# Integrating Temporality and Causality into Acyclic Argumentation Frameworks
using a Transition System

Yann Munro∗ Camilo Sarmiento∗ Isabelle Bloch  Gauvain Bourgne  Marie-Jeanne
Lesot
Sorbonne Université, CNRS, LIP6, Paris, France
<EMAIL_ADDRESS>
###### Abstract

In the context of abstract argumentation, we present the benefits of
considering temporality, i.e. the order in which arguments are enunciated, as
well as causality. We propose a formal method to rewrite the concepts of
acyclic abstract argumentation frameworks into an action language, that allows
us to model the evolution of the world, and to establish causal relationships
between the enunciation of arguments and their consequences, whether direct or
indirect. An Answer Set Programming implementation is also proposed, as well
as perspectives towards explanations.

††footnotetext: ∗ These authors contributed equally to this work.

##  1 Introduction

The abstract argumentation framework (AAF), first introduced in (?), provides
a suitable framework for representing and reasoning about contradicting
information. It makes it possible to find sets of arguments that can be
accepted together and provides explanations on why such sets have been
accepted or not. Thus, AAF provides convenient tools to model and reason about
debates. However, it is a static framework which does not include a notion of
temporality that seems crucial for modelling dialogues. Several types of
approaches have been proposed to solve this problem. One type modifies the
argumentation graph by adding or deleting attacks and arguments using specific
operators (?), and amounts to considering an AAF at each time step. Another
proposal is to transform an argumentation system into a logical formalism and
then use revision or belief change operators to update the argumentation
system (?). We propose to use another logical formalism, action languages, to
model the dynamics of a dialogue.

On the other hand, action languages offer tools to reason about action and
change and have been naturally conceived to include the notion of time. The
action language introduced by (?) has been designed to determine the evolution
of the world given a set of actions corresponding to deliberate choices of the
agent, the occurrence of which can trigger a chain reaction through external
events. We choose this action language for three reasons. First, it allows for
concurrency of events. Other languages also offering this advantage, such as
$\mathcal{C}$ (?) or PDDL+ (?), are adapted to non-deterministic or durative
actions, which increases complexity and is not useful in our framework.
Secondly, there exists a definition of actual causality that is suitable for
this action language. Finally, (?) propose a sound and complete translation
into ASP. We propose to take advantage of these properties to study the causal
relations in a dialogue, paving the way for the search of explanations.

This work constitutes a first step of a proposition to bring together
different existing tools from the Knowledge Representation and Reasoning field
to tackle issues that arise in abstract argumentation when considering
causality and temporality together. This paper is structured as follows.
Section 2 briefly recalls the principles of abstract argumentation. Section 3
describes the chosen action language and the actual causality definition
suitable for it. Section 4 proposes the main contributions of this paper: a
formalisation of acyclic abstract argumentation graphs into an action
language, with its corresponding implementation in ASP. Section 5 establishes
its formal properties, including its soundness and completeness, as well as
the relevance of the temporality inclusion. Section 6 illustrates the
exploitation of the proposed formalisation to get enriched information, as
graphical representations and causal relations. Section 7 concludes the paper.

##  2 Abstract Argumentation Framework, AAF

This section briefly recalls the basics of (?)’s AAF.

######  Definition 1.

An _abstract argumentation framework_ $AAF$ is a couple ${AF=(A,R)}$, where
$A$ is a finite set of arguments and $R$ is a set of _attacks_ corresponding
to a binary relation $A\times A$. An argument $x\in A$ attacks $y\in A$ if
$(x,y)\in R$.

As $R$ is a binary relation with a finite support, an AAF can be represented
using a graph.

######  Example 1.

To illustrate these notions, we introduce an argumentative scenario modelling
the interaction between a requesting physician, D, and a radiologist, R,
concerning an examination of a $n$ month old baby for pathology Z.
D: Can you do an X-ray scanner (CT) for this baby? ($a$)
R: It is better for a baby to avoid ionising radiations. ($b$)
R: I can suggest an MRI (magnetic resonance imaging) in two days’ time. ($c$)
D: Can Z be seen on an MRI? ($d$)
R: Yes, of course! If you want confirmation, look at the guide to good
radiology practice. ($e$)
D: But a baby might move and so you might not be able to get the information
you are looking for because the image may be artefacted. ($f$)
R: Do not worry, I am used to doing MRI for babies. ($g$)
D: Does not it cost the hospital a lot more to do an MRI? ($h$) I also have to
check with the patient’s family because it might cost them more. ($i$)
R: No problem here. The high cost includes the experience gained by my team so
that in the future this kind of delicate examination can be performed without
me. ($j$)
D: I have just spoken to the family, no problem with the MRI, the exam is
refunded. ($k$)
D: However, the family is not comfortable with the idea of having to wait two
days, could not you do the exam before? ($l$)
R: No my schedule for today is already full. My next slot is in two days, as I
told you. ($m$)

After this discussion, the decision is made to schedule an MRI in two days’
time. But later that day, the doctor receives a call from the family saying
that the baby is really not well and insisting on the urgency of the
examination. Therefore, the doctor contacts the radiologist to add a final
argument.
D: It is very urgent for the baby, we need a place today! ($n$)

From this dialogue, we can extract manually arguments and their relations to
create the AAF represented in Figure 1 with the following arguments:
$\\{\textbf{a}$: Scanner, $b$: Ionising radiation, c: MRI in two days, $d$: Z
not visible by MRI, $e$: Z visible by MRI, $f$: Difficult conditions, $g$:
High experience, $h$: High cost for the hospital, $i$: High cost for the
patient, $j$: Not problematic for the hospital, $k$: The family is covered for
an MRI, l: MRI today, $m$: No availability today, $n$: It is an
emergency!$\\}$. Arguments $a,c,l$ are called the decision variables, their
acceptance being the criterion triggering a decision: CT, MRI in two days, or
MRI today.

The obtained argumentation system is a possibility of graph that can be
extracted from this dialogue. The extraction process can be done automatically
using so-called argument mining methods (?).

$a$$b$$c$$l$$h$$d$$i$$n$$m$$e$$j$$k$$f$$g$ Figure 1:  Argumentation graph
associated with Example 1.

Note that this is a static representation of the dialogue from which any
notion of temporality has been erased. Thus, if the arguments had been stated
in a different order, it would not change the graph. This will be important to
address causality in Section 6.2.

Once an argumentation graph has been constructed, it is possible to reason on
it to determine sets of arguments that can be considered as accepted.

The _set of direct attackers_ of $x\in A$ is denoted by $Att_{x}=\\{y\in
A\mid(y,x)\in R\\}$. A set $S$ is _conflict-free_ if ${\forall(x,y)\in
S^{2}}$, $(x,y)\notin R$. An argument $x\in A$ is _acceptable_ by $S$ if
$\forall y\in Att_{x},\exists z\in S\cap Att_{y}$.

Then an _admissible set_ $S$ is defined as a conflict-free set whose elements
are all acceptable by $S$ itself. For an acyclic graph, this is the only
extension-based semantics as all others coincide with it and are therefore not
discussed here (?).

######  Example 1.

(continued) – The argument graph is acyclic. To determine the set of
acceptable arguments, it is sufficient to start from the non-attacked
arguments, here ${b,g,j,k,n}$. They are accepted by default. Then, an argument
attacked by at least one accepted argument cannot be accepted. By applying
this rule, we obtain that argument $l$ is accepted, in contrast to $a$ and
$c$. Therefore, the final decision is to perform an emergency MRI today.

##  3 Action Language and Causality

This section introduces the formal aspects of the action language proposed by
(?) and then briefly describes what is considered to be an actual cause in
this formalism. For more details refer to (?).

###  3.1 Syntax and Semantics

The purpose of the action language introduced by (?) is to determine the
evolution of the world given a set of actions corresponding to deliberate
choices of the agent. These actions might trigger some chain reaction through
external events. Therefore, in order to have a complete knowledge of the
evolution of the world, (?) keep track of both: the evolution of the states of
the world and the occurrence of events. Hence, we denote by $\mathbb{F}$ the
set of variables describing the state of the world, more precisely _ground
fluents_ representing time-varying properties, and by $\mathbb{E}$ the set of
variables describing transitions, more precisely _ground events_ that modify
fluents.

