Abstract Argumentation / Persuasion / Dynamics
Ryuta Arisaka

Ken Satoh

National Institute of Informatics
ryutaarisaka@gmail.com

National Institute of Informatics
ksatoh@nii.ac.jp

arXiv:1705.10044v2 [] 2 Jun 2017

ABSTRACT
The act of persuasion, a key component in rhetoric argumentation,
may be viewed as a dynamics modifier. Such modifiers are wellknown in other research fields: recall dynamic epistemic logic where
operators modify possible world accessibilities, or recall side effects
and concurrency in programming languages. We consider persuasion
in abstract argumentation as undertaking a similar role. We extend
Dung’s frameworks with acts of persuasion among agents into Abstract Persuasion Argumentation (APA), and set forth properties
related to arguments’ admissibilities. We show a way of enriching
our basic notion of admissibility through CTL (computation tree
logic) encoding, which also permits importation of the theoretical
results known to the logic into our argumentation frameworks.

1

INTRODUCTION

An interesting component of rhetoric argumentation is persuasion.
a 1 B:a with the
We may code an act of it into A:a 1 d B:a 2 (
3
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:
a 1 B:a in which A persuades B with a into expressing
A:a 1 (
3
1
a 3 but without conversion. Like this manner, persuasion acts as a dynamics modifier in rhetoric argumentation, allowing some argument
to appear and disappear.
Of course, these acts will not be successful if a 1 is found to be not
a 1 B:a
an acceptable argument: we may have C:a 3 → A:a 1 (
2
where a 3 attacks a 1 , in which case B is not persuaded into holding
a 2 . For judging the success of a persuasion act based on other arguments, the theoretical framework of abstract argumentation by Dung
[17] is viable, as it facilitates acceptability of a set of arguments
when attacks among them are given. We incorporate the persuasion
acts into argumentation frameworks, and study how they influence
abstract argumentation frameworks. Overall, our frameworks are
to Dung’s frameworks broadly what dynamic epistemic logic is to
epistemic logic, in the very sense that dynamic operators are embedded in a language (framework). While AGM-like argumentation
framework revisions defining a class of resulting argumentation
frameworks from an initially given argumentation framework and an
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input (which could be argument(s), attack(s) or both) are being studied, there are very few studies of a similar dynamism in the literature,
and none in the context of argumentation with persuasion as far as
our awareness goes. Bridging dynamics and statics is important for
detailed and more precise analysis of rhetoric argumentation. We
formulate the notion of admissibility in our frameworks, and then
show a way of enriching it into other types of admissibilities through
CTL (computation tree logic) embedding.

1.1

Example situations

Animal Farm (George Orwell, 1945), a satirical novella, illustrates
how easy it is for a power to control social norms. Though written
to general audience, it features many critical components of rhetoric
argumentation such as the use of pseudo-logic, scapegoating, and
half-truths [13], to persuade other agents. In the story, farm animals
chase away their farm owner Jones to govern the farm themselves
free of human tyranny. They institute norms, among which is All
animals are equal, to be followed by all animals. Pigs take on a
leading role of indoctrinating other animals and of running the farm,
gradually gaining a dominant influence over other animals, however.
Ultimately they rectify the above-stated norm into All animals are
equal but some animals are more equal than others, reinstating the
tyranny against which the farm animals once united. We extract
some arguments from the novella.
a1
a2
a3
a4
a5
a6
a7

Where Jones is, there is tyranny.
It is necessary to keep pigs in good health.
All animals are equal.
All animals are equal but pigs can have apples and milk.
Apples and milk shall be consumed by pigs.
Science proves that pigs cannot go by without apples and milk.
Surely there is no one among you who wants to see Jones back?

According to the progression of the story, a 1 is the belief firmly held
by all animals. a 3 is one of the seven principles set forth. Pigs begin
to consume all apples and milk (a 5 ), against which other animals
express dissent, which is represented with the attack arrow in the
drawing. One of the pigs demonstrate a 6 by pseudo-logic as well
as threaten them of Jones’ return (a 7 ). The three arguments a 1 , a 6

Ryuta Arisaka and Ken Satoh
and a 7 persuade animals to believe a 2 (passive persuasion act). Then
a 2 accommodates modification of a 3 into a 4 smoothly (active persuasion act). In the end, animals believe a 4 and forget a 3 . They also
keep a 1 , a 2 , a 6 and a 7 , and are content with a 5 , of course.
We have a kind of concurrency scenario when multiple persuasions into different arguments are acting on an agent. Let us consider
another 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. So 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 slides ready!
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 had 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.

