Abstract Argumentation / Persuasion / Dynamics

TL;DR

This paper extends Dung's argumentation frameworks by incorporating persuasion acts, enabling analysis of dynamic interactions among attack, persuasion, and defense, and introduces CTL encoding for enriched admissibility notions.

Contribution

It introduces a novel extension of Dung's frameworks to include persuasion acts and applies CTL encoding to analyze dynamic argumentation interactions.

Findings

Characterization of admissibility notions in the extended framework

Enrichment of argumentation frameworks through CTL encoding

Facilitation of static and dynamic argumentation coordination

Abstract

The act of persuasion, a key component in rhetoric argumentation, may be viewed as a dynamics modifier. We extend Dung's frameworks with acts of persuasion among agents, and consider interactions among attack, persuasion and defence that have been largely unheeded so far. We characterise basic notions of admissibilities in this framework, and show a way of enriching them through, effectively, CTL (computation tree logic) encoding, which also permits importation of the theoretical results known to the logic into our argumentation frameworks. Our aim is to complement the growing interest in coordination of static and dynamic argumentation.

Full text

1 Introduction

An interesting component of rhetoric argumentation is persuasion. We may code an act of it into \framebox{\text{A}:\smalla_1⃝}\dashrightarrow\framebox{\text{B}:\smalla_2⃝}\multimap^{\!\!\!\!\!\!\!a_{1}}\framebox{\text{B}:\smalla_3⃝} with the following intended meaning: some agent A’s argument $a_{1}$ persuades an agent B into holding $a_{3}$; B, being persuaded, drops $a_{2}$. There can be various reasons for the persuasive act. It may be that A is a great teacher wanting to correct some inadvisable norm of B’s, or perhaps A is a manipulator who benefits if $a_{2}$ is not present. Persuasion is popularly observed in social forums including YouTube and Twitter, and methods to represent it will help understand users’ views on topics accurately. Another less pervasive form of persuasion is possible: \framebox{\text{A}:\smalla_1⃝}\multimap\framebox{\text{B}:\smalla_3⃝} in which A persuades B with $a_{1}$ into expressing $a_{3}$ but without conversion. In either of the cases, persuasion acts as a dynamics modifier in rhetoric argumentation, allowing some argument to appear and disappear.

Of course - and this is one highlight of this paper - these acts will not be successful if $a_{1}$ is detected to be not a defensible argument: we may have \framebox{\text{C}:\smalla_3⃝}\rightarrow\framebox{\text{A}:\smalla_1⃝}\multimap\framebox{\text{B}:\smalla_2⃝} where $a_{3}$ attacks $a_{1}$. Suppose now that B is aware of $a_{3}$, then B can defend against A’s persuasion due to $a_{3}$’s attack on $a_{1}$. B is not persuaded into holding $a_{2}$ in such a case. We will care for the interactions between attack, persuasion and defence.

While AGM-like argumentation framework revisions defining a class of argumentation frameworks to result from an initially given argumentation framework and an input (which could be argument(s), attack(s) or both), and persuasion in the context of (often two-parties) dialogue games, are being studied, there are very few studies in the literature that pursue coordination of statics and dynamics. One exception is the dynamic logic for programs adapted for argumentation by Doutre et al. [20, 21], which is rich in expressiveness with non-deterministic operations, tests, sequential operations. Bridging dynamics and statics is important for detailed and more precise analysis of rhetoric argumentation. So far, however, the above-said interaction between attack, persuasion and defence has been largely unheeded. We first of all fill the gap by developing an abstract persuasion argumentation, an extension to Dung’s argumentation theory [22]. We formulate the notion of static admissibility for our theory, and then show a way of diversifying it into other types of admissibilities through, effectively, CTL (computation tree logic) embedding.

1.1 Example situations

1.1.1 Defence and reference set

One aspect that has not been shed much light on in the literature of dynamic argumentation is defence against such persuasive acts (dynamic operations). Let us consider an example.

$a_{2}$

(Mr. X) Elma does not like the music.

$a_{3}$

(Mr. Z) We should get a piano.

$a_{4}$

(Mrs. Y) We can buy Elma a Hello Kitty shoulder bag.

$a_{5}$

(Mrs. Y) We will go to Yamaha Music Communications Co., Ltd. for a piano.

The relation among them is as shown in Figure A (B): there is an attack from $a_{2}$ to $a_{3}$, and there is also a persuasion act by (Mr. Z holding) $a_{3}$ trying to convert $a_{4}$ into $a_{5}$. Suppose that $a_{5}$ is not initially on Mrs. Y’s mind, that is, that it is not visible initially. If the persuasion by Mr. Z is successful, Mrs. Y changes her mind, dropping $a_{4}$ and gaining $a_{5}$. Otherwise, she holds onto $a_{4}$. In Dung theory, defence of an argument $a_{x}$ is defined with respect to a set of arguments $A$. The reference set $A$ defends $a_{x}$ just when $A$’s members attack all arguments attacking $a_{x}$. We see that this concept may be extended also to persuasion operations. For example, if, as marked with a rectangular box in Figure A, the reference set consists of $a_{4}$ and $a_{5}$ alone, it does not detect any flaw in $a_{3}$. Thus, the persuasion is successful with respect to the reference set. However, if it also contains $a_{2}$ attacking $a_{3}$ as in Figure B, it can prevent the persuasion from taking effect on $a_{4}$.

1.1.2 Multiple persuasions

We have a kind of concurrency scenario when multiple persuasions act on an argument. Let us consider an example.

