Integrating Temporality and Causality into Acyclic Argumentation Frameworks Using a Transition System

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 on the acceptability of other arguments, whether direct or indirect. An Answer Set Programming implementation is also described. Furthermore, we lay the ground for the generation of explanations using the causal relations as well as two graphical representations to support such explanations.

1. Introduction

The abstract argumentation framework (AAF), first introduced by Dung (1995), provides a framework for representing and reasoning about contradicting information, using arguments and a binary relation between them called attack relationship. It makes it possible to find sets of arguments, called extensions, that can be accepted together, and then to give explanations on the reasons why such sets are accepted or not (Čyras et al., 2021). Now, if one wishes to model dialogs, i.e. temporally ordered arguments, and provide both on-the-spot explanations and human-centered explanations, it is essential to be able to keep track of the order in which arguments are enunciated and thus to include a notion of temporality. Indeed, this is very important to establish the causal relations in the dialog which constitute a mandatory element for building explanations (Miller, 2019). However, this is not possible with the classic version of AAF as it is a static framework. In this paper, we address the issue of modeling an entire dialog, including the order of enunciation, and then deriving the causal relationships that lead to the specific acceptability status of each argument. Note that we observe the dialog and do not take part in it. Thus, in contrast to argumentation based dialog systems (Black et al., 2021), we adopt a descriptive approach to model and reason on an AAF.

This work constitutes a first 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 considers the case of acyclic argumentation graphs. We propose to investigate the use of action description languages (ADL) (Fox & Long, 2006; Giunchiglia & Lifschitz, 1998) to solve these issues. This choice is motivated by the fact that the ADL formalism offers tools to reason about action and change, and has been conceived to include the notion of time. Indeed, it is a classical transition system: The set of events that occur at time point $t$ , denoted $E(t)$ , generates the transition from state $S(t)$ to state $S(t+1)$ . Thus, the states follow one another as events occur, simulating the evolution of the world. In order to have a complete knowledge of the evolution of this world, two traces are generated: One tracing the evolution of world states, called state trace, the other tracing the occurrence of events, the action trace. Moreover, the ADL introduced by Sarmiento et al. (2022) includes a formal study of causality: It is thus appropriate both for modeling the dynamics of a dialog and for generating explanations.

Starting from a dialog, we propose the modular framework represented in Figure 1 to reach the explanation. The process is divided into three building blocks: The action language, the causal model and the explanation model. First, the dialog is modeled using an ADL adapted for argumentation purposes. From this ADL block, the state trace and the action trace are generated. Then, from these two traces, a causal trace is generated through the causal model block using a certain definition of actual causality. Finally, thanks to the three traces and given an explanandum, i.e. here the arguments whose acceptability status is to be explained, an explanation as well as two graphical representations are produced.

Figure 1.

Modular framework for explaining temporal Abstract Argumentation Framework. The shaded boxes represent what is inferred from the dialog.

The contributions of this paper are located in the Action Language and the Explanation Model blocks. First of all, we propose a modification of the context and the semantics to translate an AAF and the admissible semantics into the chosen ADL. Its corresponding implementation in ASP is also given. Secondly, we establish the formal properties of this transformation, including its soundness and completeness, as well as the relevance of the temporality inclusion. Thirdly, we propose two different graphical representations of the dialog to improve the understanding of the argumentative process as well as a discussion on how to generate explanations. Note that this paper is an extended version of an earlier paper presented at a national conference (Munro et al., 2023).

The paper is structured as follows. Section 2 presents related works regarding the inclusion of temporality and causality in abstract argumentation. Section 3 briefly recalls the principles of abstract argumentation and describes the chosen action language and the actual causality definition suitable for it (Causal Model in Figure 1). Section 4 describes the Action Language part of Figure 1, i.e. the formalization we propose of acyclic abstract argumentation graphs into an action language, with its corresponding implementation in ASP. Section 5 establishes the formal properties of this transformation. Section 6 illustrates the exploitation of the proposed formalization to get enriched information, as graphical representations and causal relations, and discusses how to generate explanations using these relations (Explanation Model in Figure 1). Section 7 concludes the paper.

2. Related Work

The main objective of this work is the generation of explanations about the acceptability status of arguments in a dialog. Even if there is no single definition of an explanation, it is generally accepted from a social science point of view that an explanation is an answer to a why question, as reported by Miller (2019). Therefore, the notion of explanation is deeply related to the notion of causality. For that reason, this work focuses on two aspects that have been studied separately within the field of abstract argumentation: First, modeling the order of enunciation of the arguments in a dialog, and secondly, inferring the causal relationships between the arguments.

2.1. Modeling Temporality in AAF

Several extended argumentation frameworks have been proposed to include a notion of time in an AAF.

Barringer et al. (2005) propose different scales to include temporality in abstract argumentation. Among them, they suggest, without formalizing it, to include the action of adding arguments. In our paper, a possible formalization of this idea is proposed, based on an ADL. It gives both an “object level” (with the traces) and “meta level” (with the argumentative states) description of the temporality in the argumentation framework.

The YALLA logical formalism Dupin de Saint-Cyr et al. (2016) proposes to use revision or belief change operators to update the model of an argumentation system. It offers a highly expressive language that allows identifying which attack relations or arguments should be removed or added to achieve a given acceptability status in the subsequent time step. Such a reasoning can be used to build an explanation. However, the number of terms manipulated by YALLA increases exponentially with the number of arguments, which is not suited to model the whole dialog.

The approach by Doutre et al. (2017) proposes to use a dynamical propositional language to model the argumentation graph and its evolution across the dialog. By doing so, it partially answers the question we consider as it successfully integrates the notion of temporality. However, it does not include the notion of causality, a necessary notion to generate explanations.