The above figure 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 (like in preference-based argumentation), but the non-deterministic consideration leads to a more general
theory (like in probabilistic argumentation) in which the actual behaviour of a system is dependent 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 6 ’s emergence. Our interests are in obtaining varieties in
arguments admissibility so that such a query as ‘Is a 6 going to be
an admissible argument in whatever order persuasive acts may take
place?’ is possible.

2

TECHNICAL BACKGROUNDS

In Dung’s abstract argumentation [17], an argumentation framework
is a tuple (A, R) where A is (usually) a finite set of arguments (and
in the rest any A with or without a subscript is assumed to be a set
of arguments) while R is a binary relation over A, the class of all
arguments. An argument a 1 (∈ A) is said to attack another argument
a 2 (∈ A) if and only if, or iff, (a 1 , a 2 ) ∈ R. A subset A1 of A is said
to accept, synonymously to defend, a x ∈ A iff it is possible to find
some az ∈ A1 for each ay ∈ A attacking a x such that az attacks
a x . A subset A1 of A is said to be: conflict-free iff no element of
A1 attacks an element of A1 ; admissible iff it is conflict-free and
it accepts all the elements of A1 ; 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\A1 ; and grounded iff it is the set intersection of all
complete sets of A.

3

ABSTRACT ARGUMENTATION WITH
PERSUASION

Instead of the Dung binary relation, we use a ternary relation:
R : A × (A ∪ ϵ) × (A ∪ ϵ) such that (a 1 , ϵ, a 2 ) ∈ R represents
a 1 a (passive persuasion or to induce), (a , a , a ) ∈ R repa1 (
2
1 2 3
a 1 a (active persuasion or to convert), and
resents a 1 d a 2 (
3
(a 1 , a 2 , ϵ) ∈ R represents a 1 → a 2 (attack). Let us term this abstract argumentation with persuasion as APA (Abstract Persuasion
Argumentation). An APA framework is a (AI , A, R, ,→) for AI ⊆ A
and ,→⊆ 2 A × (2 A × 2 A ). AI is the set of all initially visible arguments. It corresponds to the set of initial states in a state transition
system. As in a transition system, a transition occurs in APA framework, too, modifying one set of all visible arguments to ‘a’ next
set of all visible arguments. ,→ is the transition relation. The first
component of the Cartesian product is for relativising the transition
judgement to a reference arguments set, in particular, to visible arguments of the reference set. There is nothing new in the relativisation:
we saw in Section 2 that judgement of acceptance of an argument
a x was relative to some set of arguments A1 . Elements of ,→ are
derived constructively from AI for each reference arguments set, and
to determine them, we only need to calculate all the next sets of visible arguments from a given set of all visible arguments. Therefore,
at present, let us just assume VisA to be the set of all ‘currently’
visible arguments, which equates to AI initially. If A1 ,→Ax Ay
(which is another way of stating (Ax , (A1 , Ay )) ∈,→ in this paper),
Ay becomes VisA after the transition.
To define the function that does the one-step calculation, we
first of all obtain all possible persuasion acts for each reference
arguments set A1 . We say that a persuasion act (a 1 , α, a 2 ) ∈ R for
α ∈ {ϵ } ∪ A is possible at VisA (‘at VisA ’ means when the set of all
visible arguments is currently VisA ) for A1 just when (1) a 1 ∈ VisA

Abstract Argumentation / Persuasion / Dynamics
and (2) no a ∈ A1 ∩ VisA attacks a 1 . We assume some function
NextRRelToVisA : 2 A → 2R such that NextRRelToVisA (A1 ) is
the set of all persuasion acts possible for A1 at VisA . Meanwhile,
we let execVisA : 2R → 2 A be the function that does the one-step
calculation with execVisA (B) = (VisA \{a ∈ VisA | (a x , a, ay ) ∈
B}) ∪ {a ∈ A\VisA | (a x , ϵ, a) ∈ B or (a x , ay , a) ∈ B} which updates
VisA by adding all arguments that are made visible by any persuasive act in B, and by subtracting all arguments that are made invisible
by conversion in B unless they are made visible by other persuasive acts in B (i.e. unless cancellation of the effect occurs). With
this, we define that (A1 , (VisA , execVisA (R))) ∈,→ for all A1 ⊆ A,
R ⊆ NextRRelToVisA (A1 ) and VisA , and we define also that ,→
does not include any other elements. Clearly this definition is inductive from AI .
To explain the motivation for the ‘execVisA (R) for all
R ⊆ NextRRelToVisA (A1 )’ instead of just NextRRelToVisA (A1 )
but also transitions in general, we consider some examples of persuasive acts as shown below.
a5