$a_{1}$

(Alice at London Bridge, having agreed to see Bob at 7 pm) I am going to have dinner with Bob. It is 7 pm now. He should be arriving soon.

$a_{2}$

(Tom, calling from Camden) Chris (Alice’s brother) is looking for you. He is at Camden Bar. He says there is some urgent matter, can you please get to the bar as soon as possible?

$a_{3}$

(Katie, seeing Alice by chance) Hey Alice, you’ve left your laptop at King’s library? You better go there now. Oh, and don’t forget about your presentation tomorrow morning. Make sure you have slidesready!

Having been acquainted with Bob only recently, Alice is more inclined to getting to Camden Bar or to King’s library. That is, $a_{4}$: I am going to Camden Bar, or $a_{5}$: I am going to King’s library. She knows her brother is very stern. But the assignment of which Katie reminded Alice seems to be a thing that must be prioritised, too. Whichever option she is to go for, she must, thinks she, come up with excuses to justify her choice. Therefore:

$a_{6}$

(Alice’s excuse) It is fine to skip dinner because I waited for Bob at London Bridge and he did not arrive in time. Besides, I suddenly have something urgent.

$a_{7}$

(Alice’s excuse) I cannot see Chris. For my career, it is important that I perform well at presentation tomorrow. Chris will understand.

$a_{8}$

(Alice’s excuse) I cannot go to King’s library now, because it is always urgent when Chris calls me.

Figure C represents these arguments. Now, what we have is a potentially irreversible branching. If $a_{2}$ persuades $a_{1}$ into $a_{4}$, it is no longer possible for $a_{3}$ to persuade $a_{1}$, as $a_{1}$ will not be available for persuasion. If $a_{3}$ persuades $a_{1}$ into $a_{5}$, on the other hand, it is no longer possible that $a_{2}$ persuades $a_{1}$. A certain partial order may be defined among persuasion (as in preference-based argumentation), but the non-deterministic consideration leads to a more general theory (as in probabilistic argumentation) in which the actual behaviour of a system depends on run-time executions.

Just as in program analysis, however, it may be still possible to identify certain properties, whichever an actual path may be. In this particular example, (Alice holding) $a_{1}$ may be persuaded into holding $a_{4}$ or else $a_{5}$, and we cannot tell which with certainty. However, we can certainly predict $a_{8}$’s emergence. Thus, by obtaining varieties in arguments admissibility by means of CTL, we can answer such a query as ‘Is $a_{8}$ going to be an admissible argument in whatever order persuasive acts may take place?’.

2 Technical Backgrounds

Let $\mathcal{A}$ be a class of abstract entities which we understand as arguments. We denote any member of $\mathcal{A}$ by $a$ with or without a subscript, and any finite subset of $\mathcal{A}$ by $A$ with or without a subscript. An argumentation framework [22] is a tuple $(A,R)$ where $R$ is a binary relation over $A$. Let $F^{(A,R)}(A_{1})$ denote $(A_{1},R\cap(A_{1}\times A_{1}))$, we denote by $2^{(A,R)}$ the following set: $\bigcup_{A_{1}\subseteq A}F^{(A,R)}(A_{1})$, i.e. all sub-argumentation frameworks of $(A,R)$. When confusion is unlikely to occur, we abbreviate $F^{(A,R)}(A_{1})$ for some $A_{1}$ by $F(A_{1})$.

For any $(A,R)$ an argument $a_{1}\in A$ is said to attack $a_{2}\in A$ if and only if, or iff, $(a_{1},a_{2})\in R$. A set of arguments $A_{1}\subseteq A$ is said to defend $a_{x}\in A$ iff each $a_{y}\in A$ attacking $a_{x}$ is attacked by at least one argument in $A_{1}$. A set of arguments $A_{1}\subseteq A$ is said to be: conflict-free iff no member of $A_{1}$ attacks a member of $A_{1}$; admissible iff it is conflict-free and it defends all the members of $A_{1}$; complete iff it is admissible and includes any argument it defends; preferred iff it is a set-theoretically maximal admissible set; stable iff it is preferred and attacks every argument in $A\backslash A_{1}$; and grounded iff it is the set intersection of all complete sets of $A$.

3 Abstract Persuasion Argumentation

We define our Abstract Persuasion Argumentation (APA) framework to be a tuple $(A,R,R_{\mathbf{p}},A_{0},\hookrightarrow)$ for $A_{0}\subseteq A$, for a ternary relation$R_{\mathbf{p}}:A\times(A\cup\{\epsilon\})\times A$ and for another $\hookrightarrow:2^{A}\times(2^{(A,R)}\times 2^{(A,R)})$. For $R_{\mathbf{p}}$, $(a_{1},\epsilon,a_{2})\in R_{\mathbf{p}}$ represents $a_{1}\multimap a_{2}$ (passive persuasion or to induce), and $(a_{1},a_{2},a_{3})\in R_{\mathbf{p}}$ represents $a_{1}\dashrightarrow a_{2}\multimap^{\!\!\!\!\!\!\!a_{1}}a_{3}$ (active persuasion or to convert). We refer to a subset of $R_{\mathbf{p}}$ by $\Gamma$ with or without a subscript and/or a superscript.