Finally, Kampik et al. (2024) propose to create an argumentation graph at each time step in the context of quantitative bipolar framework (QBAF). With this sequence of QBAF, different types of explanations are generated to explain the changes between two graphs. In contrast to an AAF, in a QBAF the acceptability of an argument is a numerical value computed using an acceptability function. For that reason, a first difference with our approach lies in the nature of what is explained. Secondly, the main objective of Kampik et al. (2024) is to explain the changes between two graphs and not necessarily to model the temporality of a dialog. Finally, in contrast to this work, our approach is based on an ADL. Therefore, it provides the causal foundations necessary for the explanation process.

2.2. Causality in AAF

Bochman (2005) establishes an equivalence between abstract argumentation and a causal reasoning system. This system has later been extended to more complex causal reasoning (Bochman, 2021). However, these works apply to classical AAFs without temporality which is a central notion in our work.

Recently, Bengel et al. (2022) used a knowledge-based formalism to define counterfactual reasoning for structured argumentation. Their definition is the direct application of Pearl’s definition for causal model (Pearl, 2000), similar to what is called a but-for test. However, first we are dealing with abstract argumentation. Secondly, the definition of causality based on but-for test is not always adapted (see Section 3.2.2 for more details).

2.3. Argumentative eXplainable Artificial Intelligence

There exist numerous work on explaining the acceptability status of arguments, which is called argumentative eXplainable Artificial Intelligence, see e.g. Čyras et al. (2021) and Vassiliades et al. (2021). However, to the best of our knowledge, they are not grounded in a framework that includes a notion of time. For example, Borg and Bex (2022, 2024) study different forms of explanation related to causality, such as a definition of contrastive explanation. The latter is based on the definition of acceptance and non-acceptance explanation (Borg & Bex, 2021), which groups all arguments that play a role in the acceptability status of an argument. Such a set is similar to the causal chain, a key element of an explanation (see section 6.2 for more details). Nevertheless, their work is based on AAFs or structured argumentation frameworks that do not include a notion of the order of enunciation.

3. Preliminaries

This section recalls the basics of Dung’s AAF (Dung, 1995) and illustrates them with an example. It then recalls the formal aspects of the Action Language proposed in Sarmiento et al. (2022) that is used in this paper.

3.1. Abstract Argumentation Framework

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 in $A\times A$ . An argument $x\in A$ attacks argument $y\in A$ if $(x,y)\in R$ .

As $R$ is a binary relation with finite support, an AAF can be represented using a graph where $A$ is the set of vertices and $R$ defines the edges.

Example 1

Throughout this paper, we use the following fictitious illustrative example modeling the interaction between a requesting physician, P, and a radiologist, R, concerning an examination of a young baby for pathology Z.

P: 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. ( $c$ )

P: Can Z be seen on an MRI? ( $d$ )

R: Yes, of course. If you want confirmation, look at the guide to good radiology practice. ( $e$ )

P: 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. ( $g$ )

P: Does not it cost the hospital a lot more to do an MRI rather than a CT? ( $h$ ) 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 type of delicate examination can be performed without me. ( $j$ )

P: I have just spoken to the family, no problem with the MRI, the exam is refunded. ( $k$ )

P: However, the family is not comfortable with the idea of having to wait two days, could not you do the exam earlier? ( $l$ )

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 physician receives a call from the family saying that the baby is really not well and insisting on the emergency of the examination. Therefore, the physician contacts the radiologist to add a final argument.

P: It is very urgent for the baby, we need a place today. ( $n$ )

From this dialog, arguments and their relations are extracted to create the AAF represented in Figure 2 with the following arguments: $\{\mathbf{\text{a}}$ : CT Scanner, $b$ : Ionising radiation, $\mathbf{\text{c}}$ : 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, $\mathbf{\text{l}}$ : 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. Even if these three arguments have a special role compared to the others, they are abstract entities like every other argument in the graph. For that reason, we do not add any formal restrictions on them, in contrast to what can be done in settings such as argument-based negotiation (Amgoud et al., 2007).

Figure 2.

Argumentation graph associated with Example 1.

The obtained argumentation system shown in Figure 2 is a possible graph that can be extracted from this dialog. Here the extraction process has been done manually to create this AAF. However, this process can be done automatically using so-called argument mining methods (Lippi & Torroni, 2016).

Note that the graph is a static representation of the dialog from which any notion of temporality has been erased. Thus, if the arguments had been stated in a different order, the graph would be the same. This is 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 $At{t}_x=\{y\in A\mid (y,x)\in R\}$ . A set $S$ is conflict-free if $\mathrm{\forall }(x,y)\in S^2$ , $(x,y)\notin R$ . An argument $x\in A$ is acceptable by $S$ if $\mathrm{\forall }y\in At{t}_x,\mathrm{\exists }z\in S\cap At{t}_y$ .

Then an admissible set $S$ is defined as a conflict-free set whose elements are all acceptable by $S$ itself. For acyclic graphs, most of the commonly used semantics coincide with the maximum admissible sets therefore they are not discussed here (Rahwan & Simari, 2009). Indeed, in the paper, the argumentation graph is supposed to be acyclic.

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, contrary to $a$ and $c$ . Therefore, the final decision is to perform an emergency MRI today.

3.2. Action Language and Causality

In this paper, we propose to investigate the use of a transition system to include the order in which arguments are enunciated in a dialog formalization. We do not consider a transition system as a language for reasoning about action and change, but rather as a general way of adapting to a dynamic environment. Thus, in practice, we choose an action language as a formalism to include a notion of temporality in an argumentation graph. In particular, we select the action language developed in Sarmiento et al. (2022) due to its adapted tools for reasoning about causality, which is the first step towards explanations. This section introduces the formal aspects of the action language proposed in Sarmiento et al. (2022) and then briefly describes what is considered to be an actual cause in this formalism. For more details refer to Sarmiento et al. (2023).

3.2.1. Syntax and Semantics

Transition systems generally define dynamical systems, describing the state of the world at any time step as a collection of variables that describe the properties of the world. Furthermore, they assert that the evolution of the world can be described as a series of states transitioning from one to another as a result of a set of events. An action language is a formalism for describing such a transition system (Gelfond & Lifschitz, 1998). The purpose of the action language introduced in Sarmiento et al. (2022) is to describe the evolution of the world given a set of actions corresponding to the 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, the method in Sarmiento et al. (2022) keeps track of both: The evolution of the states of the world and the occurrence of events.