A _fluent literal_ is either a fluent $f\in\mathbb{F}$, or its negation $\neg
f$. The set of fluent literals in $\mathbb{F}$ is denoted by
$Lit_{\mathbb{F}}$, i.e. $Lit_{\mathbb{F}}=\mathbb{F}\cup\left\\{\neg f\mid
f\in\mathbb{F}\right\\}$. The complement of a fluent literal $l$ is defined as
$\overline{l}=\neg f$ if $l=f$ or $\overline{l}=f$ if $l=\neg f$.

######  Definition 2 (state).

A set $L\subseteq Lit_{\mathbb{F}}$ is a _state_ if it is:

  * •

Coherent: $\forall l\in L,\overline{l}\not\in L$;

  * •

Complete: $\forall f\in\mathbb{F},f\in L$ or $\neg f\in L$.

A state thus gives the value of each of the fluents describing the world. Time
is modelled linearly and in a discrete way to associate a state $S(t)$ to each
time point $t$ of a set $\mathbb{T}=\left\\{-1,0,\dots,N\right\\}$. $S(0)$ is
the _initial state_. Using a bounded past formalisation, all states before
$t=0$ are gathered in a state $S(-1)=\mathbb{F}\setminus S(0)$.

An event $e\in\mathbb{E}$ is an atomic formula. Each event is characterised by
three elements: preconditions and triggering conditions give conditions that
must be satisfied by a state $S$ for the event to be triggered (their
difference is detailed later in the section); effects indicate the changes to
the state that are expected if the event occurs. Note the deliberate use of
the term ‘expected’ as an event may have fewer effects than those formalised.

The preconditions and effects are represented as formulas of the languages
$\mathcal{P}\Coloneqq l|\psi_{1}\wedge\psi_{2}|\psi_{1}\vee\psi_{2}$ and
$\mathcal{E}\Coloneqq l|\varphi_{1}\wedge\varphi_{2}$, respectively. The
functions which associate preconditions, triggering conditions and effects
with each event are respectively defined as:
$pre:\mathbb{E}\rightarrow\mathcal{P}$,
$tri:\mathbb{E}\rightarrow\mathcal{P}$, and
$e\mbox{{ff}}:\mathbb{E}\rightarrow\mathcal{E}$. $\mathbb{E}$ is partitioned
into two disjoint sets: $\mathbb{A}$ contains the actions carried out by an
agent and thus subjected to their volition; $\mathbb{U}$ contains the
exogenous events which are triggered as soon as all the $pre$ conditions are
fulfilled, therefore without the need for an agent to perform them. Thus, for
exogenous events $pre$ and $tri$ are the same. By contrast, for actions, $tri$
conditions necessarily include $pre$ conditions but those are not sufficient:
the $tri$ conditions of an action also include the volition of the agent or
some kind of manipulation by another agent.

The set of all events which occur at time point $t$ is denoted by $E(t)$.
Allowing concurrency of events (meaning that more than one event can occur at
each time point) is one of the main advantages of this action language.

These definitions lead to a classical transition system: $E(t)$ generates the
transition between the states $S(t)$ and $S(t+1)$. Thus, the states follow one
another as events occur, simulating the evolution of the world.

With a bounded past formalisation, events that occurred before $t=0$ must be
represented in order to obtain causal results that are consistent with the
philosophical conception of causality. Thus, for each fluent literal $l\in
S(0)$ an event $ini_{l}\in\mathbb{E}$ is introduced, such that
$e\mbox{{ff}}(ini_{l})=l$. Then, $E(-1)=\left\\{ini_{l},l\in S(0)\right\\}$
which satisfies ${e\mbox{{ff}}(E(-1))=S(0)}$.

To solve potential conflicts or to prioritise between events, a strict partial
order $\succ_{\mathbb{E}}$ is introduced, which ensures the triggering primacy
of one event over another.

######  Definition 3 (context $\kappa$).

The _context_ , denoted as $\kappa$, is the octuple
$(\mathbb{E},\mathbb{F},pre,tri,e\mbox{{ff}},S(0),\succ_{\mathbb{E}},\mathbb{T})$,
where $\mathbb{E}$, $\mathbb{F}$, $pre$, $tri$, $e\mbox{{ff}}$, $S(0)$,
$\succ_{\mathbb{E}}$, and $\mathbb{T}$ are as defined above.

######  Definition 4 (valid execution).

An _execution_ is a sequence $E(-1),S(0),E(0),\dots,E(N),S(N+1)$. Such an
execution is _valid_ given $\kappa$ if $\forall t\in\mathbb{T}$:

  1. 1.

$S(t)\subseteq Lit_{\mathbb{F}}$ is a state according to Definition 2.

  2. 2.

$E(t)\subseteq\mathbb{E}$ verifies:

    1. 2.a

$\forall e\in E(t)$, $S(t)\models pre(e)$;

    2. 2.b

$\nexists(e,e^{\prime})\in E(t)^{2},~{}e\succ_{\mathbb{E}}e^{\prime}$;

    3. 2.c

$\forall e\in\mathbb{E}$ such that $S(t)\models tri(e)$,
$\forall e\in\mathbb{E}$ $e\in E(t)$ or ${\exists e^{\prime}\in E(t),}$
$e^{\prime}\succ_{\mathbb{E}}e$;

  3. 3.

$S(t+1)=\left\\{l\in S(t),\forall e\in E(t),\overline{l}\not\in
e\mbox{{ff}}(e)\right\\}\cup$
$S(t+1)=$ $\left\\{l\in Lit_{\mathbb{F}},\exists e\in E(t),l\in
e\mbox{{ff}}(e)\right\\}$.

There is potentially more than one valid execution for a given context
$\kappa$. In fact, there is no specification of when actions are performed in
$\kappa$. Adding a set of timed actions
$\sigma\subseteq\mathbb{A}\times\mathbb{T}$ which models volition of agents as
an input, called scenario, leads to a unique valid execution. From this unique
execution the event trace and state trace we are interested in, denoted by
$\tau_{\sigma,\kappa}^{e}$ and $\tau_{\sigma,\kappa}^{s}$, respectively, can
be extracted.

######  Definition 5 (traces $\tau_{\sigma,\kappa}^{e}$ and
$\tau_{\sigma,\kappa}^{s}$).

Given a scenario $\sigma$ and a context $\kappa$, the _event trace_
$\tau_{\sigma,\kappa}^{e}$ is the sequence of events $E(-1),E(0),\dots,E(N)$
from the execution which is valid given $\kappa$, such that: $\forall
t\in\mathbb{T},{\forall e\in
E(t)},{e\in\mathbb{A}}\Leftrightarrow(e,t)\in\sigma$. The _state trace_
$\tau_{\sigma,\kappa}^{s}$ is the sequence of states $S(0),S(1),\dots,S(N+1)$
corresponding to $\tau_{\sigma,\kappa}^{e}$.

###  3.2 Actual Causality

The actual causation definition proposed by (?) is an action language suitable
formalisation of Wright’s NESS test. Introduced by (?), this test states that:
‘A particular condition was a cause of a specific consequence if and only if
it was a necessary element of a set of antecedent actual conditions that was
sufficient for the occurrence of the consequence.’