APA is a dynamic argumentation framework where arguments can appear (go visible) or disappear (go invisible). As in a transition system, it comes with an initial state and a transition relation $\hookrightarrow$. For any APA framework $(A,R,R_{\mathbf{p}},A_{0},\hookrightarrow)$, we define a state to be a member $F(A_{x})$ of $2^{(A,R)}$, and we say any argument that occurs in a state visible and any that does not occur in the state invisible, in each case at that particular state.111We assume the standard notion of occurrence. We define $F(A_{0})$ to be the initial state.

Example 1

In Elma example, we assumed $A_{0}=\{a_{2},a_{3},a_{4}\}$ and $F(A_{0})=(A_{0},\{(a_{2},a_{3})\})$. In Alice example, $A_{0}=\{a_{1},a_{2},a_{3}\}$ and $F(A_{0})=(A_{0},\emptyset)$.

Definition 1 (Reachable states)

For APA $(A,R,R_{\mathbf{p}},A_{0},\hookrightarrow)$, for a set of arguments $A_{x}\subseteq A$, and for states $F(A_{1})$ and $F(A_{2})$, we say that there is a transition from $F(A_{1})$ to $F(A_{2})$ with respect to $A_{x}$ iff it holds that $(A_{x},(F(A_{1}),F(A_{2})))\in\hookrightarrow$, which we alternatively state either as $(F(A_{1}),F(A_{2}))\in\hookrightarrow^{A_{x}}$ or as $F(A_{1})\hookrightarrow^{A_{x}}F(A_{2})$. We say that a state $F(A_{x})$ is reachable iff $F(A_{x})$ either is the initial state or else is such that $F(A_{0})\hookrightarrow^{A_{i_{1}}}\cdots\hookrightarrow^{A_{i_{n}}}F(A_{x})$, $1\leq n$.

A reachable state is a static snapshot of an APA framework at one moment, which is a Dung argumentation framework. To enumerate all reachable states, it suffices to define $\hookrightarrow$ in specific detail. And this is where the notion of defence against persuasive acts with respect to a reference set at a state - specifically visible arguments of the set at the state - comes into play:

Definition 2 (Possible persuasion acts)

For APA $(A,R,R_{\mathbf{p}},A_{0},\hookrightarrow)$, we say that a persuasion act $(a_{1},\alpha,a_{2})\in R_{\mathbf{p}}$, $\alpha\in\{\epsilon\}\cup A$, is possible with respect to: (i) a reference set $A_{x}\subseteq A$; and (ii) a state $F(A_{u})$ iff $a_{1},\alpha\in A_{u}\cup\{\epsilon\}$ and $a_{1}$ is not attacked by any member of $A_{x}\cap A_{u}$. We denote the set of all members of $R_{\mathbf{p}}$ that are possible with respect to a reference set $A_{x}\subseteq A$ and a state $F(A_{u})$ by $\Gamma^{A_{x}}_{F(A_{u})}$.

Example 2

(Continued) In Elma example with $A_{0}=\{a_{2},a_{3},a_{4}\}$, there is one argument, $a_{2}$, which is in $A_{0}$ (thus visible), and which attacks $a_{3}$, so $(a_{2},a_{3},a_{4})\in R_{\mathbf{p}}$ is possible with respect to $A_{x}\subseteq A$ and $F(A_{0})$ only if $a_{2}\not\in A_{x}$. $\Gamma^{A_{x}}_{F(A_{0})}$ is: $\{(a_{2},a_{3},a_{4})\}$ if $a_{2}\not\in A_{x}$; $\emptyset$, otherwise. In Alice example with $A_{0}=\{a_{1},a_{2},a_{3}\}$, $(a_{2},a_{1},a_{4})$ and $(a_{3},a_{1},a_{5})$ are both possible with respect to any $A_{x}\subseteq A$ and $F(A_{0})$, because for no $(a_{x},\alpha,a_{y})\in R_{\mathbf{p}}$ there is $(a_{z},a_{x})\in R$. $\Gamma^{A_{x}}_{F(A_{0})}=\{(a_{2},a_{1},a_{4}),(a_{3},a_{1},a_{5})\}$.

Since transition as we consider is non-deterministic, each persuasion act possible in a state may or may not execute for transition. Therefore, for any APA $(A,R,R_{\mathbf{p}},A_{0},\hookrightarrow)$, any reference set $A_{x}\subseteq A$ and any state $F(A_{1})$, there are $2^{|\Gamma^{A_{x}}_{F(A_{1})}|}-1$ transitions, though some of them may be identical.

Definition 3 (Non-deterministic transition)

For APA $(A,R,R_{\mathbf{p}},A_{0},\hookrightarrow)$, for $A_{1}\subseteq A$ and for $\Gamma\subseteq R_{\mathbf{p}}$, let $\textsf{neg}^{A_{1}}(\Gamma)$ be $\{a_{x}\in A_{1}\ |\ \exists a_{1},a_{2}\in A_{1}.(a_{1},a_{x},a_{2})\in\Gamma\}$, and let $\textsf{pos}^{A_{1}}(\Gamma)$ be $\{a_{2}\in A_{1}\ |\ \exists a_{1},\alpha\in A_{1}\cup\{\epsilon\}.(a_{1},\alpha,a_{2})\in\Gamma\}$. For $A_{x}\subseteq A$ and states $F(A_{1})$ and $F(A_{2})$, we define: $F(A_{1})\hookrightarrow^{A_{x}}F(A_{2})$ iff there is some $\emptyset\subset\Gamma\subseteq\Gamma^{A_{x}}_{F(A_{1})}\subseteq R_{\mathbf{p}}$ such that $A_{2}=(A_{1}\backslash\textsf{neg}^{A_{1}}(\Gamma))\cup\textsf{pos}^{A_{1}}(\Gamma)$.