a4

a2

a4
a1

a2

a1

a3

a1

a4
a1
a2

a3

a4

a2

a5

a3
a1

a3

a4

Assume that a square argument is initially visible, so that we have
AI = {a 1 , a 2 , a 4 } in the top-left figure, AI = {a 1 , a 2 } in the topright figure, and AI = {a 1 , a 3 , a 4 } in the bottom figure. In case we
have more than one persuasion, a change to be incurred to VisA
may be synchronous or asynchronous. In the former, all persuasive acts that are currently possible are checked before any changes
are effected, and all the changes occur simultaneously. If persuasion is synchronous in the top-left figure, we obtain both a 5 as
incurred by a 4 and a 3 as incurred by a 1 . In asynchronous active
persuasion (as we saw earlier in the Alice example), not all possible persuasive acts may be simultaneously executed, although
that is one possibility. It may be that a 2 is persuaded by a 1 into
a 3 . Then there is nothing that a 4 may persuade, and a 5 remains invisible if not already visible (so under the present assumption that
AI = {a 1 , a 2 , a 4 }, it remains invisible). It may be that a 2 is persuaded
by a 4 into a 5 , instead, in which case a 3 remains invisible if not already visible. Using the formal notations we introduced, we have
NextRRelToVisA (A1 ) = {(a 1 , a 2 , a 3 ), (a 4 , a 2 , a 5 )} for each A1 ⊆
{a 1 , a 2 , a 3 , a 4 , a 5 } when VisA is AI , as this simple example contains
no attacks. Now, to reflect the concurrency aspect of persuasion acts,
we need a transition for each subset of NextRRelToVisA for each reference arguments’ set A1 , and this explains the‘execVisA (R) for all

R ⊆ NextRRelToVisA (A1 )’. For each A1 ∈ {a 1 , a 2 , a 3 , a 4 , a 5 }, we
have execVisA ({(a 1 , a 2 , a 3 )}) = {a 1 , a 3 , a 4 } (by executing the active
persuasion act (a 1 , a 2 , a 3 ), a 2 becomes invisible, and a 3 becomes
visible), execVisA ({(a 4 , a 2 , a 5 )}) = {a 1 , a 4 , a 5 } (by executing the active persuasion act (a 1 , a 4 , a 5 ), a 4 becomes invisible and a 5 becomes
visible), and execVisA ({(a 1 , a 2 , a 3 ), (a 4 , a 2 , a 5 )}) = {a 1 , a 3 , a 4 , a 5 }
(by executing the two persuasion acts simultaneously, both a 3 and a 5
become visible, and a 2 becomes invisible). So (A1 , (AI , {a 1 , a 3 , a 4 })),
(A1 , (AI , {a 1 , a 4 , a 5 })), and (A1 , (AI , {a 1 , a 3 , a 4 , a 5 })) for each A1 ⊆
{a 1 , a 2 , a 3 , a 4 , a 5 } is a member of ,→. No other transitions are possible after any of the three possibilities, so they are the only elements
of ,→.
The top-right figure contains looping persuasion links. From AI ,
it may be that (a 2 , a 1 , a 4 ) executes, and we have AI ,→A1 {a 2 , a 4 },
and then we have {a 2 , a 4 } ,→A1 {a 2 , a 4 } (fixpoint). Similarly when
(a 1 , a 2 , a 3 ) executes. Or it may be that at first AI ,→A1 {a 3 , a 4 },
and then {a 3 , a 4 } ,→A1 {a 1 , a 3 , a 4 }, {a 3 , a 4 } ,→A1 {a 2 , a 3 , a 4 }, or
{a 3 , a 4 } ,→A1 {a 1 , a 2 , a 3 , a 4 }, depending on which persuasion act(s)
take place. From {a 1 , a 3 , a 4 }, we have {a 1 , a 3 , a 4 } ,→A1 {a 1 , a 3 , a 4 }
(if (a 3 , ϵ, a 1 ) or (a 1 , ϵ, a 4 ) or both of them take place) or {a 1 , a 3 , a 4 } ,→A1
{a 1 , a 2 , a 3 , a 4 }. Similarly for {a 2 , a 3 , a 4 }. From {a 1 , a 2 , a 3 , a 4 }, we
have {a 1 , a 2 , a 3 , a 4 } ,→A1 {a 1 , a 2 , a 3 , a 4 } or {a 1 , a 2 , a 3 , a 4 } ,→A1
{a 2 , a 3 , a 4 } (if (a 2 , a 1 , a 4 ) takes place but not (a 3 , ϵ, a 1 )), or
{a 1 , a 2 , a 3 , a 4 } ,→A1 {a 1 , a 3 , a 4 } (if (a 1 , a 2 , a 3 ) takes place but not
(a 4 , ϵ, a 2 )). And these are all the possible transitions in the top-right
figure.
The bottom figure contains persuasive acts and, unlike in the two
previous figures, also an attack. For any A1 such that a 2 < A1 , we
have {a 1 , a 3 , a 4 } ,→A1 {a 1 , a 3 , a 5 } (if conversion takes place before
inducement) or {a 1 , a 3 , a 4 } ,→A1 {a 1 , a 2 , a 3 , a 4 } (if inducement
takes place before conversion) or {a 1 , a 3 , a 4 } ,→A1 {a 1 , a 2 , a 3 , a 5 }
(if both persuasion acts take place at the same time). From {a 1 , a 3a 5 },
we have {a 1 , a 3 , a 5 } ,→A1 {a 1 , a 2 , a 3 , a 5 }. From {a 1 , a 2 , a 3 , a 4 }, we
have {a 1 , a 2 , a 3 , a 4 } ,→A1 {a 1 , a 2 , a 3 , a 4 } (if conversion does not
take place) or {a 1 , a 2 , a 3 , a 4 } ,→A1 {a 1 , a 2 , a 3 , a 5 }. However, for
any A2 such that a 2 ∈ A2 , the conversion will not take place when
a 2 ∈ VisA . While, if VisA does not contain a 2 , the transitions from
VisA are the same as before, from {a 1 , a 2 , a 3 , a 4 }, we do not have
{a 1 , a 2 , a 3 , a 4 } ,→A2 {a 1 , a 2 , a 3 , a 5 }.