A causal relation links a cause to an effect. Since action languages represent
the evolution of the world as a succession of states produced by the
occurrence of events, states are introduced between events. Therefore, in
addition to the actual causality relation that links two occurrences of
events, as commonly accepted by philosophers, it is necessary to define causal
relations where causes are occurrences of events and effects are formulas of
the language $\mathcal{P}$ that are true at a given time. These intermediate
relations are established on the basis of Wright’s NESS test of causation. In
order to give an actual causality definition suitable for action languages,
three causal relations are introduced by (?): (i)  _Direct NESS-causes_ give
essential information about causal relations by looking at the effects that
the occurrence of an event has actually had, which are not necessarily the
same as those expected. Direct NESS-causes relate occurrences of events and
formulas of $\mathcal{P}$ being true at a specific time point. However, the
set of direct NESS-causes of a formula of $\mathcal{P}$ may include exogenous
events that are not necessarily relevant. It is therefore essential to
establish a causal chain by going back in time in order to find the set of
actions that led to the formula truthfulness. (ii)  _NESS-causes_ allow for
such a causal chain to be found. If we denote by $\psi\in\mathcal{P}$ the
formula true at $t_{\psi}$ we are interested in, and $C$ the set of direct
NESS-causes of $(\psi,t_{\psi})$, finding the NESS-causes means finding what
causes $(tri(C),t)$ necessarily, where $t<t_{\psi}$. Note that direct NESS-
causes are by definition a special case of NESS-causes. (iii) The occurrence
of a first event $e$ is considered an _actual cause_ of the occurrence of a
second event $e^{\prime}$ if and only if the occurrence of $e$ is a NESS-cause
of the triggering of $e^{\prime}$. From this we can deduce that, if the
occurrence $(e^{\prime},t_{2})$ is a direct NESS-cause of $(\psi,t_{3})$ and
the occurrence $(e,t_{1})$ is an actual cause of $(e^{\prime},t_{2})$, with
$t_{1}<t_{2}<t_{3}$, then the occurrence $(e,t_{1})$ is a NESS-cause of
$(\psi,t_{3})$. These three causal relations are illustrated using Example 1
in Section 6.2.

##  4 From AAF to Action Languages

This section presents our first contribution: a formalisation of acyclic AAF
into the action language introduced above. Section 4.1 presents the definition
of the argumentative context $\kappa$, Section 4.2 provides the modified
definitions of the action language semantics, and Section 4.3 briefly sketches
the structure of the ASP implementation.

In contrast to AAF, we propose to take into account the order of enunciation
of arguments. Instead of having only a couple $(A,R)$, the input is a couple
$(\Delta,R)$, where $\Delta$ is a dialogue, i.e. a sequence of statements in
natural language:

######  Definition 6 (dialogue $\Delta$).

A _dialogue_ is ${\Delta=\\{(a,o)\mid a\in A,o\in\mathbb{N}\\}}$, where each
argument $a$ is associated to its order of enunciation, $o$.

###  4.1 Instantiating the Context

In order to formalise an AAF in the action language described in Section 3,
let us first define the variables necessary to describe the world, i.e. the
AAF. These variables correspond to the fluents $\mathbb{F}$. As introduced in
Section 2, there are two elements to consider: the arguments and the attack
relation. First, to describe an argument $x$, we create two fluents:
$p_{x}\in\mathbb{F}$ and $a_{x}\in\mathbb{F}$ expressing whether the argument
is present in the graph and whether it is acceptable. Regarding $R$, we use
the fluent $cA_{y,x}\in\mathbb{F}$ to model that $y$ can attack argument $x$.
As we only deal with acyclic AAF, $\nexists(x_{1},\dots,x_{n})\in A$ such that
$\left(cA_{x_{1},x_{2}},\dots,cA_{x_{n-1},x_{n}},cA_{x_{n},x_{1}}\right)\in\mathbb{F}$.
We call this property acyclicity of the fluents $cA$.

In an AAF, the only deliberate action is to enunciate an argument, which leads
to ${\mathbb{A}=\\{enunciate_{x}\mid x\in A\\}}$. For this action to be
possible, argument $x$ must not have already been said. $x$ then becomes
present and acceptable by default. This choice is justified by the fact that
its acceptability is evaluated in the next state before it has an impact on
the rest of the graph. Formally:

$\displaystyle pre(enunciate_{x})\equiv\neg p_{x}$ $\displaystyle
e\mbox{{ff}}(enunciate_{x})\equiv p_{x}\wedge a_{x}$

Remark – None of the events described below has the effect of making an
argument not present. This implies that it is not possible to enunciate an
argument that is already present. This assumption does not contradict the
framework of classical argumentation. Indeed, a repeated argument would be
manifested by an identical but differently named argument in the graph, which
is obviously possible with our transformation. However, since the action
language we use provides tools for taking temporality into account, there may
be a better approach, but it would require further study. Nevertheless, as a
first step, this article aims to lay a solid foundation at the cost of some
simplifying assumptions.

Before enunciating the next argument, we choose to update the acceptability of
all other arguments present after the enunciation of a new argument. This
defines a state which we call _argumentative state_.

######  Definition 7 (argumentative state).

A state $S(t)$ is an _argumentative state_ if:
i) $\forall x,y,\left[S(t)\models a_{x}\wedge p_{y}\wedge cA_{y,x}\Rightarrow
S(t)\models\neg a_{y}\right]$;
ii) $\forall x,\left[S(t)\models p_{x}\wedge\left(\bigwedge_{y}\neg
a_{y}\vee\neg cA_{y,x}\right)\Rightarrow S(t)\models a_{x}\right]$.

After an argument is enunciated, we want updates to be triggered
automatically. We represent them with two exogenous events:
${makesUnacc_{y,x}\in\mathbb{U}}$ and ${makesAcc_{x}\in\mathbb{U}}$. An
argument is acceptable only if it is unattacked or attacked only by
unacceptable arguments. Hence, it is enough for one of the attackers to be
acceptable to make the attacked argument unacceptable. The two cases are
considered:

Acceptability update: Suppose that an argument $y$ just enunciated can attack
argument $x$, and that $x$ and $y$ are acceptable. Then, $x$ being attacked by
an acceptable argument $y$, it becomes unacceptable. Formally, the exogenous
event $makesUnacc_{y,x}$ can be written as:

$\displaystyle tri(makesUnacc_{y,x})\equiv$ $\displaystyle a_{x}\wedge
a_{y}\wedge cA_{y,x}$ $\displaystyle e\mbox{{ff}}(makesUnacc_{y,x})\equiv$
$\displaystyle\neg a_{x}$

This definition also allows dealing with cases where a new argument $z$ makes
an attacker $y$ of $x$ acceptable again. In this case, $x$ becomes
unacceptable.

Non-acceptability update: Suppose that argument $x$ is not acceptable and that
an argument $z$ has just been enunciated. This argument has no direct link
with $x$ but may impact the acceptability of some attackers of $x$. We
therefore check whether all the arguments that can attack $x$ are acceptable
or not. If none of them are indeed acceptable, then $x$ becomes acceptable
again. In the action language, this is expressed by the exogenous event
$makesAcc_{x}$:

$\displaystyle tri(makesAcc_{x})\equiv$ $\displaystyle p_{x}\wedge\neg
a_{x}\wedge\left(\bigwedge_{y}\neg cA_{y,x}\vee\neg a_{y}\right)$
$\displaystyle e\mbox{{ff}}(makesAcc_{x})\equiv$ $\displaystyle a_{x}$

Finally, when an argument $x$ is enunciated, it must be checked that it has
not become unacceptable because of an argument $y$ already present before it
makes other arguments unacceptable. This is reflected in the following
priority rule:

$\displaystyle makesUnacc_{y,x}\succ_{\mathbb{E}}makesUnacc_{x,z}$

Note that adding an argument to the graph can only directly impact the other
arguments by making them unacceptable. For this reason, it is not necessary to
establish a priority rule of the form
$makesUnacc_{y,x}\succ_{\mathbb{E}}makesAcc_{z}$ as this situation is already
addressed by the previous rule.