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 $\mathrm{\neg }f$ . The set of fluent literals in $\mathbb{F}$ is denoted by $Li{t}_{\mathbb{F}}$ , i.e. $Li{t}_{\mathbb{F}}=\mathbb{F}\cup \{\mathrm{\neg }f\mid f\in \mathbb{F}\}$ . The complement of a fluent literal $l$ is defined as $\overline{l}=\mathrm{\neg }f$ if $l=f$ or $\overline{l}=f$ if $l=\mathrm{\neg }f$ .

Definition 2 (state)

A set $L\subseteq Li{t}_{\mathbb{F}}$ is a state if it is:

Coherent: $\mathrm{\forall }l\in L,\overline{l}\notin L$ ;

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

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

An event $e\in \mathbb{E}$ is an atomic formula. When studying causality, it is important to understand why an event $e$ can occur and what are its consequences. Therefore, three elements are used to characterize such an event: Preconditions and triggering conditions give conditions that must be satisfied by a state $S$ for event $e$ to be triggered (their difference is detailed below); effects indicate the changes to the state that are expected if event $e$ occurs. Note the deliberate use of the term “expected” as an event may have fewer effects than those formalized.

The preconditions and triggering conditions on the one hand and the effects on the other hand are represented as formulas of the languages $\mathcal{P}::=l|{\psi }_1\wedge {\psi }_2|{\psi }_1\vee {\psi }_2$ and $\mathcal{E}::=l|{\phi }_1\wedge {\phi }_2$ , respectively. The functions that associate preconditions, triggering conditions and effects with each event are respectively defined as: $pre:\mathbb{E}\to \mathcal{P}$ , $tri:\mathbb{E}\to \mathcal{P}$ , and $e\mathit{\text{ff}}:\mathbb{E}\to \mathcal{E}$ . $\mathbb{E}$ is partitioned into two disjoint sets: The set $\mathbb{A}$ contains the actions carried out by an agent and thus subjected to their volition; the set $\mathbb{U}$ 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 that occur at time point $t$ is denoted by $E(t)$ . Allowing for 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 (Sarmiento et al., 2022).

With a bounded past formalization, events that occurred before $t=0$ must be represented in order to obtain causal relations that are consistent with the philosophical conception of causality. Indeed, it might happen that one of the reasons why a formula of $\mathcal{P}$ is true is that one of its fluent literals was true in the initial state and remains so. Thus, for each fluent literal $l\in S(0)$ an event $in{i}_l\in \mathbb{E}$ is introduced, such that $e\mathit{\text{ff}}(in{i}_l)=l$ . Then, $E(-1)=\{in{i}_l,l\in S(0)\}$ which satisfies $e\mathit{\text{ff}}(E(-1))=S(0)$ .

As the considered action language allows for the concurrency of events, it requires tools to solve potential conflicts or to prioritise among events: This is achieved by introducing a strict partial order ${\succ }_{\mathbb{E}}$ to ensure the triggering primacy of one event over another.

Definition 3 (context $\kappa$ )

The context, denoted as $\kappa$ , is the octuple $(\mathbb{E},\mathbb{F},pre,tri,e\mathit{\text{ff}},S(0),{\succ }_{\mathbb{E}},\mathbb{T})$ , where $\mathbb{E}$ , $\mathbb{F}$ , $pre$ , $tri$ , $e\mathit{\text{ff}}$ , $S(0)$ , ${\succ }_{\mathbb{E}}$ , and $\mathbb{T}$ are as defined above.

As mentioned previously, when considering causality, it is important to understand what are the consequences of the occurrence of an event $e$ . The predicate $actualE\mathit{\text{ff}}$ keeps track of basic causal information: The direct effects an event has at the time it occurs.

Definition 4 (actual effects)

Given a context $\kappa$ , a partial state $L$ and a set of events $E$ , the predicate $actualE\mathit{\text{ff}}(E,L)$ is defined as:

$$actualE\mathit{\text{ff}}(E,L)=\bigcup _{e\in E}actualE\mathit{\text{ff}}(\left\{e\right\},L)=\{l_i\mid \mathrm{\exists }e\in E,l_i\in e\mathit{\text{ff}}(e),\text{and}{l}_i\notin L\}$$

Given a partial state $L$ , this predicate represents the actual effects of a set of events $E$ if occurring in this state $L$ . Note that here we use the term actual effect, and not expected, as this predicate returns the set of all the changes that really happen.

For the sake of conciseness, we adopt an update operator giving the resulting state when an event occurs at a given state. Given a state $S(t)$ and a set of events $E\in \mathbb{E}$ , the resulting state is obtained by adding the actual effects of $E$ in $S(t)$ , having first removed its complement to keep the state coherence.

Definition 5 (update operator $▹$ )

Given a context $\kappa$ , a state $S(t)$ and a set of events $E\in \mathbb{E}$ , the update operator denoted by $▹$ defines a new state $S(t)▹E=S(t)\setminus \overline{actualE\mathit{\text{ff}}(E,S(t))}\cup actualE\mathit{\text{ff}}(E,S(t))$ .

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

Definition 6 (valid execution)

An execution is a sequence alternating sets of events and states at each time stamp $E(-1),S(0),E(0),\dots ,E(N),S(N+1)$ .

An execution is valid given $\kappa$ , if $\mathrm{\forall }t\in \mathbb{T}$ :

(1)

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

(2)

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

• 2.a

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

2.b

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

2.c

$\mathrm{\forall }e\in \mathbb{E}$ such that $S(t)\models tri(e)$ , $e\in E(t)$ or $\mathrm{\exists }{e}^{\prime }\in E(t),$ $e^{\prime }{\succ }_{\mathbb{E}}e$ .