3.1

Admissibilities

Now, what distinct properties are there for an admissible arguments
set in APA frameworks? Just as conflict-freeness and defendedness
in Dung’s frameworks, reasonable criteria may be set forth.
Conflict-freeness We say that A1 ⊆ A is currently conflict-free iff
no visible element of A1 attacks a visible element of A1 at VisA .
Mutation-freeness We say that A1 ⊆ A is currently mutation-free
iff no visible element of A1 converts a visible element of A1 at VisA .
Defendedness We say that A1 ⊆ A is currently defended iff for each
(a 2 , a 1 , α) ∈ R for a 2 ∈ VisA \A1 , a 1 ∈ VisA ∩ A1 and α ∈ {ϵ } ∪ A,
there exists some a x ∈ VisA ∩ A1 such that (a x , a 2 , ϵ) ∈ R.
Properness We say that A1 ⊆ A is currently proper iff A1 ⊆ VisA .
Mutation-freeness is a natural variation of conflict-freeness for internal conversions, and defendedness is a natural extension of Dung’s
defendedness with conversions. Properness ensures that we will not

Ryuta Arisaka and Ken Satoh
be talking of invisible arguments. The major difference from Dung’s
properties is that, because of transitions of currently visible arguments, all these properties are dependent on VisA . We naturally find
characterisation of ‘current’ admissibility with these notions: A1 ⊆ A
is currently admissible iff A1 is currently conflict-free, mutation-free,
defended and proper.
For general admissibilities, one way of describing more varieties is to
embed this 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.
Let the grammar of ϕ be:
ϕ := ⊥ | > | Pδ (Ax ) | ¬ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ ⊃ ϕ | AXϕ | EXϕ |
AFϕ | EFϕ | AGϕ | EGϕ
where Pδ (Ax ) is the atomic expression that some set of arguments
Ax is currently admissible in an argumentation framework δ :=
(AI , A, R, ,→). We did not include Until operators for certain semantic awkwardness, as to be mentioned later. We denote the class of all
atomic predicates for δ by Pδ . For semantics, let S be the class of
∀ , s ∃ , s ∀ , s ∃ , s ∀ , s ∃ }. We refer
all sequences of the elements of X:{s N
N F F G G
to an element of S by Γ. Γ may be empty. Let L : 2 A → 2 Pδ be a labelling function such that L(Ax ) = {Pδ (Ay ) ∈ Pδ | Ay is currently
admissible when VisA is Ax }. We define a head and a tail operator h : S → X, t : S → S, such that for any non-empty Γ, we
have h(Γ).t(Γ) = Γ. Here ‘.’ is a sequence concatenator. We define
M := (S, δ, L) to be a transition system with the following forcing
relations.1
•
•
•
•
•
•