Remark – In the above transformation, we do not distinguish between the
notions of potential and real attack, because such a difference disappears in
the equations. Indeed, let us consider a fluent ${att_{y,x}\in\mathbb{F}}$
translating the fact that argument $y$ actually attacks argument $x$. Let us
define the exogenous event $isAttacking_{y,x}\in\mathbb{U}$ as:

$\displaystyle tri(isAttacking_{y,x})\equiv$ $\displaystyle p_{x}\wedge
p_{y}\wedge cA_{y,x}$ $\displaystyle e\mbox{{ff}}(isAttacking_{y,x})\equiv$
$\displaystyle att_{y,x}$

From this definition, an argument $y$ attacks an argument $x$ if both are
present and $y$ can attack $x$. However, for this attack to be taken into
account, the attacker $y$ must be acceptable. We obtain conditions of the form
$a_{y}\wedge att_{y,x}$, i.e. $a_{y}\wedge p_{y}\wedge cA_{y,x}$. However, an
argument cannot be acceptable without being present, i.e. $a_{y}\wedge
p_{y}\equiv a_{y}$. Thus, taking this new fluent into account, we would have:
$tri(makesUnacc_{y,x})\equiv a_{y}\wedge a_{x}\wedge att_{y,x}=a_{y}\wedge
a_{x}\wedge cA_{y,x}$. The same precondition applies as without the
introduction of $isAttacking\text{ and }att$. Therefore, we use only $cA$.

###  4.2 Semantics Adapted to AAF

Having an adapted $\kappa$ for the argumentative framework, we propose to
modify the action language semantics to produce traces that are representative
of the reality. For this purpose, arguments will be stated from argumentative
states step by step in the order determined by the dialogue $\Delta$.

The current form of scenario $\sigma$ is not ideal for this task. Indeed, it
implies that we need to know in advance how many steps each chain of
admissibility update events will take to plan at which time the next argument
should be stated. To solve this issue we introduce a set of ranked actions
$\varsigma\subseteq\mathbb{A}\times\mathbb{N}$ which is called _sequence_. The
input to obtain unique traces will no longer be the scenario $\sigma$ but the
sequence $\varsigma$. These modifications require changes to Definitions 4 and
5.

######  Definition 8 (argumentative setting $\chi$).

The _argumentative setting_ of the action language, denoted by $\chi$, is the
couple $(\varsigma,\kappa)$ with $\varsigma$ a sequence and $\kappa$ a
context.

Definition 9 is the result obtained after modifying Definition 4. Conditions
2.d and 2.e are added and $\forall e\in\mathbb{E}$ is replaced by $\forall
e\in\mathbb{U}$ in condition 2.c. These modifications respectively express
that an action in the sequence can be triggered only if no exogenous event is
triggered at the same time point, and that event sets in the event trace
cannot be empty. Conditions 1, 2.a, 2.b, and 3 remain unchanged. So the
triggering of exogenous events remains unchanged.

######  Definition 9 (valid execution in an argumentative context).

Given an argumentative context $\kappa$, a sequence
$E(-1),S(0),E(0),\dots,E(N),S(N+1)$ is a _valid execution_ w.r.t. $\kappa$ if,
in addition to conditions 1, 2.a, 2.b, and 3 of Definition 4, the following
conditions are satisfied $\forall t\in\mathbb{T}$:

  1. 2

$E(t)\subseteq\mathbb{E}$ satisfies:

    1. 2.c

$\forall e\in\mathbb{U}$ such that $S(t)\models tri(e)$,
$e\in E(t)$ or ${\exists e^{\prime}\in
E(t),~{}e^{\prime}\succ_{\mathbb{E}}e}$;

    2. 2.d

If $\exists e\in E(t)\cap\mathbb{A}$, then $\forall e^{\prime}\in\mathbb{U}$,
$S(t)\not\models tri(e^{\prime})$;

    3. 2.e

$E(t)\not=\varnothing$.

In Definition 5 traces were defined as extracts of a valid execution given
$\kappa$ and additional conditions related to $\sigma$. Instead of defining
directly traces, Definition 10 corresponds to a valid execution given
$\chi=(\varsigma,\kappa)$. Traces are simply extracts from such valid
executions.

######  Definition 10 (valid execution given $\chi$).

Given an argumentative setting $\chi=(\varsigma,\kappa)$, a valid execution
w.r.t. $\kappa$, is _valid w.r.t. to $\chi$_ if:

  1. 1.

${\forall t\in\mathbb{T}}$, ${E(t)\subset\left(\left\\{a,\exists
o\in\mathbb{N},(a,o)\in\varsigma\right\\}\cup\mathbb{U}\right)}$;

  2. 2.

${\forall\left(\left(e,o\right),\left(e^{\prime},o^{\prime}\right)\right)\in\varsigma^{2}}$
such that $o<o^{\prime}$,
$\exists t,t^{\prime}$ such that $e\in E(t)\mbox{ and }e^{\prime}\in
E(t^{\prime})\mbox{ and }t<t^{\prime}$;

  3. 3.

${\forall\left(\left(e,o\right),\left(e^{\prime},o^{\prime}\right)\right)\in\varsigma^{2}}$
such that $o=o^{\prime}$,
$\exists t$ such that $(e,e^{\prime})\in E(t)^{2}$.

Given a valid execution given $\chi$, its _event trace_ $\tau_{\chi}^{e}$ is
its sequence of events $E(-1),E(0),\dots,E(N)$, its _state trace_
$\tau_{\chi}^{s}$ is its sequence of states $S(0),S(1),\dots,S(N+1)$.

###  4.3 ASP Implementation

We propose an adapted implementation in ASP based on the sound and complete
one described in (?). The ASP program $\pi_{con}(\kappa)$ and
$\pi_{seq}(\varsigma)$ are obtained by the translation of the context $\kappa$
and the sequence $\varsigma$ respectively. $\pi_{\mathbb{A}}$ is obtained by
the translation of the action language semantics introduced in Section 3.1 and
modified in Section 4.2. $\pi_{\mathbb{C}}$ is obtained by the translation of
the causal relations definitions introduced by (?). The entire program
$\Pi(\chi)=\pi_{sce}(\varsigma)\cup\pi_{con}(\kappa)\cup\pi_{\mathbb{A}}\cup\pi_{\mathbb{C}}$
is available111https://gitlab.lip6.fr/sarmiento/kr_2023.git.

##  5 Formal Properties

This section establishes formal properties of the proposed transformation.
First, we prove that a notion of temporality is captured by the
transformation. Then we prove its soundness and completeness. Finally, we
introduce the propositions that pave the way to the discussion of Section 6.

###  5.1 Preliminary Property on a Valid Execution

We start by showing that, although valid executions given $\kappa$ are not
unique, valid executions given $\chi$ are, and thus are the corresponding
traces $\tau_{\chi}^{e}$ and $\tau_{\chi}^{s}$.

######  Proposition 1.

Given an argumentative setting $\chi=(\varsigma,\kappa)$, the traces
$\tau_{\chi}^{e}$ and $\tau_{\chi}^{s}$ are unique.

###### Proof.

Let us prove by contradiction the unicity of valid executions given $\chi$.
Let $\chi=(\varsigma,\kappa)$ be the argumentative setting and $\epsilon$,
$\epsilon^{\prime}$ two valid executions given $\chi$. By way of a reductio ad
absurdum, we suppose that $\epsilon\neq\epsilon^{\prime}$.

According to Definition 4, $S(t+1)$ is derived from $S(t)$ and the events in
$E(t)$, and similarly for $S^{\prime}$. Hence, given that $\kappa$ is common
to $\epsilon$ and $\epsilon^{\prime}$, $E(-1)=E^{\prime}(-1)$ and
$S(0)=S^{\prime}(0)$, the first discrepancy between $\epsilon$, and
$\epsilon^{\prime}$ is not to be found in a set of states, but in a set of
events, which is not empty as the executions are valid. Let $t_{0}$ be the
minimal date at which a difference between $\epsilon$ and $\epsilon^{\prime}$
is observed. We have $E(t_{0})\neq E^{\prime}(t_{0})$, $\forall t<t_{0}$,
$E(t)=E^{\prime}(t)$, and $\forall t\leq t_{0}$, $S(t)=S^{\prime}(t)$. Thus,
$\forall e\in\mathbb{E}$, $S(t_{0})\models pre(e)\Leftrightarrow
S^{\prime}(t_{0})\models pre(e)$. Without loss of generality, let us consider
an event $e_{0}$ such that $e_{0}\not\in E(t_{0})$ and $e_{0}\in
E^{\prime}(t_{0})$. Two cases can occur, $e_{0}\in\mathbb{U}$ or
$e_{0}\in\mathbb{A}$.