(3)

$S(t+1)=(S(t)▹E(t)).$

There is potentially more than one valid execution for a given context $\kappa$ . Indeed, $\kappa$ does not specify which deliberate events, i.e. the so-called actions, are performed. Therefore, a set of timed actions $\sigma \subseteq \mathbb{A}\times \mathbb{T}$ called scenario is added to model the agent volition. For example, considering an illustrative case where the world is composed of an agent, a switch and a light bulb. A time should be specified for when the agent closes the electrical switch, i.e. performs an action. However, it is not necessary to specify when the light bulb is switched on, i.e. when an exogenous event is triggered, as this happens automatically when the switch is closed, i.e. when the triggering conditions are satisfied. This leads to a unique valid execution. From this unique execution the event trace and state trace, denoted by ${\tau }_{\sigma ,\kappa }^e$ and ${\tau }_{\sigma ,\kappa }^s$ , respectively, can be extracted.

Definition 7 (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:

$\mathrm{\forall }t\in \mathbb{T},\mathrm{\forall }e\in E(t),e\in \mathbb{A}\iff (e,t)\in \sigma$ .

The state trace ${\tau }_{\sigma ,\kappa }^s$ is the sequence of states $S(0),S(1),\dots ,S(N+1)$ corresponding to ${\tau }_{\sigma ,\kappa }^e$ .

3.2.2. Actual Causality

In the taxonomy proposed in the case of argumentation in Čyras et al. (2021), explaining the acceptability status of an argument can be related to the identification of arguments that must be removed from an argumentation graph to make a non-acceptable argument acceptable (Fan & Toni, 2015). 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 (Menzies & Beebee, 2020). To address this issue, the Necessary Element for the Sufficiency of a Set test, NESS-test, has been proposed by Wright in Wright (1985) leading to a different definition of actual causality. 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.” In this section, we recall the actual causation definition proposed in Sarmiento et al. (2022) which is a suitable formalization in an action language of Wright’s NESS test.

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 (Andreas & Guenther, 2021; Menzies & Beebee, 2020), 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. In order to give an actual causality definition suitable for action languages, three causal relations are introduced in Sarmiento et al. (2022) on the basis of Wright’s NESS test of causation:

(i) Direct NESS-causes are the first building block to find long distance causal relations. Indeed, they look at the effects that the occurrence of an event actually had, which are not necessarily the same as the expected ones. 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 time $t_{\psi }$ 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.

The formal definition of the three different causal relations can be found in Sarmiento et al. (2023). In practice, depending on the form of the formula we are interested in, it is possible to determine its direct NESS-causes with simple conditions. The propositions giving these conditions are enough to understand the cases of causality in an argumentative context. The proofs for these two propositions can be found in Sarmiento et al. (2023).

In the case where $\psi$ is a fluent literal $l$ , the direct NESS-causes are the last occurrences of events that made $l$ true before or at $t_{\psi }$ .

Proposition 1 (direct NESS-cause of a literal Sarmiento et al., 2023)

Given a couple $\chi =(\sigma ,\kappa )$ called causal setting, times $t,t_l\in \mathbb{T}$ , $e\in E^{\chi }(t)$ , and a fluent literal $l\in Li{t}_{\mathbb{F}}$ true at time $t_l$ , $(e,t)$ is a direct NESS-cause of $(l,t_l)$ iff

( $c_1$ ) $(S^{\chi }(t)▹\{e\})\models l$ ;

( $c_2$ ) $\mathrm{\forall }{t}^{\prime },t<t^{\prime }\le t_l$ , $S^{\chi }(t^{\prime })\models l$ ;

( $c_3$ ) $S^{\chi }(t)⊭l$ .

In the case where $\psi$ is a conjunction $\psi =l_1\wedge \cdots \wedge l_m$ of fluent literals, the direct NESS-causes are all the occurrences of events that are direct NESS-causes of the truthfulness of one of the literals $l_j$ in the conjunction at $t_{\psi }$ .

Proposition 2 (direct NESS-cause of a literal conjunction Sarmiento et al., 2023)

Given a causal setting $\chi =(\sigma ,\kappa )$ , times $t,t_l\in \mathbb{T}$ , $e\in E^{\chi }(t)$ , and a formula $\psi =l_1\wedge \cdots \wedge l_m$ of $\mathcal{P}$ true at $t_{\psi }$ , $(e,t)$ is a direct NESS-cause of $(\psi ,t_{\psi })$ iff $\mathrm{\exists }j\in \{1,\dots ,m\}$ ,

( $c_1$ ) $(S^{\chi }(t)▹\{e\})\models l_j$ ;

( $c_2$ ) $\mathrm{\forall }{t}^{\prime },t<t^{\prime }\le t_{\psi }$ , $S^{\chi }(t^{\prime })\models l_j$ ;

( $c_3$ ) $S^{\chi }(t)⊭l_j$ .

Structural equations Halpern and Pearl (2005) constitute another formal model of causality that addresses the over-determination issue and can be exploited in the argumentation framework, e.g. using the transformation of acyclic abstract argumentation graphs to that formalism, as proposed in Munro et al. (2022). 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 (Andreas & Guenther, 2021), whereas Halpern’s definitions (Halpern & Pearl, 2005) belong to the family of counterfactual approaches (Menzies & Beebee, 2020). Secondly, the use of action languages makes it possible to model and to take into account temporality and the dynamics of the dialog, which is a crucial component. From a mathematical point of view, according to Beckers (2021), Halpern’s definition of causality can be described as “contrastive actual weak sufficiency,” whereas the one used in this paper would be “minimal actual strong sufficiency”: In a nutshell, whereas the former emphasizes that a cause must be necessary for an effect, hence the contrastive aspect, the latter emphasizes 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 nor interventionism, mechanisms that are computationally expensive and criticised for introducing subjectivity into causal enquiry (Sarmiento et al., 2022; Wright, 1985).

4. From AAF to Action Languages

This section presents our first contribution: A formalization of acyclic AAF into the action language introduced in Section 3.2. Section 4.1 presents the translation of an AAF in the form of the octuplet $\kappa$ defining the context called the argumentative context. Section 4.2 provides the modified definitions of the action language semantics adapted to the argumentative context, and Section 4.3 briefly sketches the structure of the ASP implementation based on an adaption of the one proposed in Sarmiento et al. (2023).