•

•

•

•
•
•
•
•
•
1

M, A1 , Γ |= >.
M, A1 , Γ 6 |= ⊥.
M, A1 , Γ |= Pδ (Ax ) with an empty Γ iff Pδ (Ax ) ∈ L(A1 ).
∀ iff M, A , t(Γ) |= P (A ) for
M, A1 , Γ |= Pδ (Ax ) with h(Γ) = s N
y
δ x
each transition A1 ,→Ax Ay .
∃ iff there exists a transition
M, A1 , Γ |= Pδ (Ax ) with h(Γ) = s N
A1 ,→Ax Ay such that M, Ay , t(Γ) |= Pδ (Ax ).
M, A1 , Γ |= Pδ (Ax ) with h(Γ) = s F∀ iff there exists some i ≥ 1
for each transition A1 ,→Ax · · · ,→Ax Ai (,→Ax · · · ) such that
M, Ai , t(Γ) |= Pδ (Ax ).
M, A1 , Γ |= Pδ (Ax ) with h(Γ) = s F∃ iff there exists some transition
A1 ,→Ax · · · ,→Ax Ai (,→Ax · · · ) and some i ≥ 1 such that
M, Ai , t(Γ) |= Pδ (Ax ).
∀ iff M, A , t(Γ) |= P (A ) for
M, A1 , Γ |= Pδ (Ax ) with h(Γ) = sG
k
δ x
each k ≥ 1 for each transition A1 ,→Ax · · · such that Ak is in the
sequence.
∃ iff there exists a transition
M, A1 , Γ |= Pδ (Ax ) with h(Γ) = sG
A1 ,→Ax · · · such that M, Ak , t(Γ) |= Pδ (Ax ) for each k ≥ 1 such
that Ak is in the sequence.
M, A1 , Γ |= ¬ϕ iff M, A1 , Γ 6 |= ϕ.
M, A1 , Γ |= ϕ 1 ∧ ϕ 2 iff M, A1 , Γ |= ϕ 1 and M, A1 , Γ |= ϕ 2 .
M, A1 , Γ |= ϕ 1 ∨ ϕ 2 iff M, A1 , Γ |= ϕ 1 or M, A1 , Γ |= ϕ 2 .
M, A1 , Γ |= ϕ 1 ⊃ ϕ 2 iff M, A1 , Γ 6 |= ϕ 1 or M, A1 , Γ |= ϕ 2 .
∀ .Γ |= ϕ.
M, A1 , Γ |= AXϕ iff M, A1 , s N
∃
M, A1 , Γ |= EXϕ iff M, A1 , s N .Γ |= ϕ.

The liberty of allowing arguments into M causes no confusion, much less issues. If
one is so inclined, he/she may choose to consider that components of δ that appear in
M are semantic counterparts of those that appear in the syntax of CTL with one-to-one
correspondence between them.

•
•
•
•

M, A1 , Γ
M, A1 , Γ
M, A1 , Γ
M, A1 , Γ

|= AFϕ iff M, A1 , s F∀ .Γ |= ϕ.
|= EFϕ iff M, A1 , s F∃ .Γ |= ϕ.
∀ .Γ |= ϕ.
|= AGϕ iff M, A1 , sG
∃ .Γ |= ϕ.
|= EGϕ iff M, A1 , sG