i) Let us first show by contradiction that $e_{0}\not\in\mathbb{U}$. Let us
suppose $e_{0}\in\mathbb{U}$. As $e_{0}\in E^{\prime}(t_{0})$ and
$S(t_{0})=S^{\prime}(t_{0})$, $S(t_{0})\models tri(e_{0})$. Then from 2.c in
Definition 9, $e_{0}\not\in E(t_{0})$ implies $\exists e\in E(t_{0})$ such
that ${e\succ_{\mathbb{E}}e_{0}}$. Then from 2.b applied to
$E^{\prime}(t_{0})$, we get ${e\not\in E^{\prime}(t_{0})}$. Now, either
$e\in\mathbb{A}$ or $e\in\mathbb{U}$. If $e\in\mathbb{A}$, as $e\in E(t_{0})$,
condition 2.d would imply that $S^{\prime}(t_{0})\not\models tri(e_{0})$ which
contradicts our assumption. In the second case, if $e\in\mathbb{U}$, then
$e\in E(t_{0})$ and $e\not\in E^{\prime}(t_{0})$ because of the same reasons
behind ${e_{0}\not\in E(t_{0})}$ and ${e_{0}\in E^{\prime}(t_{0})}$. If we
apply the same reasoning to $e$, we get $\exists e^{\prime}\in\mathbb{U}$ such
that $e^{\prime}\in E(t_{0})$, $e^{\prime}\not\in E^{\prime}(t_{0})$, and
$e^{\prime}\succ_{\mathbb{E}}e$. This reasoning can be repeated again on
$e^{\prime}$ and so on. As $\mathbb{U}$ is finite, either the chain will be
broken, or an event will be used a second time. The first case means
$\exists\tilde{e}\in\mathbb{U}$ such that $\tilde{e}\not\in E(t_{0})$ and
$\tilde{e}\in E^{\prime}(t_{0})$ is false, which makes all the chain false. In
the second case, by transitivity we get
$\tilde{e}\succ_{\mathbb{E}}\tilde{e}$. This leads to a contradiction as
$\succ_{\mathbb{E}}$ is a strict partial order. Thus, $e_{0}\in\mathbb{U}$ is
not possible.

ii) As from (i) $e_{0}\in\mathbb{A}$, by condition 1. in Definition 10,
${e_{0}\in E^{\prime}(t_{0})}$ implies ${e_{0}\in\left\\{a,\exists
o\in\mathbb{N},(a,o)\in\varsigma\right\\}}$. $\varsigma$ being the same for
$\epsilon$ and $\epsilon^{\prime}$, the rank $o_{0}\in\mathbb{N}$ associated
to $e_{0}$ is the same. Hence, given that $\forall t<t_{0}$,
$\epsilon(t)=\epsilon^{\prime}(t)$ and $e_{0}\in E^{\prime}(t_{0})$,
$e_{0}\not\in E^{\prime}(t)$ implies $e_{0}\not\in E(t)$. The only possibility
left is the procrastination of actions. In the case where
$\nexists(e,o_{0})\in\varsigma$ such that $e\in E(t_{0})$,
$E(t_{0})=\varnothing$ which is in contradiction with condition 2.e in
Definition 9. Otherwise, we have a contradiction with condition 3. in
Definition 10. Thus, $e_{0}\in\mathbb{A}$ is not possible.
Since the assumption of the existence of $t_{0}$ leads to a contradiction for
all above cases, we have no alternative but to reject the existence of a
minimal daten$t_{0}$ at which a difference between $\epsilon$ and
$\epsilon^{\prime}$ is observed, and thus to reject that
$\epsilon\neq\epsilon^{\prime}$. The unicity of traces is accordingly
established. ∎

From now on, when reference is made to events and states, they will be those
from the unique $\tau_{\chi}^{e}$ and $\tau_{\chi}^{s}$, respectively. Thus,
the set of all events which actually occurred at time point $t$ is
$E^{\chi}(t)=\tau_{\chi}^{e}(t)$. Following the same reasoning, the actual
state at time point $t$ is $S^{\chi}(t)=\tau_{\chi}^{s}(t)$.

###  5.2 Soundness and Completeness

In this section, we establish the soundness and completeness of our
transformation. For that, we first introduce the notion of associated graph as
follows:

######  Definition 11.

Given a state $S^{\chi}(t)$, $AF^{\prime}=(A^{\prime},R^{\prime})$, where
$A^{\prime}=\\{x\mid S^{\chi}(t)\models p_{x}\\}$ and $R^{\prime}=\\{(y,x)\mid
S^{\chi}(t)\models cA_{y,x}\\}$, is called the _associated graph_ of
$S^{\chi}(t)$.

From the acyclicity property of the fluent $cA$, the associated graph is
acyclic.

Now, we focus on the notion of acceptability. We first characterise
argumentative states using $tri$.

######  Lemma 1.

Let $S^{\chi}(t)$ be a state. The two following propositions are equivalent:

$\bullet\;\forall e\in\mathbb{U}$, $S^{\chi}(t)\not\models tri(e)$

$\bullet\;S^{\chi}(t)$ is an argumentative state as defined in Def. 7.

###### Proof.

In our context, $\mathbb{U}=\\{makesAcc,makesUnacc\\}$. We prove that
$S^{\chi}(t)\not\models tri(makesAcc_{x})$ is equivalent to (ii) of Definition
7 and $S^{\chi}(t)\not\models tri(makesUnacc_{y,x})$ is equivalent to (i) of
Definition 7: for any $x,y$

$\bullet$ $\neg tri(makesAcc_{x})=\neg(p_{x}\wedge\neg
a_{x}\wedge(\bigwedge_{y}\neg a_{y}\vee\neg cA_{y,x}))$
$=\neg(p_{x}\wedge(\bigwedge_{y}\neg a_{y}\vee\neg cA_{y,x}))\vee a_{x}$
$=p_{x}\wedge(\bigwedge_{y}\neg a_{y}\vee\neg cA_{y,x})\Rightarrow a_{x}$,
which leads to the desired equivalence with (ii) in Definition 7.

$\bullet$$\neg tri(makesUnacc_{y,x})=\neg(a_{x}\wedge a_{y}\wedge cA_{y,x})$
$=\neg(a_{x}\wedge p_{y}\wedge a_{y}\wedge cA_{y,x})\mbox{ as $a_{y}$ implies
$p_{y}$}\\\ =\neg(a_{x}\wedge p_{y}\wedge cA_{y,x})\vee\neg a_{y}$
$=a_{x}\wedge p_{y}\wedge cA_{y,x}\Rightarrow\neg a_{y}$, which leads to the
desired equivalence with (i) in Definition 7. ∎

An argumentative state can therefore be seen as a state where nothing happens
until a voluntary action is made. Now we prove that it is always possible to
reach such a state from an argumentative state in which an $x$ is enunciated.

######  Proposition 2.

Given an argumentative state $S^{\chi}(t)$ and ${x\in A}$, if
$enunciate_{x}\in E^{\chi}(t)$, then $\exists t^{\prime}\in\mathbb{T}$,
$t<t^{\prime}$ such that $S^{\chi}(t^{\prime})$ is an argumentative state.

###### Proof.

Given an argumentative state $S^{\chi}(t)$ and $x\in A$ such that
$enunciate_{x}\in E(t)$, let us prove that $\exists t^{\prime}\in\mathbb{T}$,
${t<t^{\prime}}$ such that no trigger is a logical consequence of
$S^{\chi}(t^{\prime})$, which leads to the desired result using Lemma 1.

As $\mathbb{U}$ and $\\{S^{\chi}(t)\models cA_{y,x}\mid(x,y)\in A^{2}\\}$ are
finite sets, there is a finite number of possible triggering for
$E^{\chi}(t)$. Moreover, as there is a finite number of arguments, there is a
finite number of paths in the associated graph. The graph being acyclic, each
path length is finite. Therefore, there is a finite maximum number of sets of
events ($M$) and so $\exists t^{\prime}\leq(t+M+1)$ such that ${\forall
e\in\mathbb{U},S^{\chi}(t^{\prime})\nvDash tri(e)}$. ∎

Finally, this proposition allows us to prove that an acceptable argument in
the argumentative state is acceptable in the associated graph and vice-versa,
as the triggering rules have been made in order to model how acceptability is
computed. We first prove a useful lemma.