For $\Gamma\subseteq R_{\mathbf{p}}$, if $\Gamma\subseteq\Gamma^{A_{x}}_{F(A_{1})}$, it is a (non-deterministically) chosen set of possible persuasion acts at $F(A_{1})$. Thus, $\textsf{neg}^{A_{1}}(\Gamma)$ is the set of all visible arguments that are to be converted, and $\textsf{pos}^{A_{1}}(\Gamma)$ is that of all visible arguments that are to be generated, in the transition. As clear from this definition, while every member of $\textsf{pos}^{A_{1}}(\Gamma)$, if not visible in $F(A_{1})$, will be visible in $F(A_{2})$, not necessarily every member of $\textsf{neg}^{A_{1}}(\Gamma)$ will be invisible in $F(A_{2})$ in case it also belongs to $\textsf{pos}^{A_{1}}(\Gamma)$, in which case the effect is offset.

Example 3

Consider the argumentation in the figures below, in each of which visible arguments are marked with a black border around the circle. Suppose $A_{0}=\{a_{1},a_{2}\}$ as in Figure D. At $F(A_{0})$, there are more than one possible persuasion acts: $\Gamma^{A_{x}}_{F(A_{0})}=\{(a_{1},a_{2},a_{4}),(a_{2},a_{1},a_{3})\}$ for any reference set $A_{x}\subseteq A$. There are three transitions for $F(A_{0})$, depending on which one(s) execute simultaneously. If just $(a_{1},a_{2},a_{4})$, $a_{2}$ will go invisible, while $a_{4}$ will be visible, so we have $F(A_{0})\hookrightarrow^{A_{x}}F(A_{1})$ (Figure E) for any $A_{x}$. If just $(a_{2},a_{1},a_{3})$, we have $F(A_{0})\hookrightarrow^{A_{y}}F(A_{2})$ (Figure F) for any $A_{y}$. Or both of them may execute at once, in which case both $a_{1}$ and $a_{2}$ will be invisible, and $a_{3}$ and $a_{4}$ meanwhile will be visible, so we have $F(A_{0})\hookrightarrow^{A_{z}}F(A_{3})$ (Figure G) for any $A_{z}$. Reasoning similarly for the new states, we eventually enumerate all reachable states and all transitions among them:

• •

$F(A_{0})\hookrightarrow^{A_{x}}F(A_{1})$, $F(A_{0})\hookrightarrow^{A_{y}}F(A_{2})$, $F(A_{0})\hookrightarrow^{A_{z}}F(A_{3})$.

• •

$F(A_{1})\hookrightarrow^{A_{u}}F(A_{1})$.

• •

$F(A_{2})\hookrightarrow^{A_{v}}F(A_{2})$.

• •

$F(A_{3})\hookrightarrow^{A_{p}}F(A_{4})$, $F(A_{3})\hookrightarrow^{A_{q}}F(A_{5})$, $F(A_{3})\hookrightarrow^{A_{r}}F(A_{6})$.

• •

$F(A_{4})\hookrightarrow^{A_{i}}F(A_{4})$, $F(A_{4})\hookrightarrow^{A_{j}}F(A_{6})$.

• •

$F(A_{5})\hookrightarrow^{A_{k}}F(A_{5})$, $F(A_{5})\hookrightarrow^{A_{c}}F(A_{6})$.

• •

$F(A_{6})\hookrightarrow^{A_{d}}F(A_{4})$, $F(A_{6})\hookrightarrow^{A_{f}}F(A_{5})$, $F(A_{6})\hookrightarrow^{A_{g}}F(A_{6})$.

The reference sets for the transitions are any subset of $\{a_{1},a_{2},a_{3},a_{4}\}$. Notice, apart from the trivial self-transitions, states could oscillate infinitely between $F(A_{4}),F(A_{5})$ and $F(A_{6})$.

Proposition 1

Suppose an APA framework $\delta$ with a finite number of arguments. It is necessary that the number of (reachable) states is finite. It is, however, not necessary that the number of transitions in $\delta$ is finite.

3.1 Admissibilities

We now define the static notion of admissibility in APA frameworks, based on three criteria. For APA $(A,R,R_{\mathbf{p}},A_{0},\hookrightarrow)$,

Conflict-freeness We say that $A_{1}\subseteq A$ is conflict-free in a (reachable) state $F(A_{x})$ iff no member of $A_{1}\cap A_{x}$ attacks a member of $A_{1}\cap A_{x}$.

Defendedness We say that a reference set $A_{1}\subseteq A$ defends $a\in A$ in a state $F(A_{x})$ iff either $a\not\in A_{x}$ or else both of the conditions below hold.

Every $a_{u}\in A_{x}$ attacking $a$ is attacked by at least one member of $A_{x}\cap A_{1}$ (counter-attack). 2. 2.

There is no state $F(A_{y})$ such that both $F(A_{x})\hookrightarrow^{A_{1}}F(A_{y})$ and $a\not\in A_{y}$ at once (no elimination).

We say that $A_{1}\subseteq A$ is defended in a state $F(A_{x})$ iff $A_{1}$ as a reference set defends every member of its in $F(A_{x})$.

Properness We say that $A_{u}\subseteq A$ is proper in a state $F(A_{x})$ iff $A_{u}\subseteq A_{x}$.