We say that ϕ is true in δ iff M, AI , (empty) |= ϕ. It is straightforward to see the following well-known equivalences of CTL in
our semantics. ¬AFϕ ≡ EG¬ϕ, ¬EFϕ ≡ AG¬ϕ, ¬AXϕ ≡ EG¬ϕ
(De Morgan’s Laws), AGϕ ≡ ϕ ∧ AXAGϕ, EGϕ ≡ ϕ ∧ EXEGϕ,
AFϕ ≡ ϕ ∨ AXAFϕ, EFϕ ≡ ϕ ∨ EXEFϕ (Expansion Laws) by
induction on the size (the number of symbols) of ϕ.
As readers notice, we are dealing with all state transitions only at
entailments of an atomic expression. This is because semantic interpretation of an expression of the kind EX(Pδ (A1 ) ∧ Pδ (A2 )) with a
temporal operator over more than two atomic predicates are possibly
taking a distinct arguments set. In words, this expression says: there
exists a transition such that after the transition, Pδ (A1 ) and Pδ (A2 )
are both true in the next state. However, the meaning of the ‘next
state’ actually depends on the parameter to Pδ , as a transition by a
persuasive act is relativised to a reference arguments set. As such, until the recursive entailment reaches an atomic expression, we cannot
really pinpoint which states the next states are. The consequence is
that EX, EF, EG distribute over inner expressions, e.g. EX(ϕ 1 ∧ ϕ 2 )
and EXϕ 1 ∧ EXϕ 2 have the same semantic meaning. The reason we
dropped Until operators is due to this semantic interpretation that
it does not reflect an ordinary expectation out of the operations. Of
course, when it is known that all atomic predicates accept the same
arguments set, a more standard interpretation may be given, and the
Until operators will behave as anticipated. We shall discuss these
semantic interpretations in a greater detail in a future work.
There are some notable admissibility types we can take out of the
CTL sub-logic.
(1) EXPδ (Ax ) and EFPδ (Ax ). An example situation where these
properties are useful: a sales person X can attempt persuasion of
his/her unwilling customer A just once (EXPδ (Ax )) or as many
as he/she likes (EFPδ (Ax )) before A delivers a final decision. Is
it possible to influence A’s decision to include Ax , favourable
decisions to X? The duals of these expressions deal with negative
statements. Enforcement [9] is somehow related to a particular
instance of these kinds of queries.
(2) AGPδ (Ax ): An example: a patient is under treatment for one
morbidity based on a clinical guideline. His/her clinician sees a
need of prescribing more actions to him/her for another morbidity
he/she sustains. Is it possible that those actions (described with the
persuasion operators) will keep the currently attained desirable
effects of the treatment (Ax ) intact? The dual of this expression
deals with negative statements.
Adaptation of Dung’s complete, preferred, stable and grounded semantics should also diversify our admissibilities. It must be noted,
however, that the presence of our relativised dynamic transitions
do not permit step-wise construction of a preferred set from the
grounded set as meaningfully as in Dung’s.

3.2

Conclusion with related work

We have shown a new research direction for abstract argumentation
with dynamic operators of persuasive acts on Dung’s frameworks.

Abstract Argumentation / Persuasion / Dynamics
We set forth important properties and notions, and showed embedding of ‘current admissibility’ into CTL for various admissibilities.
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 is highly expected. Studying concurrent aspects of argumentation are in fact 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 are likely
to form exciting research. Our present plan is to take into account
nuances of persuasive acts like pseudo-logic, scapegoating, threat,
and half-truths in addition to the future work we mentioned earlier.
For dynamic aspects, [6, 11, 12, 14, 15, 26] calculate a modified
argumentation framework (post-state) from an original argumentation framework (pre-state) given some input. In spirit, they are
similar to the AGM belief revision [1]. Coordination of dynamics
and statics, on the other hand, is still under-investigated. A kind was
studied in [3] for coalition profitability and formability semantics.
More closely, propositional dynamic logic [16] was used to represent
abstract argumentation so that in the language addition and removal
of arguments and attacks may be expressed. In an extended work,
we will detail comparisons with their work.
Argumentation theories that accommodate aspects of persuasion
have been noted across several papers. In [7], argumentation frameworks were augmented with values that controlled defeat judgement.
Unlike their work, our persuasion acts 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 [2, 8, 10, 18–20, 22–25]. While
we do not assume any dialogue here, our admissibilities should
provide means of describing many types of argumentation queries.
Studies on temporal arguments include [4, 5, 21]. These actually
consider arguments that may be time-dependent. APA frameworks
keep arguments abstract, but observe temporal progress through
actual execution of persuasive acts. Our use of temporal logic is for
describing admissibilities rather than arguments (recall that Pδ (Ax )
is a formula on admissibility and not an argument). The idea of
visible and invisible arguments is similar to usable and unusable
arguments in [5]. However, while their usable/unusable judgement
is coupled directly with defendedness of arguments, our visibility
changes are in accordance to executions of persuasive acts.

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