######  Lemma 2.

Given an argumentative state $S^{\chi}(t)$, for any $x$,
$S^{\chi}(t)\models p_{x}\wedge\neg a_{x}\Leftrightarrow
S^{\chi}(t)\models\exists y,p_{x}\wedge a_{y}\wedge cA_{y,x}$.

###### Proof.

$[\Rightarrow]:$ For any $x$ such that $S^{\chi}(t)\models p_{x}\wedge\neg
a_{x}$, (ii) of Definition 7 implies that $S^{\chi}(t)\models\neg
p_{x}\vee(\bigvee_{y}a_{y}\wedge cA_{y,x})$. Therefore, as $S^{\chi}(t)\models
p_{x},S^{\chi}(t)\models\exists y,p_{x}\wedge a_{y}\wedge cA_{y,x}$.

$[\Leftarrow]:$ Let us prove it by contraposition: let $x_{0}$ be such that
$S^{\chi}(t)\models\neg p_{x_{0}}\vee a_{x_{0}}$. If $S^{\chi}(t)\models\neg
p_{x_{0}}$, then $x_{0}$ is such that $S^{\chi}(t)\models\forall y,\neg
p_{x_{0}}\vee\neg a_{y}\vee\neg cA_{y,x_{0}}$, which ends the proof.
Otherwise, $S^{\chi}(t)\models p_{x_{0}}\wedge a_{x_{0}}$. If
$S^{\chi}(t)\models p_{x_{0}}\wedge(\exists y,a_{y}\wedge cA_{y,x_{0}})$ then
$S^{\chi}(t)\models tri(makesUnacc_{y,x_{0}})$, which is not possible as
$S^{\chi}(t)$ is argumentative. Thus both cases lead to a contradiction.∎

The next proposition establishes the correspondence between acceptability in
argumentation and argumentative states.

######  Proposition 3.

Given an argumentative state $S^{\chi}(t)$ and its associated graph
${AF=(A,R)}$ according to Definition 11, then for any $x$, $x\in A$ acceptable
by $A\iff S^{\chi}(t)\models a_{x}$.

###### Proof.

$[\Rightarrow]:$ Let $x_{0}\in A$ such that $x_{0}$ is acceptable by A, let us
prove that $S^{\chi}(t)\models a_{x_{0}}$.

Let us suppose that $S^{\chi}(t)\models\neg a_{x_{0}}$. Moreover, by
construction of AF $S^{\chi}(t)\models p_{x_{0}}$. $S^{\chi}(t)$ is an
argumentative state so according to Lemma 2 $S^{\chi}(t)\models
p_{x_{0}}\wedge\neg a_{x_{0}}\Leftrightarrow S^{\chi}(t)\models\exists
y,p_{x_{0}}\wedge a_{y}\wedge cA_{y,{x_{0}}}$. (i) of Definition 7 applied to
$a_{y}$ says that $\forall z,S^{\chi}(t)\models a_{y}\wedge p_{z}\wedge
cA_{z,y}\Rightarrow S^{\chi}(t)\models\neg a_{z}$. As there is a finite number
of arguments, it is possible to repeat the process we applied for ${x_{0}}$ on
$z$ until one of the two scenarios:

$\bullet$ $S^{\chi}(t)\models\nexists y,p_{z}\wedge a_{y}\wedge cA_{y,z}$.
This leads to trigger the exogenous event $makesAcc_{z}$ which is not possible
as $S^{\chi}(t)$ is an argumentative state according to Lemma 1.

$\bullet$ $\forall z,S^{\chi}(t)\models a_{y}\wedge p_{z}\wedge
cA_{z,y}\Rightarrow S^{\chi}(t)\models\neg a_{z}$ where $p_{z}\wedge cA_{z,y}$
is false. Then in $AF$, $Att_{y}=\emptyset$. Therefore, $y$ is acceptable
which contradicts that ${x_{0}}$ is acceptable.

So, $S^{\chi}(t)\models a_{x_{0}}$.

$[\Leftarrow]:$ Let ${x_{0}}\in A$ such that $S^{\chi}(t)\models a_{x_{0}}$.
Let us prove that ${x_{0}}$ is acceptable by $A$.

As $S^{\chi}(t)\models a_{x_{0}}$ and $S^{\chi}(t)$ is argumentative, we have
that $\forall y,S^{\chi}(t)\models a_{x_{0}}\wedge p_{y}\wedge
cA_{y,{x_{0}}}\Rightarrow S^{\chi}(t)\models\neg a_{y}$. Then, for any $y$
satisfying the premise, by definition of $AF$, $({x_{0}},y)\in A^{2}$ and
$(y,{x_{0}})\in R$.

If such a $y$ is acceptable by $A$, then according to $[\Rightarrow]$,
$S^{\chi}(t)\models a_{y}$. In that case, $S^{\chi}(t)\models
tri(makesUnacc_{y,{x_{0}}})$ which contradicts the fact that $S^{\chi}(t)$ is
an argumentative state.

So, as $\forall y\in Att_{x}$, $y$ is not acceptable by $A$, ${x_{0}}$ is
acceptable by $A$. ∎

We proved that there is an equivalence between an argumentative state and its
associated graph. Now, from a dialogue and the attack relation, the traces are
generated as well as an AAF. From that point, we establish the existence of a
state whose associated graph is equal to the initial AAF. Such a state is
called the _final argumentative state_ and is defined as an argumentative
state $S^{\chi}(t)$ such that $\forall x\in A$, $\exists
t^{\prime}\in\mathbb{T}$ such that $t^{\prime}<t$ and $enunciate_{x}\in
E^{\chi}(t^{\prime})$.

######  Theorem 1 (Soundness and Completeness).

Given a dialogue $\Delta$ and a set of attack $R$, given the argumentative
setting $\chi$, the associated argumentative graph $AF^{\prime}$ of the final
argumentative state $S^{\chi}(t)$, and ${AF=(A,R)}$ obtained from
$(\Delta,R)$, it holds that $AF^{\prime}=AF$.

###### Proof.

As $AF^{\prime}$ is associated to a final argumentative state, $\forall x\in
A$, $\exists t^{\prime}\in\mathbb{T}$ such that $t^{\prime}<t$ and
${enunciate_{x}\in E^{\chi}(t^{\prime})}$. Now,
$eff(enunciate_{x})=p_{x}\wedge a_{x}$. So, $A^{\prime}=A$.

Moreover, by construction of $cA_{y,x}$ and $R^{\prime}$, $R=R^{\prime}$. So
$AF=AF^{\prime}$.

Finally, from Proposition 3 as $S^{\chi}(t)$ is argumentative, $\forall x\in
A=A^{\prime},S^{\chi}(t)\models a_{x}\Leftrightarrow x$ is acceptable by $A$.
∎

###  5.3 On Temporality and Causality

The preliminary Proposition 1 highlights the fact that temporality is captured
by the proposed transformation. Indeed, given an order of enunciation, as
expressed by sequence $\varsigma$, there exists a unique trace of states
corresponding to a unique way to traverse a graph. When only given a context
$\kappa$, this unicity property does not hold. This section shows that this
temporality does not impact the final argumentative state but impacts the
causal relations.

######  Proposition 4.

Let $\varsigma$ and $\varsigma^{\prime}$ be sequences such that
$\varsigma^{\prime}$ is a permutation of the ranks of $\varsigma$. Given the
final argumentative states $S^{\varsigma,\kappa}(t)$,
$S^{\varsigma^{\prime},\kappa}(t^{\prime})$, belonging to
$\tau_{\varsigma,\kappa}^{s}$ and $\tau_{\varsigma^{\prime},\kappa}^{s}$, with
$(t,t^{\prime})\in\mathbb{T}\times\mathbb{T^{\prime}}$,
$S^{\varsigma,\kappa}(t)=S^{\varsigma^{\prime},\kappa}(t^{\prime})$.