Differing from classical 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 $(\mathrm{\Delta },R)$ , where $\mathrm{\Delta }$ is a dialog, i.e. a sequence of abstract arguments, such as statements in natural language in the case of Example 1. Formally:

Definition 8 (dialog $\mathrm{\Delta }$ )

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

4.1. Instantiating the Context

In order to formalize an AAF in the action language described in Section 3.2, 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 3.1, there are two elements to consider: The arguments and the attack relation $R$ . 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 $c{A}_{y,x}\in \mathbb{F}$ to model that $y$ can attack argument $x$ . This fluent models a notion of potential attack instead of real attack as argument $b$ attacks argument $a$ if and only if it can attack $a$ , i.e. $(b,a)\in R$ , and if both arguments are present. We discuss this distinction later in the paper after the whole instantiation of the context has been presented. As we consider acyclic AAF, $\nexists (x_1,\dots ,x_n)\in A$ such that $(c{A}_{x_1,x_2},\dots ,c{A}_{x_{n-1},x_n},c{A}_{x_n,x_1})\in \mathbb{F}$ . We call this property acyclicity of the fluents $cA$ .

Example 1

(continued) – Applying these definitions, the set of fluents is given by $\mathbb{F}={\displaystyle (\bigcup _{*\in A}\{p_{*},a_{*}\})\cup \{c{A}_{b,a},c{A}_{d,c},c{A}_{d,l},c{A}_{e,d},}$ $c{A}_{f,e},c{A}_{g,f},c{A}_{h,c},c{A}_{h,l},c{A}_{i,c},c{A}_{i,l},c{A}_{j,h},c{A}_{k,i},c{A}_{m,l},c{A}_{n,c},c{A}_{n,m}\}$ .

In an AAF, the only deliberate action is to enunciate an argument, which leads to $\mathbb{A}=\{enunciat{e}_x\mid x\in A\}$ . For this action to be possible, argument $x$ must not have already been enunciated. Argument $x$ 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:

$$\begin{array}{rl}& pre(enunciat{e}_x)\equiv \mathrm{\neg }{p}_x\\ & e\mathit{\text{ff}}(enunciat{e}_x)\equiv p_x\wedge a_x\end{array}$$

Remark. None of the events described below has the effect of making an argument not present. This implies that if an argument repetition is planned in the dialog, it is ignored and the next argument enunciation is processed. Indeed, the effect of repeating an argument would be to reset its acceptability status to acceptable. However, in the case of an acyclic graph, as there is a single acceptability status for each argument, this will not have any effect after the acceptability update rules described below are applied.

When a new argument is enunciated, before enunciating the next argument, we choose to update the acceptability of all other arguments present after the enunciation of this new argument until a suitable state is reached. Such a state refers to one in which the AAF principles for acceptability of arguments are applied. In particular, if an argument attacks an acceptable argument, then the attacker is necessarily unacceptable. Similarly, if an argument is not attacked, or is attacked only by unacceptable arguments, then it is acceptable. This defines a state which we call argumentative state.

Definition 9 (argumentative state)

A state $S(t)$ is an argumentative state if:

i) $\mathrm{\forall }x,y,[S(t)\models a_x\wedge p_y\wedge c{A}_{y,x}\Rightarrow S(t)\models \mathrm{\neg }{a}_y]$ ;

ii) $\mathrm{\forall }x,[S(t)\models p_x\wedge (\underset{y}{\bigwedge }\mathrm{\neg }{a}_y\vee \mathrm{\neg }c{A}_{y,x})\Rightarrow S(t)\models a_x]$ .

After an argument is enunciated, we want updates to be triggered automatically. We represent them with two exogenous events: $makesUnac{c}_{y,x}\in \mathbb{U}$ and $makesAc{c}_x\in \mathbb{U}$ . 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 in turn:

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 $makesUnac{c}_{y,x}$ is written as:

$$\begin{array}{rl}tri(makesUnac{c}_{y,x})\equiv & \phantom{\mathrm{\neg }}{a}_x\wedge a_y\wedge c{A}_{y,x}\\ e\mathit{\text{ff}}(makesUnac{c}_{y,x})\equiv & \mathrm{\neg }{a}_x\end{array}$$

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 $makesAc{c}_x$ :

$$\begin{array}{rl}tri(makesAc{c}_x)\equiv & p_x\wedge \mathrm{\neg }{a}_x\wedge (\underset{y}{\bigwedge }\mathrm{\neg }c{A}_{y,x}\vee \mathrm{\neg }{a}_y)\\ e\mathit{\text{ff}}(makesAc{c}_x)\equiv & a_x\end{array}$$

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 $z$ unacceptable. In other words, $x$ makes $z$ unacceptable if and only if there is no acceptable argument $y$ that makes $x$ unacceptable. In practice, for $makesUnac{c}_{x,z}$ to be triggered, no events of the form $makesUnac{c}_{y,x}$ have to be triggered. This is reflected in the following priority rule:

$$makesUnac{c}_{y,x}{\succ }_{\mathbb{E}}makesUnac{c}_{x,z}$$

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 $makesUnac{c}_{y,x}{\succ }_{\mathbb{E}}makesAc{c}_z$ 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 $at{t}_{y,x}\in \mathbb{F}$ translating the fact that argument $y$ actually attacks argument $x$ . Let us define the exogenous event $isAttackin{g}_{y,x}\in \mathbb{U}$ as:

$$\begin{array}{rl}tri(isAttackin{g}_{y,x})\equiv & p_x\wedge p_y\wedge c{A}_{y,x}\\ e\mathit{\text{ff}}(isAttackin{g}_{y,x})\equiv & at{t}_{y,x}\end{array}$$

From this definition, argument $y$ attacks 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 $a_y\wedge at{t}_{y,x}$ , i.e. $a_y\wedge p_y\wedge c{A}_{y,x}$ . 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: $tri(makesUnac{c}_{y,x})\equiv a_y\wedge a_x\wedge at{t}_{y,x}=a_y\wedge a_x\wedge c{A}_{y,x}$ . The same precondition applies without the introduction of $isAttacking\text{ and }att$ . Therefore, we use only $cA$ .