Defendedness above extends Dung’s defendedness naturally for $R_{\mathbf{p}}$. Properness ensures that we will not be talking of invisible arguments. With these properties, we say $A_{u}\subseteq A$ is: admissible in a state $F(A_{1})$ iff $A_{u}$ is conflict-free, defended and proper in $F(A_{1})$; complete in a state $F(A_{1})$ iff $A_{u}$ is admissible in $F(A_{1})$ and includes all arguments it defends; preferred iff no $A_{v}\subseteq A$ that is complete in a state $F(A_{1})$ is a strict superset of $A_{u}$; stable iff it is preferred and attacks every member of $A_{1}\backslash A_{u}$; and grounded in a state $F(A_{1})$ iff it is the set intersection of all complete sets in $F(A_{1})$. Since each state is a Dung argumentation framework, we have:

Proposition 2

For APA $(A,R,R_{\mathbf{p}},A_{0},\hookrightarrow)$, for a state $F(A_{1})$ and for $A_{x}\subseteq A_{1}$: if $A_{x}$ is stable, then $A_{x}$ is preferred; if $A_{x}$ is preferred, then $A_{x}$ is complete; if $A_{x}$ is complete, then $A_{x}$ is admissible; there exists at least one complete set; and there may not exist any stable set.

For general admissibilities across transition, one way of describing more varieties is to embed this state-wise admissibility and transitions into computation tree logic (CTL) or other branching-time logic, by which model-theoretical results known to the logic become available to APA frameworks, too. We consider CTL with some path restrictions. Denote $\{\textsf{ad},\textsf{co},\textsf{pr},\textsf{st},\textsf{gr}\}$ by $\Omega$, and refer to a member of $\Omega$ by $\omega$. Let the grammar of $\phi$ be:

$\phi:=\bot\ |\ \top\ |\ a\dot{\in}_{\delta}A_{x}\ |\ P_{\delta}(\omega,A_{x})\ |\ \neg\phi\ |\ \phi\wedge\phi\ |\ \phi\vee\phi\ |\ \phi\supset\phi\ |\ \textbf{AX}^{\Sigma}\phi\ |\\ {\ }\quad\quad\textbf{EX}^{\Sigma}\phi\ |\ \textbf{AF}^{\Sigma}\phi\ |\ \textbf{EF}^{\Sigma}\phi\ |\ \textbf{AG}^{\Sigma}\phi\ |\ \textbf{EG}^{\Sigma}\phi\ |\ \textbf{E}^{\Sigma}[\phi\textbf{U}\phi]\ |\ \textbf{A}^{\Sigma}[\phi\textbf{U}\phi]$

where both $a\dot{\in}_{\delta}A_{x}$ and $P_{\delta}(\omega,A_{x})$ are atomic predicates for an APA framework $\delta:=(A,R,R_{\mathbf{p}},A_{0},\hookrightarrow)$, with $A_{x}\subseteq A$, with $a\in A$ and with $\Sigma\subseteq 2^{A}$. A is ‘in all branches’, E is ‘in some branch’, X is ‘next state’, F is ‘future state’, G is ‘all subsequent states’, and U is ‘until’. The superscripts restrict paths to only those reachable with member(s) of $\Sigma$ as reference set(s). See below for the exact semantics. We denote the class of all atomic predicates for $\delta$ by $\mathcal{P}_{\delta}$. For semantics, let $L:\Omega\times 2^{\mathcal{A}}\rightarrow 2^{\mathcal{P}_{\delta}}$ be a valuation function such that $L(\omega,A_{x})$ is:

• •

$\{P_{\delta}(\textsf{ad},A_{y})\in\mathcal{P}_{\delta}\ |\ A_{y}\text{ is admissible in }F(A_{x})\}$ if $\omega=\textsf{ad}$.

• •

$\{P_{\delta}(\textsf{co},A_{y})\in\mathcal{P}_{\delta}\ |\ A_{y}\text{ is complete in }F(A_{x})\}$ if $\omega=\textsf{co}$.

• •

$\{P_{\delta}(\textsf{pr},A_{y})\in\mathcal{P}_{\delta}\ |\ A_{y}\text{ is preferred in }F(A_{x})\}$ if $\omega=\textsf{pr}$.

• •

$\{P_{\delta}(\textsf{st},A_{y})\in\mathcal{P}_{\delta}\ |\ A_{y}\text{ is stable in }F(A_{x})\}$ if $\omega=\textsf{st}$.

• •

$\{P_{\delta}(\textsf{gr},A_{y})\in\mathcal{P}_{\delta}\ |\ A_{y}\text{ is grounded in }F(A_{x})\}$ if $\omega=\textsf{gr}$.

We define $\mathcal{M}:=(\delta,L)$ to be a transition system with the following forcing relations.222 The liberty of allowing arguments into $\mathcal{M}$ causes no confusion, let alone issues. If one is so inclined, he/she may choose to consider that components of $\delta$ that appear in $\mathcal{M}$ are semantic counterparts of those that appear in the syntax of CTL with one-to-one correspondence between them.

• •

$\mathcal{M},A_{1}\models\top$.

• •

$\mathcal{M},A_{1}\not\models\bot$.

• •

$\mathcal{M},A_{1}\models a\dot{\in}_{\delta}A_{x}$ iff $a\in A_{x}$.

• •

$\mathcal{M},A_{1}\models P_{\delta}(\omega,A_{x})$ iff $P_{\delta}(\omega,A_{x})\in L(\omega,A_{1})$ (in plain terms, this says $A_{x}$ is admissible / complete / preferred / stable / grounded in a state $F(A_{1})$).