###### Proof.

Let us call $AF$ and $AF^{\prime}$, the associated graphs of the final
argumentative states $S^{\varsigma,\kappa}(t)$ and
$S^{\varsigma^{\prime},\kappa}(t^{\prime})$, respectively. Given that they
have the same actions in the sequence, then $A=A^{\prime}$. They also share
the same context so $R=R^{\prime}$. Therefore $AF=AF^{\prime}$.

Now, according to Proposition 3, $x\in A$ acceptable by $A\Leftrightarrow
S^{\varsigma,\kappa}(t)\models a_{x}$. So $\forall
x,S^{\varsigma,\kappa}(t)\models a_{x}\Leftrightarrow\forall
x,S^{\varsigma^{\prime},\kappa}(t^{\prime})\models a_{x}$. ∎

This property implies that the final argumentative state does not depend on
$\varsigma$, but only on the set of arguments it contains: no matter the order
in which arguments are enunciated, the final argumentative state is always the
same. This immediately leads to the following unicity corollary:

######  Corollary 1.

Given an ${AF=(A,R)}$, $\exists!S^{\chi}(t)$ final argumentative state which
associated argumentative graph is ${AF=(A,R)}$.

Proposition 4 and its corollary are in accordance with AAF. The relevance of
temporality integration comes from the intermediate states, as illustrated in
the next section, and from the causal relations that can be derived from it:

######  Proposition 5.

Causal relations depend on the sequence $\varsigma$.

###### Proof.

This proposition is proved by example, commented in details in the next
section that illustrated the effect of considering Example 1, and a
modification thereof in Example 2. Let $\Pi(\chi)$ be the program obtained
given $\kappa,\varsigma$ of Example 1, as described in Section 4.3, and let
$\Pi(\chi^{\prime})$ be the program similarly obtained given
$\kappa,\varsigma^{\prime}$ of Example 2. Given the NESS-cause definition in
(?), ${\Pi(\chi)\models ness(o(enunciate_{d},4),h(neg(a_{c}),31))}$, where
occurrence of events $(e,t)\in\mathbb{E}\times\mathbb{T}$ are represented by
the predicate $o(e,t)$ and the truthfulness of $\mathcal{P}$ formulas
$(\psi,t)\in\mathbb{F}\times\mathbb{T}$ by the predicate $h(\psi,t)$, but
$\nexists t,t^{\prime}\in\mathbb{T}^{2}$, ${\Pi(\chi^{\prime})\models
ness(o(enunciate_{d},t),h(neg(a_{c}),t^{\prime}))}$. ∎

This proposition shows that causal relations depend on the order in which
arguments are enunciated. Thus, even if, as in the classical framework of
argumentation, the acceptability of an argument in the final argumentative
state does not depend on it (cf Theorem 1), it is still essential to take
temporality into account when dealing with notions close to causality,
especially in explicability.

##  6 Application to the Example and Discussion

This section proposes to illustrate the proposed transformation from AAF to
action language for Example 1, highlighting its exploitation to get enriched
information about the modelled dialogue, more precisely for providing visual
representations justifying the acceptance or rejection of arguments. It
considers successively two classes of argumentation explanations in (?)’s
taxonomy: it first shows how it can lead to graphical representations of the
processes of accepting/rejecting arguments, it then discusses the case of
causal explanations.

###  6.1 Graphical Representation and Explanation

According to (?), argumentation explanations can consist in extracting
argumentative subgraphs to justify the acceptance or rejection of an argument
for a given AAF semantics, producing a graphical representation of the
underlying process.

The transformation proposed in the previous section makes it possible to
derive graphical representations of the argumentative process. Indeed, the
traces of events and states can be used to obtain a narrative of the
interaction that can be represented graphically. The visualisation we propose
is illustrated in Figure 2, in a simplified form, for Example 1. It is
enriched in the next section using causality relations.

$t=6$$enu_{e}$$una_{e,d}$$t=7$$t=8$$t=9$$b$$b$$b$$\mathbf{c}$$e$$e$$d$$e$$d$$acc_{c}$$enu_{f}$$d$$d$$b$

Figure 2: Partial graphical representation associated to Example 1. Hexagons represent fluents and rectangles events.  | $a$ | $b$ | $c$ | $d$ | $e$ | $f$ | $g$ |  $h$,$i$ | $j$ | $k$ | $l$ | $m$ | $n$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$a$ | $\bullet$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$
$b$ |  | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
$c$ |  |  | $\bullet$ | $\circ$ | $\bullet$ | $\circ$ | $\bullet$ | $\circ$ | $\circ$ | $\bullet$ | $\bullet$ | $\bullet$ | $\circ$
$d$ |  |  |  | $\bullet$ | $\circ$ | $\bullet$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$
$e$ |  |  |  |  | $\bullet$ | $\circ$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
$f$ |  |  |  |  |  | $\bullet$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$
$g$ |  |  |  |  |  |  | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
$h$ |  |  |  |  |  |  |  | $\bullet$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$
$i$ |  |  |  |  |  |  |  | $\bullet$ | $\bullet$ | $\circ$ | $\circ$ | $\circ$ | $\circ$
$j$ |  |  |  |  |  |  |  |  | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
$k$ |  |  |  |  |  |  |  |  |  | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
$l$ |  |  |  |  |  |  |  |  |  |  | $\bullet$ | $\circ$ | $\bullet$
$m$ |  |  |  |  |  |  |  |  |  |  |  | $\bullet$ | $\circ$
$n$ |  |  |  |  |  |  |  |  |  |  |  |  | $\bullet$
Table 1: Tabular representation of the entire interaction.

Given an event and a state traces $\tau_{\chi}^{e}$ and $\tau_{\chi}^{s}$, we
propose to display the consecutive states, showing fluents as hexagons and the
triggered events as rectangles. Since the acceptability of arguments is what
mainly matters, we propose to represent only the fluents $a_{x}$, using the
argument names for the sake of readability. Moreover, we do not show fluents
when their negation is true in the state, except when the occurrence of a
represented event results in the negation of the fluent. In this case, the
negation is represented by a lighter shade. The events $enunciate_{x}$,
$makesUnacc_{y,x}$, and $makesAcc_{x}$ are shortened as $enu_{x}$,
$una_{y,x}$, and $acc_{x}$, respectively.

######  Example 1.

(continued) – Figure 2 shows a partial representation of the state trace
obtained for Example 1 using the ASP implementation described in Section 4.3.

The first represented state corresponds to $S(6)$, an argumentative state in
the sense of Def. 7, which allows for the enunciation of the next argument:
since all arguments preceding $e$ have already been enunciated, the action
$enunciate_{e}$ can be performed. The occurrence of this event is the
transition to the next state $S(7)$ where, as shown in Figure 2, argument $e$
is acceptable. Unlike $S(6)$, $S(7)$ is not an argumentative state: condition
(i) of Def. 7 is not satisfied because ${(a_{d}\wedge cA_{e,d})}$ and
${a_{e}\in S(7)}$. Therefore, the next argument cannot be enunciated. However,
since the triggering conditions of $makesUnacc_{e,d}$ are satisfied, this
exogenous event is triggered, leading to a new state transition. Since
argument $d$ is no longer acceptable in $S(8)$, condition (i) of Def. 7 is now
satisfied. Still, condition (ii) is not satisfied by $S(8)$, preventing the
next argument from being enunciated. Instead, $makesAcc_{c}$ is triggered,
leading to the following state $S(9)$. Here, as shown in Figure 2, argument
$c$ is acceptable. As this new state is argumentative, the next argument, $f$,
can be stated. The dialogue continues step by step and ends at state $S(31)$.