In this section, we defined the context $\kappa$ adapted for AAF. In particular, we defined:

the set of fluents $\mathbb{F}=\{p_x,a_x\mid x\in A\}\cup \{c{A}_{y,x}\mid (y,x)\in R\}$

the set of events $\mathbb{E}=\{enunciat{e}_x\mid x\in A\}\cup \{makesAc{c}_x,makesUnac{c}_{y,x}\mid x,y\in A\}$ with $pre,tri,e\mathit{\text{ff}}$ and ${\succ }_{\mathbb{E}}$ defined above.

4.2. Semantics Adapted to AAF

Having an adapted $\kappa$ for the argumentative framework, we modify the action language semantics to produce traces that are representative of the argumentative process. For this purpose, arguments are stated from argumentative states step by step, in the order determined by the dialog $\mathrm{\Delta }$ .

The definition of a scenario $\sigma$ as a set of timed actions ( $\sigma \subseteq \mathbb{A}\times \mathbb{T}$ , see Section 3.2) is not ideal for this task. Indeed, the actions are performed at their corresponding time, regardless of what other events are happening. But when modeling a dialog, we want the action, i.e. the enunciation of the arguments, to be performed in a defined order, but also when no more acceptability updates are triggered. This implies that we need to know in advance how many steps each chain of acceptability update events will take in order to plan when 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. Now, the natural number associated with each action does not represent when the action should be performed, but rather after what other actions. The input to obtain unique traces is no longer the scenario $\sigma$ but the sequence $\varsigma$ .

Definition 10 (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.

These modifications require changes to Definitions 6 and 7. Definition 11 below is the result obtained after modifying Definition 6. Conditions 2.d and 2.e are added and $\mathrm{\forall }e\in \mathbb{E}$ is replaced by $\mathrm{\forall }e\in \mathbb{U}$ 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. As a consequence the triggering of exogenous events remains unchanged.

Definition 11 (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 given $\kappa$ if the following conditions are satisfied, $\mathrm{\forall }t\in \mathbb{T}$ :

(1)

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

(2)

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

• 2.a

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

2.b

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

2.c

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

2.d

If $\mathrm{\exists }e\in E(t)\cap \mathbb{A}$ , then $\mathrm{\forall }{e}^{\prime }\in \mathbb{U}$ , $S(t)⊭tri(e^{\prime })$ ;

2.e

$E(t)\ne \varnothing$ .

(3)

$S(t+1)=(S(t)▹E(t)).$

In Definition 7 traces are defined as extracts of a valid execution given $\kappa$ that satisfy additional conditions related to $\sigma$ . Instead of defining directly traces, Definition 12 below corresponds to a valid execution given $\chi =(\varsigma ,\kappa )$ . Traces are simply extracts from such valid executions. The modifications include: Replacing the scenario $\sigma$ by the sequence $\varsigma$ ; specifying that the only actions that can be in the event trace are those in $\varsigma$ ; specifying that an action in the sequence can be triggered only if all actions with an inferior rank have happened; and specifying that two actions with the same rank must happen at the same time in the event trace.

Definition 12 (valid execution given $\chi$ )

Given an argumentative setting $\chi =(\varsigma ,\kappa )$ , a valid execution given $\kappa$ $E(-1),S(0),E(0),\dots ,E(N),S(N+1)$ , is valid given $\chi$ if:

(1)

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

(2)

$\mathrm{\forall }((e,o),(e^{\prime },o^{\prime }))\in {\varsigma }^2$ such that $o<o^{\prime }$ , $\mathrm{\exists }t,t^{\prime }\in {\mathbb{T}}^2$ such that $e\in E(t)and{e}^{\prime }\in E(t^{\prime })andt<t^{\prime }$ ;

(3)

$\mathrm{\forall }((e,o),(e^{\prime },o^{\prime }))\in {\varsigma }^2$ such that $o=o^{\prime }$ , $\mathrm{\exists }t\in \mathbb{T}$ such that $e\in E(t)and{e}^{\prime }\in E(t)$ .

For 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

To generate the traces used to illustrate the paper, we have adapted a version of the sound and complete implementation in ASP of the action language we consider (Sarmiento et al., 2023). This implementation is described in details in Sarmiento et al. (2023). The ASP programs ${\pi }_{con}(\kappa )$ and ${\pi }_{seq}(\varsigma )$ are obtained by the translation of the context $\kappa$ and the sequence $\varsigma$ respectively. ${\pi }_{\mathbb{C}}$ is exactly the same as in Sarmiento et al. (2023). Finally, ${\pi }_{\mathbb{A}}$ is obtained by the translation of the action language semantics introduced in Section 3.2.1 and modified in Section 4.2.

The most important change concerns when an action can be performed. First, we move from a scenario to a sequence. Unlike the original program, where actions are performed in the state determined by the scenario, in our program an action ordered $O$ can only be performed if the previous action, i.e. ordered $O-1$ in the sequence, has been completed. To do this, a rule called $done$ is added to check whether an action has been completed. As a consequence, the triggered rule is modified to take into account that the previous action must be completed before the next one can be started. Secondly, an action can be performed if and only if no exogenous events, called auto, are triggered. This results in the following change:

$done(O,T):-happens(A,\mathrm{\_},T1),performs(A,O),time(T),T1<T.$

$triggered(A,GD,T):-action(A,GD,E\mathit{\text{ff}}ect),performs(A,O),done(O-1,T),$

$holds(GD,T),time(T),nottriggered(U,\mathrm{\_},T):auto(U,\mathrm{\_},\mathrm{\_}).$

Definition 13 (actual causality program $\mathrm{\Pi }(\chi )$ )

Given a scenario ${\pi }_{seq}(\varsigma )$ , a context ${\pi }_{con}(\kappa )$ , an action language semantics ${\pi }_{\mathbb{A}}$ , and causal relations definitions ${\pi }_{\mathbb{C}}$ , the actual causality program is $\mathrm{\Pi }(\chi )={\pi }_{seq}(\varsigma )\cup {\pi }_{con}(\kappa )\cup {\pi }_{\mathbb{A}}\cup {\pi }_{\mathbb{C}}$ .