• •

$\mathcal{M},A_{1}\models\neg\phi$ iff $\mathcal{M},A_{1}\not\models\phi$.

• •

$\mathcal{M},A_{1}\models\phi_{1}\wedge\phi_{2}$ iff $\mathcal{M},A_{1}\models\phi_{1}$ and $\mathcal{M},A_{1}\models\phi_{2}$.

• •

$\mathcal{M},A_{1}\models\phi_{1}\vee\phi_{2}$ iff $\mathcal{M},A_{1}\models\phi_{1}$ or $\mathcal{M},A_{1}\models\phi_{2}$.

• •

$\mathcal{M},A_{1}\models\phi_{1}\supset\phi_{2}$ iff $\mathcal{M},A_{1}\not\models\phi_{1}$ or $\mathcal{M},A_{1}\models\phi_{2}$.

• •

$\mathcal{M},A_{1}\models\textbf{AX}^{\Sigma}\phi$ iff $\mathcal{M},A_{2}\models\phi$ for each transition $F(A_{1})\hookrightarrow^{A_{x}}F(A_{2})$, $A_{x}\in\Sigma$.

• •

$\mathcal{M},A_{1}\models\textbf{EX}^{\Sigma}\phi$ iff there is some transition $F(A_{1})\hookrightarrow^{A_{x}}F(A_{2})$, $A_{x}\in\Sigma$, such that $\mathcal{M},A_{2}\models\phi$.

• •

$\mathcal{M},A_{1}\models\textbf{AF}^{\Sigma}\phi$ iff there is some $i\geq 0$ for each transition $F(A_{1})\hookrightarrow^{A_{j1}}\cdots\hookrightarrow^{A_{ji}}F(A_{i+1})(\hookrightarrow^{A_{x}}\cdots)$, $A_{jk}\in\Sigma$ for $1\leq k\leq{i+1}$, such that $\mathcal{M},A_{i+1}\models\phi$.

• •

$\mathcal{M},A_{1}\models\textbf{EF}^{\Sigma}\phi$ iff there are some $i\geq 1$ and a transition $F(A_{1})\hookrightarrow^{A_{j1}}\cdots\hookrightarrow^{A_{ji}}F(A_{i+1})(\hookrightarrow^{A_{x}}\cdots)$, $A_{jk}\in\Sigma$ for $1\leq k\leq{i+1}$, such that $\mathcal{M},A_{i+1}\models\phi$.

• •

$\mathcal{M},A_{1}\models\textbf{AG}^{\Sigma}\phi$ iff $\mathcal{M},A_{k}\models\phi$ for each transition $F(A_{1})\hookrightarrow^{A_{j1}}\cdots$, $A_{jn}\in\Sigma$ for $1\leq n$, such that $F(A_{k})$ occurs in the transition sequence.

• •

$\mathcal{M},A_{1}\models\textbf{EG}^{\Sigma}\phi$ iff there is some transition $F(A_{1})\hookrightarrow^{A_{j1}}\cdots$, $A_{jn}\in\Sigma$ for $1\leq n$, such that $\mathcal{M},A_{k}\models\phi$ and that $F(A_{k})$ occurs in the transition sequence.

• •

$\mathcal{M},A_{1}\models\textbf{A}^{\Sigma}[\phi_{1}\textbf{U}\phi_{2}]$ iff there exists some $i\geq 0$ for each transition $F(A_{1})\hookrightarrow^{A_{j1}}\cdots\hookrightarrow^{A_{ji}}F(A_{i+1})(\hookrightarrow^{A_{x}}\cdots)$ such that $\mathcal{M},A_{i+1}\models\phi_{2}$ and that $\mathcal{M},A_{k}\models\phi_{1}$ for all $k<{i+1}$.

• •

$\mathcal{M},A_{1}\models\textbf{E}^{\Sigma}[\phi_{1}\textbf{U}\phi_{2}]$ iff there exists some $i\geq 0$ and a transition $F(A_{1})\hookrightarrow^{A_{j1}}\cdots\hookrightarrow^{A_{ji}}F(A_{i+1})(\hookrightarrow^{A_{x}}\cdots)$ such that $\mathcal{M},A_{i+1}\models\phi_{2}$ and that $\mathcal{M},A_{k}\models\phi_{1}$ for all $k<{i+1}$.

We say that $\phi$ is true (in $\delta$) iff $(\delta,L),A_{0}\models\phi$.

While this logic appears more graded than CTL for the superscripts $\Sigma$, there is an obvious encoding of it into the standard CTL with an additional atomic predicate in the grammar of $\phi$ that judges whether an argument is visible. That is, we can for example replace $\textbf{EX}^{\{A_{x}\}}\phi$ with $\textbf{EX}(\phi_{1}\wedge\phi)\vee\cdots\vee\textbf{EX}(\phi_{n}\wedge\phi)$ if we can express by the expression that, for any transition $F(A_{c})\hookrightarrow^{A_{x}}F(A_{d})$ such that $F(A_{c})$ is the state with respect to which the expression is evaluated, there exists some $1\leq i\leq n$ such that $\phi_{i}$ holds good just when all and only members of $A_{d}$ are visible, and that for every $\phi_{i}$, $1\leq i\leq n$, there exists some $A_{d}$ such that $F(A_{c})\hookrightarrow^{A_{x}}F(A_{d})$ and that $\phi_{i}$ holds good just when all members of $A_{d}$ are visible, which confirms that our logic is effectively CTL. It is straightforward to see the following well-known equivalences in our semantics:

Proposition 3 (De Morgan’s Laws and Expansion Laws)

$\neg\textbf{AF}^{\Sigma}\phi\equiv\textbf{EG}^{\Sigma}\neg\phi$, $\neg\textbf{EF}^{\Sigma}\phi\equiv\textbf{AG}^{\Sigma}\neg\phi$, $\neg\textbf{AX}^{\Sigma}\phi\equiv\textbf{EG}^{\Sigma}\neg\phi$ (De Morgan’s Laws), $\textbf{AG}^{\Sigma}\phi\equiv\phi\wedge\textbf{AX}^{\Sigma}\textbf{AG}^{\Sigma}\phi$, $\textbf{EG}^{\Sigma}\phi\equiv\phi\wedge\textbf{EX}^{\Sigma}\textbf{EG}^{\Sigma}\phi$, $\textbf{AF}^{\Sigma}\phi\equiv\phi\vee\textbf{AX}^{\Sigma}\textbf{AF}^{\Sigma}\phi$, $\textbf{EF}^{\Sigma}\phi\equiv\phi\vee\textbf{EX}^{\Sigma}\textbf{EF}^{\Sigma}\phi$, $\textbf{A}^{\Sigma}[\phi_{1}\textbf{U}\phi_{2}]\equiv\phi_{2}\vee(\phi_{1}\wedge\textbf{AX}^{\Sigma}\textbf{A}^{\Sigma}[\phi_{1}\textbf{U}\phi_{2}])$, $\textbf{E}^{\Sigma}[\phi_{1}\textbf{U}\phi_{2}]\equiv\phi_{2}\vee(\phi_{1}\wedge\textbf{EX}^{\Sigma}\textbf{E}^{\Sigma}[\phi_{1}\textbf{U}\phi_{2}])$ (Expansion Laws) .

Proof is by induction on the size (the number of symbols) of $\phi$ for each $\Sigma$. Other well-known general properties of CTL immediately hold true, such as existence of a sound and complete axiomatisation of CTL. Atomic entailments are decidable for any APA $\delta$ (with a finite number of arguments), since each state is a Dung argumentation framework.

Example 4

For Elma example (re-listed above in Figure X that marks initially visible arguments), recall $A_{0}=\{a_{2},a_{3},a_{4}\}$. Denote the argumentation by $\delta$. By stating that $(a_{5}\dot{\in}_{\delta}A_{1}\wedge\textbf{EF}^{\{A_{1}\}}P_{\delta}(\textsf{ad},A_{1}))\supset\neg a_{2}\dot{\in}_{\delta}A_{1}$ is true, we have stated that if $a_{2}$ is a member of a reference set $A_{1}$, and if the same reference set is used for all transitions, $A_{1}$ that contains $a_{5}$ is never admissible.

For Alice example (re-listed above in Figure Y that marks initially visible arguments), recall $A_{0}=\{a_{1},a_{2},a_{3}\}$. Denote the argumentation by $\delta$. By stating that $(\textbf{AF}^{\{A_{1}\}}\textbf{AG}^{\{A_{1}\}}P_{\delta}(\textsf{co},A_{1}))\supset a_{8}\dot{\in}_{\delta}A_{1}$ is true, we have stated that if a set of arguments $A_{1}$ is such that, in all branches with $A_{1}$ as the reference set, it will be permanently complete from some state on, then it must include $a_{8}$.

For the example in Figure D (re-listed above), recall $A_{0}=\{a_{1},a_{2}\}$ within $A=\{a_{1},a_{2},a_{3},a_{4}\}$. Assume: $\textsf{Exct}(\{a_{i_{1}},\ldots,a_{i_{n}}\},A_{2})\equiv\linebreak((a_{i_{1}}\dot{\in}_{\delta}A_{2})\wedge\cdots\wedge(a_{i_{n}}\dot{\in}_{\delta}A_{2}))\wedge(\neg a_{i_{n+1}}\dot{\in}_{\delta}A_{2}\wedge\cdots\wedge\neg a_{i_{m}}\dot{\in}_{\delta}A_{2})$ where $A_{2}\subseteq A$ and where $\{a_{i_{1}},\ldots,a_{i_{m}}\}\equiv\{a_{1},\ldots,a_{4}\}\equiv A$. Assume also that $\Sigma=2^{A}$.

By $\textsf{Exct}(\{a_{2},a_{3},a_{4}\},A_{4})\wedge\textsf{Exct}(\{a_{1},a_{3},a_{4}\},A_{5})\wedge\\ \indent\indent\textbf{AG}^{\Sigma}((P_{\delta}(\textsf{ad},A_{4})\supset\textbf{EF}^{\Sigma}(P_{\delta}(\textsf{ad},A_{5})\wedge\neg P_{\delta}(\textsf{ad},A_{4})))\\ \indent\indent\indent\wedge(P_{\delta}(\textsf{ad},A_{5})\supset\textbf{EF}^{\Sigma}(P_{\delta}(\textsf{ad},A_{4})\wedge\neg P_{\delta}(\textsf{ad},A_{5}))))$,

we have described that when either of $A_{4}$ and $A_{5}$ is admissible in some reachable state, there is always a branch where the other becomes admissible and that becomes not admissible in some future state, that is to say, there can be an infinite number of oscillation among states that admit different sets of arguments.