A second, more compact, tabular visualisation is proposed, illustrated in
Table 1: the arguments are represented in the first column, the order of the
performed actions in the first row. For the sake of readability,
$enunciate_{x}$ is shortened as $x$. In each table cell, $\bullet$ means that
the argument is acceptable while $\circ$ means that it is not. If an argument
has not been enunciated yet, its acceptability cannot be evaluated, which is
represented by the shaded boxes. In contrast to the previous representation
where the updating stages are shown, this second form has the advantage of
being more compact and allows the display the whole dialogue. It also makes it
possible to see quickly the direct and indirect impacts of the argument
enunciation on the other argument acceptability. In particular, the
enunciation order effect can be observed, as illustrated by the graphical
comparison of Example 1 and its modification given in Example 2.

######  Example 2.

Let us consider the same dialogue as in Example 1, starting with the
enunciation of arguments $a,b,c$, but considering that the physician then
directly asks if it is possible to do the MRI today ($l$). The radiologist
replies that he can only do it in two days at the earliest ($m$). The
physician then specifies that it is an emergency ($n$). The remaining
arguments are then enunciated in the same order ad in the initial example.

Table 2 displays the proposed compact visualisation for the evolution of the
acceptability of the decision variable $c$, starting from its enunciation.
Even if the final state of the argumentation graph is identical, as expected
according to Proposition 4 established in the previous section, with $c$ being
rejected, the display makes it easy to observe the very important impact that
the order of the actions can have on the intermediate stages that lead to it:
in the new scenario, $c$ is not accepted from the $6^{th}$ action, i.e. $n$
enunciation, with no modification until the end.

This visualisation thus also illustrates the relevance of the temporality
integration in the argumentation framework: the differences between the two
scenarios cannot be captured by classical AAFs.

$\varsigma_{1}$ | c | d | e | f | g | h,i | j | k | l | m | n
---|---|---|---|---|---|---|---|---|---|---|---
c | $\bullet$ | $\circ$ | $\bullet$ | $\circ$ | $\bullet$ | $\circ$ | $\circ$ | $\bullet$ | $\bullet$ | $\bullet$ | $\circ$
$\varsigma_{2}$ | c | l | m | n | d | e | f | g | h,i | j | k
c | $\bullet$ | $\bullet$ | $\bullet$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$
Table 2: Impact of the order in which arguments are enunciated (lines 1, 3) on
the acceptability of the arguments (lines 2, 4).

###  6.2 On Causality and Explanation

Beyond the graphical representation of the acceptance/rejection process, the
proposed formalisation of AAF into action models provides tools for richer
explanations, allowing us to transfer the notion of actual causality recalled
in Section 3.2 to the argumentation framework. Indeed, the extraction of
causal chains has been shown to be an important property for explanations (?).
In the taxonomy proposed in the case of argumentation in (?), such a causal
explanation can be related to the identification of arguments that must be
removed from an argumentation graph to make a non-acceptable argument
acceptable (?). In causal terminology, this corresponds to the search for a
but-for cause of the non-acceptability of an argument.

However, this test does not solve cases where the occurrence of one of two
events would have been sufficient to cause an effect in the absence of the
other, called over-determination (?). Among others, the definition of
causality underlying the NESS test, as briefly recalled in Section 3.2 and
implemented for the considered action language, makes it possible to solve
this issue.

Structural equations (?) constitute another formal model of causality that
addresses the over-determination issue and can be exploited in the
argumentation framework, using the transformation of acyclic abstract
argumentation graphs to that formalism proposed in (?). The main differences
are as follows: from a philosophical point of view, the definition of
causality underlying the NESS test belongs to the family of regularity
approaches (?), whereas Halpern’s definitions belong to the family of
counterfactual approaches (?). Secondly, the use of action languages makes it
possible to model and take into account temporality and the dynamics of the
dialogue, which is a crucial component. From a mathematical point of view,
according to (?), Halpern’s definition of causality can be described as
‘Contrastive actual weak sufficiency’, whereas the one used here would be
‘Minimal actual strong sufficiency’: in a nutshell, whereas the former
emphasises that a cause must be necessary for an effect, hence the contrastive
aspect, the latter emphasises sufficiency and subordinates necessity to it.
From a practical point of view, the advantage of the causal approach used here
is that it does not require counterfactual reasoning or interventionism,
mechanisms that are computationally onerous and criticised for introducing
subjectivity into causal enquiry (?; ?).

These causal relations, that may lead later to causal explanations, can be
represented graphically, enriching the proposed visualisation illustrated in
Figure 2 by different types of causes. This principle is illustrated in Figure
3, commented below.

$t=28$$enu_{n}$$una_{n,c}$$t=29$$t=30$$t=31$$b$$b$$b$$acc_{l}$$\mathbf{c}$$\mathbf{c}$$e$$g$$j$$k$$m$$\mathbf{c}$$e$$g$$j$$k$$m$$n$$e$$g$$j$$k$$n$$e$$g$$j$$k$$\mathbf{l}$$n$$una_{n,m}$$\mathbf{c}$$m$$m$$b$

Figure 3: Enriched graphical representation of Example 1, with causal
relations extracts: Direct NESS-causes ( ), NESS-causes ( ), and actual causes
( ).

######  Example 1.

(continued) – Figure 3 graphically displays the last four states in the trace
of Example 1, corresponding to the enunciation of argument $n$ and the
subsequent update mechanisms. Argument $n$, which states the urgency of the
examination, is the one that closes the debate. As represented in Figure 3 its
enunciation in state $S(28)$ is a direct NESS-cause (dNc) of its acceptability
in the following states, a relation we denote by $(enunciate_{n},28)$ dNc
$(a_{n},29-31)$. Similarly, we have $(makesUnacc_{n,c},29)$ dNc $(\neg
a_{c},{30-31})$, $(makesUnacc_{n,m},29)$ dNc ${(\neg a_{m},30-31)}$, and
$(makesAcc_{l},30)$ dNc $(a_{l},31)$. As these examples show, this first
relationship is the basic building block of causality, which is concerned with
causal relationships given the actual effects of the occurrence of an event.
Yet this relationship is not enough. If we want to know why argument $l$ is
acceptable at the end of the dialogue (i.e. why the decision to have an MRI on
the same day is made), it is not satisfactory to simply say that it is because
of the event $(makesAcc_{l},30)$.

To find out why the latter happens, we need to look at the NESS causes and the
actual causes to construct the causal chain that lead to it. By transitivity
we get that $(makesUnacc_{n,m},29)$ is a cause of the fact that $makesAcc_{l}$
was triggered, and therefore of the effects that triggering may have had.
Going back even further and looking for the causes for which the occurrence
$(makesUnacc_{n,m},29)$ took place, we find $(enunciate_{n},28)$ actual cause
$(makesUnacc_{n,m},29)$ and therefore $(enunciate_{n},28)$ NESS-cause $(\neg
a_{m},30-31)$. By transitivity we can derive $(enunciate_{n},28)$ NESS-cause
$(a_{l},31)$. This new relation allows us to say that the physician
enunciating that it is an emergency is one of the causes of the final
decision, an answer that already seems more satisfactory and can be included
in an explanation. The same reasoning can be applied to find the causes of
$(\neg a_{c},31)$, the other decision variable.

##  7 Conclusion

This paper has proposed a formalisation of acyclic abstract argumentation
systems in the action language of (?), establishing its formal properties: it
first allows increasing the expressiveness of these models, through the
integration of temporality, making it possible to examine the effect of the
order of the argument enunciation. Moreover, it allows us to exploit the
notion of causality associated to the action language, offering the
possibility to give rich information about the argument acceptance or
rejection and justifications about the latter. The paper has proposed two
types of graphical representations of the argumentation process that can be
used as visual support, opening the way for new forms of argumentation
explanations.

Future works will aim at developing such explanations, applying the principles
developed in the context of eXplainable Artificial Intelligence (XAI), e.g.
detailed in (?): causal chains are established as essential for explanations,
but they must also be short. The question of which relations to emphasise
remains open, as well as the way in which they can be used to define
contrastive explanations, requiring to be able to reason about counterfactual
scenarios.

#### Acknowledgements

The authors would like to thank Professor Catherine Adamsbaum, pediatric
radiologist, for the enlightening discussions about the examples. This work
was partly supported by the 3rd author’s chair in Artificial Intelligence
(Sorbonne Université and SCAI).

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