The entire program $\mathrm{\Pi }(\chi )={\pi }_{seq}(\varsigma )\cup {\pi }_{con}(\kappa )\cup {\pi }_{\mathbb{A}}\cup {\pi }_{\mathbb{C}}$ is available ¹ .

5. Formal Properties of the Proposed Transformation

This section establishes formal properties of the proposed transformation described in the previous section. 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 on explanation 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 so are the corresponding traces ${\tau }_{\chi }^e$ and ${\tau }_{\chi }^s$ .

Proposition 3

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 uniqueness of valid executions given $\chi$ . Let $\chi =(\varsigma ,\kappa )$ be the argumentative setting and $ϵ$ , ${ϵ}^{\prime }$ two valid executions given $\chi$ . Assume, ad absurdum, that $ϵ\ne {ϵ}^{\prime }$ .

According to Definition 6, $\mathrm{\forall }t,S(t+1)$ is derived from $S(t)$ and the events in $E(t)$ , and similarly for $S^{\prime }$ . Hence, given that $\kappa$ is common to $ϵ$ and ${ϵ}^{\prime }$ , $E(-1)=E^{\prime }(-1)$ and $S(0)=S^{\prime }(0)$ , the first discrepancy between $ϵ$ , and ${ϵ}^{\prime }$ 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 first date at which a difference between $ϵ$ and ${ϵ}^{\prime }$ is observed. We have $E(t_0)\ne E^{\prime }(t_0)$ , $\mathrm{\forall }t<t_0$ , $E(t)=E^{\prime }(t)$ , and $\mathrm{\forall }t\le t_0$ , $S(t)=S^{\prime }(t)$ . Thus, $\mathrm{\forall }e\in \mathbb{E}$ , $S(t_0)\models pre(e)\iff S^{\prime }(t_0)\models pre(e)$ . Without loss of generality, let us consider an event $e_0$ such that $e_0\notin E(t_0)$ 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 $e_0\notin \mathbb{U}$ . 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 11, $e_0\notin E(t_0)$ implies $\mathrm{\exists }e\in E(t_0)$ such that $e{\succ }_{\mathbb{E}}{e}_0$ . Then from 2.b applied to $E^{\prime }(t_0)$ , we get $e\notin E^{\prime }(t_0)$ . 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 $S^{\prime }(t_0)⊭tri(e_0)$ which contradicts our assumption. In the second case, if $e\in \mathbb{U}$ , then $e\in E(t_0)$ and $e\notin E^{\prime }(t_0)$ because of the same reasons behind $e_0\notin E(t_0)$ and $e_0\in E^{\prime }(t_0)$ . If we apply the same reasoning to $e$ , we get $\mathrm{\exists }{e}^{\prime }\in \mathbb{U}$ such that $e^{\prime }\in E(t_0)$ , $e^{\prime }\notin E^{\prime }(t_0)$ , 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 $\mathrm{\exists }\tilde{e}\in \mathbb{U}$ such that $\tilde{e}\notin E(t_0)$ 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) From (i) $e_0\in \mathbb{A}$ . Therefore, by condition 1 in Definition 12, $e_0\in E^{\prime }(t_0)$ implies $e_0\in \{a\mid \mathrm{\exists }o\in \mathbb{N},(a,o)\in \varsigma \}$ . $\varsigma$ being the same for $ϵ$ and ${ϵ}^{\prime }$ , the rank $o_0\in \mathbb{N}$ associated to $e_0$ is the same. Hence, given that $\mathrm{\forall }t<t_0$ , $ϵ(t)={ϵ}^{\prime }(t)$ and $e_0\in E^{\prime }(t_0)$ , $e_0\notin E^{\prime }(t)$ implies $e_0\notin E(t)$ . 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 11. Otherwise, we have a contradiction with condition 3 in Definition 12. 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 date $t_0$ at which a difference between $ϵ$ and ${ϵ}^{\prime }$ is observed, and thus to reject that $ϵ\ne {ϵ}^{\prime }$ . The uniqueness of traces is accordingly established. □

From now on, when references are 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 occur 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 the transformation proposed in Section 4. For that, we first introduce the notion of associated graph as follows:

Definition 14

Given a state $S^{\chi }(t)$ its associated graph is the AAF $A{F}^{\prime }=(A^{\prime },R^{\prime })$ , where $A^{\prime }=\{x\mid S^{\chi }(t)\models p_x\}$ and $R^{\prime }=\{(y,x)\in A^{\prime }\times A^{\prime }\mid S^{\chi }(t)\models c{A}_{y,x}\}$ .

From the acyclicity property of the fluent $cA$ (see Section 4.1), the associated graph is acyclic.

Now, we focus on the notion of acceptability. We first characterize argumentative states using the triggering condition functions $tri$ .

Lemma 1

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

$\mathrm{\forall }e\in \mathbb{U}$ , $S^{\chi }(t)⊭tri(e)$

$S^{\chi }(t)$ is an argumentative state as defined in Definition 9.

Proof.