Straightforwardly:

Proposition 4 (Non-monotonicity of admissibility)

Suppose APA $(A,R,R_{\mathbf{p}},A_{0},\hookrightarrow)$, and suppose that a set of arguments $A_{x}\subseteq A$ is admissible in a reachable state $F(A_{1})$. It is not necessary that $A_{x}$ be admissible in a state $F(A_{2})$ which satisfies $F(A_{1})\hookrightarrow^{A_{y}}F(A_{2})$ for some $A_{y}\subseteq A$.

4 Discussion and Related Work

For dynamics of argumentation, adaptation of the AGM-like belief revision [1, 28] to argumentation systems [9, 19, 18, 15, 16, 34] is popularly studied. In these studies, the focus is on restricting the class of resulting argumentation frameworks (post-states) by means of postulates for a given argumentation-framework (pre-state) and some action (add/remove an argument/attack/argumentation framework). In APA, generation by inducement and modification by conversion are primarily defined. Removal of an argument, however, is easily emulated through conversion by setting $a_{1}=a_{3}$ in $(a_{1},a_{2},a_{3})\in R_{\mathbf{p}}$. In the literature of belief revision theory, some consider selective revision [26], where a change to a belief set takes place if the input that is attempting a change is accepted. While such screening should be best assumed to have taken place beforehand within belief revision, a similar idea is critical in argumentation theory where defence of an argument is foundationally tied to a reference set of arguments. Since any set of arguments may be chosen to be a reference set, and since which arguments in the set are visible non-monotonically changes, it is not feasible to assume some persuasion acts successful and others not in all states.

Coordination of dynamics and statics is somehow under-investigated in the literature of argumentation theory. A kind for coalition profitability and formability semantics with what are termed conflict-eliminable sets of arguments [4] focuses on the interaction between sets of arguments before and after coalition formation. Doutre et al. show the use of propositional dynamic logic in program analysis/verification for encoding Dung theory and addition/removal of attacks and arguments [20, 21]. The logic comes with sequential operations, non-deterministic operations, tests. In comparison to their logic, our theory is an extension to Dung theory, which already provides a sound theoretical judgement for defence against attacks, which we extended also to persuasion acts. As far as we could fathom, such interaction between attack, persuasion, and defence has not been primarily studied in the literature. For another, a Dung-based theory has a certain appeal as a higher-level specification language. Consider the argumentation in Figure D. APA requires 4 arguments, its subset as the set of initially visible arguments, 2 inducements and 2 conversions for specification of the dynamic argumentation. By contrast, specification of a dynamic argumentation in the dynamic logic can be exponentially long as the number of non-deterministic branches increases; for the same dynamic argumentation in Figure D, it requires descriptions of all possible reachable states and transitions among them for the specification. We might take an analogy in chess here. While the number of branches in a chess game is astronomical, the game itself is specifiable in a small set of rules. For yet another, the dynamic logic facilitates dynamic changes to attacks in addition to arguments, which we did not study in this paper. The reason is mostly due to such consideration bound to lead to recursive persuasions and attacks (for recursive attacks/supports, see [8, 24, 3, 6, 14]) in our theory, which we believe will be better detailed in a separate paper for more formal interest.

Argumentation theories that accommodate aspects of persuasion have been noted across several papers. In [10], argumentation frameworks were augmented with values that controlled defeat judgement. Compared to their work, persuasion acts in APA are stand-alone relations which may be ‘executed’ non-deterministically and concurrently, may irreversibly modify visible arguments, and may produce loops. In most of argumentation papers on this topic, persuasion or negotiation is treated in a dialogue game [11, 2, 12, 23, 25, 27, 30, 31, 32, 33] where proponent(s) and opponent(s) take turns to modify an argumentation framework. APA does not assume the turn-based nature. In real-time rhetoric argumentation, as also frequently seen in social forums, more than one dialogue or more than one line of persuasive act may be running simultaneously. In this work, we were more interested in modelling those situations. The various admissibility judgement enabled by (effectively) CTL (and other branching-time logic) should provide means of describing many types of argumentation queries.

Studies on temporal arguments include [29, 7, 5, 13]. Most of these actually consider arguments that may be time-dependent. APA frameworks keep arguments abstract, and observe temporal progress through actual execution of persuasive acts. We use temporal logic for describing admissibilities rather than arguments (recall that $P_{\delta}(\omega,A_{x})$ is a formula on admissibility, not an argument). In timed argumentation frameworks [13], arguments are available for set periods of time. Combined with APA, it should become possible to explain how and why arguments are available for the durations of time in the frameworks, the explanatory power incidentally having been the strength of argumentation theory.

5 Conclusion

We have shown a direction for abstract argumentation with dynamic operators extending Dung’s theory. We set forth important properties and notions, and showed embedding of state-wise admissibility into CTL for various admissibilities across transitions. Many technical developments are expected to follow. Our contribution is promising for bringing together knowledge of abstract argumentation in AI and techniques and issues of concurrency in program analysis in a very near future. Cross-studies in the two domains are highly expected. Study in concurrent aspects of argumentation is important for evaluation of opinion transitions, which influences development of more effective sales approaches and better marketing in business, and consensus control tactics in politics. Harnessing our study with probabilistic methods is likely to form exciting research. For future work, we plan to: take into account nuances of persuasive acts such as pseudo-logic, scapegoating, threat, and half-truths [17]; and extend APA with multi-reference sets.

Acknowledgements

We thank anonymous reviewers for helpful comments. There was one suggestion concerning terms: to say to “convince” instead of “actively persuade” or “convert”. We seriously contemplated the suggested modification, and only in the end chose to leave the text as it was.