_{d_{k} \Rightarrow a} |^{p} sgn (1 - a) | 1 - a |^{p - 1} + w_{l} | 1 - e_{l} |^{p} | V_{e_{l} ↪ a} |^{p} sgn (a) | a |^{p - 1} - μ_{g_{m}, a} sgn (g_{m} - a) | g_{m} - a |^{p - 1} + ν_{a} sgn (a) | a |^{p - 1} - λ_{a} sgn (1 - a) | 1 - a |^{p - 1})$ . The right-hand expression of $d a / d t$ is expressed as a function of $f (a; w_{i}, b_{i}, V_{b_{i} \to a}; w_{k}, d_{k}, V_{d_{k} \Rightarrow a}; w_{l}, e_{l}, V_{e_{l} ↪ a}; μ_{g_{m}, a}, g_{m}; ν_{a}; λ_{a})$ . Note that $w_{i}$ , $w_{k}$ , $w_{l}$ , $μ_{g_{m}, a}$ , $ν_{a}$ , and $λ_{a}$ are constants. For vector $x$ , this expression is collectively denoted as $d x / d t = f (x)$ .

To simplify the proofs, we introduce some lemmas.

Hereafter, $p = 2$ . For $p = 2$ , the cost that affects $a$ is $Φ_{a} = w_{i} {\bar{b_{i}}}^{2} a^{2} {\bar{V_{b_{i} \to a}}}^{2} + w_{k} {\bar{d_{k}}}^{2} (1 - a)^{2} {\bar{V_{d_{k} \Rightarrow a}}}^{2} + w_{l} (1 - \bar{e_{l}})^{2} a^{2} {\bar{V_{e_{l} ↪ a}}}^{2} + μ_{g_{m}, a} (g_{m} - a)^{2} + ν_{a} a^{2} + λ_{a} (1 - a)^{2}$ . The differential equation for $a$ is $d a / d t = - 2 (w_{i} b_{i}^{2} V_{b_{i} \to a}^{2} a - w_{k} d_{k}^{2} V_{d_{k} \Rightarrow a}^{2} (1 - a) + w_{l} (1 - e_{l})^{2} V_{e_{l} ↪ a}^{2} a - μ_{g_{m}, a} (g_{m} - a) + ν_{a} a - λ_{a} (1 - a))$ .

In the following, indexes of attackers and supporters of $a$ and $b$ are aligned according to the mapping associated with $=_{*}$ , $\geq_{*}$ , or $>_{*}$ .

$B_{i}$ , $D_{k}$ , and $E_{l}$ are just for readability. They are not uniquely determined and dependent on time $T$ .

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Differential equation based on cost for recursive argumentation graph (DECRAG): Continuous framework for recursive bipolar argumentation with similarity

Abstract

Keywords

1. Introduction

1.1. Background

1.2. Approach

1.3. Organization

2. Formalism

2.1. Argumentation graph

Definition 1 Argumentation graph

Definition 2 Isomorphism

2.2. Partially clasped cost model

Definition 3 Cost

2.3. Differential equation

Definition 4 Differential equation

2.4. Averaging

2.5. Average of variable

2.6.1. Approximate time average calculation

3. Examples

3.1. Small example

Example 1 Extension with a cyclic structure

Example 2 Extension with contradiction

Example 3 Order semantics example

Example 4 Weight in WAF

Example 5 Recursive attack in AFRA

3.6. Attack-support AF (ASAF)

Example 7 Recursive attack in ASAF

3.7.1. Even-length cycle (ai→ai+1 ( 1 ≤ i ≤ n ) even n , cyclic index)

3.7.2. Odd-length cycle (ai→ai+1 ( 1 ≤ i ≤ n ) odd n , cyclic index)

Property 1 Existence of solution 41

Property 2 Anonymity 20

Property 3 Reconstructability ( p > 1 )

Definition 7 Dependency graph

Property 4 Nondilution 20 and bivariate directionality 15

Property 5 Value range 41

Property 6 Value range at t → ∞

Property 7 Existence of fixed point 41

Property 8 Continuity

Property 9 Constructibility

Property 10 Set of numerical solutions

Property 11 Time average of fixed point and limit cycle

Property 12 Time average of claim relations and similarities

Property 13 Even-length cycle solution

Property 14 Odd-length cycle solution

Property 15 Aperiodic oscillation by multiple odd-length cycles

Property 16 Existence of time averages for aperiodic oscillation by multiple odd-length cycles

Property 17 a for ν a → ∞ or λ a → ∞

Property 18 Convergence for sufficiently large λ

Property 19 Stable extension

4.2. Group property

Property 20 GP1

Property 21 GP2

Property 22 GP3

Property 23 GP4

Property 24 GP5

Property 25 GP6

Property 26 GP7

Property 27 GP8

Property 28 GP9

Property 29 GP10

Property 30 GP11

4.3. Property of attack and support

Property 31 Balanced

Property 32 Monotonic

Property 33 Strictly monotonic

Property 34 Duality 51

4.4. Property of attack

Property 35 Independence (In) 4

Property 36 Void precedence (VP)

Property 37 Counter-transitivity (CT)

Property 38 Strict counter-transitivity (SCT)

Property 39 Defense precedence (DP)

Property 40 Distributed-defense precedence (DDP)

Property 41 Increase of defense branch ( ↑ DB)

Property 42 Addition of attack branch (+AB)

Property 43 Increase of attack branch ( ↑ AB)

Property 44 Total (Tot)

Property 45 Nonattacked equivalence (NaE)

4.5. Property of acyclic dependency graph

Property 46 Unique fixed point 52

3.7.1. Even-length cycle (a_i→a_i+1 ( $1 \leq i \leq n$ ) even $n$ , cyclic index)

3.7.2. Odd-length cycle (a_i→a_i+1 ( $1 \leq i \leq n$ ) odd $n$ , cyclic index)

Property 1 Existence of solution⁴¹

Property 2 Anonymity²⁰

Property 3 Reconstructability ( $p > 1$ )

Property 4 Nondilution²⁰ and bivariate directionality¹⁵

Property 5 Value range⁴¹

Property 6 Value range at $t \to \infty$

Property 7 Existence of fixed point⁴¹

Property 17 $a$ for $ν_{a} \to \infty$ or $λ_{a} \to \infty$

Property 18 Convergence for sufficiently large $λ$

Property 34 Duality⁵¹

Property 35 Independence (In)⁴

Property 41 Increase of defense branch ( $↑$ DB)

Property 43 Increase of attack branch ( $↑$ AB)

Property 46 Unique fixed point⁵²

Property 48 Extreme similarity⁴¹