In our context, $\mathbb{U}=\{makesAcc,makesUnacc\}$ . We prove that $S^{\chi }(t)⊭tri(makesAc{c}_x)$ is equivalent to (ii) of Definition 9 and $S^{\chi }(t)⊭tri(makesUnac{c}_{y,x})$ is equivalent to (i) of Definition 9: for any $x,y$ $\bullet$ $\mathrm{\neg }tri(makesAc{c}_x)\equiv \mathrm{\neg }(p_x\wedge \mathrm{\neg }{a}_x\wedge (\underset{y}{\bigwedge }\mathrm{\neg }{a}_y\vee \mathrm{\neg }c{A}_{y,x}))$

$\equiv \mathrm{\neg }(p_x\wedge (\underset{y}{\bigwedge }\mathrm{\neg }{a}_y\vee \mathrm{\neg }c{A}_{y,x}))\vee a_x$

$\equiv (p_x\wedge (\underset{y}{\bigwedge }\mathrm{\neg }{a}_y\vee \mathrm{\neg }c{A}_{y,x})\Rightarrow a_x)$ , which leads to the desired equivalence with (ii) in Definition 9. $\bullet \mathrm{\neg }tri(makesUnac{c}_{y,x})\equiv \mathrm{\neg }(a_x\wedge a_y\wedge c{A}_{y,x})$

$$\begin{gathered} \equiv \mathrm{\neg }(a_x\wedge p_y\wedge a_y\wedge c{A}_{y,x})as{a}_y\text{ implies }{p}_y \\ \equiv \mathrm{\neg }(a_x\wedge p_y\wedge c{A}_{y,x})\vee \mathrm{\neg }{a}_y \end{gathered}$$

$\equiv (a_x\wedge p_y\wedge c{A}_{y,x})\Rightarrow \mathrm{\neg }{a}_y$ , which leads to the desired equivalence with (i) in Definition 9. □

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 argument $x$ is enunciated.

Proposition 4

Given an argumentative setting $\chi$ , a time $t$ , an argumentative state $S^{\chi }(t)$ and an argument $x\in A$ , if $enunciat{e}_x\in E^{\chi }(t)$ , then $\mathrm{\exists }{t}^{\prime }\in \mathbb{T}$ , $t<t^{\prime }$ such that $S^{\chi }(t^{\prime })$ is an argumentative state.

Proof.

Given an argumentative state $S^{\chi }(t)$ and $x\in A$ such that $enunciat{e}_x\in E^{\chi }(t)$ , let us prove that $\mathrm{\exists }{t}^{\prime }\in \mathbb{T}$ , $t<t^{\prime }$ such that $\mathrm{\forall }e\in \mathbb{U},S^{\chi }(t^{\prime })⊭tri(e)$ , which leads to the desired result using Lemma 1.

As $\mathbb{U}$ and $\{(x,y)\in A^2\mid S^{\chi }(t)\models c{A}_{y,x}\}$ are finite sets, there is a finite number of possible triggerings 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 therefore $\mathrm{\exists }{t}^{\prime }\le (t+M+1)$ such that $\mathrm{\forall }e\in \mathbb{U},S^{\chi }(t^{\prime })⊭tri(e)$ . □

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 designed so as to model how acceptability is computed. We first prove a useful lemma.

Lemma 2

Given an argumentative state $S^{\chi }(t)$ , for any $x\in A$ , $S^{\chi }(t)\models p_x\wedge \mathrm{\neg }{a}_x\iff S^{\chi }(t)\models \mathrm{\exists }y,p_x\wedge a_y\wedge c{A}_{y,x}$ .

Proof.

$[\Rightarrow ]:$ For any $x$ such that $S^{\chi }(t)\models p_x\wedge \mathrm{\neg }{a}_x$ , (ii) of Definition 9 implies that $S^{\chi }(t)\models \mathrm{\neg }{p}_x\vee (\underset{y}{\bigvee }{a}_y\wedge c{A}_{y,x})$ . Therefore, as $S^{\chi }(t)\models p_x,S^{\chi }(t)\models \mathrm{\exists }y,p_x\wedge a_y\wedge c{A}_{y,x}$ .

$[\Leftarrow ]:$ Let us prove it by contraposition: let $x_0$ be such that $S^{\chi }(t)\models \mathrm{\neg }{p}_{x_0}\vee a_{x_0}$ . If $S^{\chi }(t)\models \mathrm{\neg }{p}_{x_0}$ , then $x_0$ is such that $S^{\chi }(t)\models \mathrm{\forall }y,\mathrm{\neg }{p}_{x_0}\vee \mathrm{\neg }{a}_y\vee \mathrm{\neg }c{A}_{y,x_0}$ , which ends the proof. Otherwise, $S^{\chi }(t)\models p_{x_0}\wedge a_{x_0}$ . If $S^{\chi }(t)\models p_{x_0}\wedge (\mathrm{\exists }y,a_y\wedge c{A}_{y,x_0})$ then $S^{\chi }(t)\models tri(makesUnac{c}_{y,x_0})$ , which is not possible as $S^{\chi }(t)$ is argumentative using Lemma 1. Thus both cases lead to a contradiction. □

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

Proposition 5

Given an argumentative state $S^{\chi }(t)$ and its associated graph $AF=(A,R)$ according to Definition 14, for any $x\in A$ , $x$ acceptable by $A⟺S^{\chi }(t)\models a_x$ .

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 $S^{\chi }(t)\models \mathrm{\neg }{a}_{x_0}$ . By construction of AF $S^{\chi }(t)\models p_{x_0}$ . $S^{\chi }(t)$ is an argumentative state so, according to Lemma 2, $S^{\chi }(t)\models p_{x_0}\wedge \mathrm{\neg }{a}_{x_0}\iff S^{\chi }(t)\models \mathrm{\exists }y,p_{x_0}\wedge a_y\wedge c{A}_{y,x_0}$ . (i) of Definition 9 applied to $a_y$ implies that $\mathrm{\forall }z,S^{\chi }(t)\models a_y\wedge p_z\wedge c{A}_{z,y}\Rightarrow S^{\chi }(t)\models \mathrm{\neg }{a}_z$ . If there is no $z$ satisfying the premise of this implication, then for any $z$ , $p_z\wedge c{A}_{z,y}$ is false, while implies that, in AF, $At{t}_y=\mathrm{\varnothing }$ . Therefore, $y$ that attacks $x_0$ , is acceptable which contradicts that $x_0$ is acceptable.

Otherwise, as there is a finite number of arguments, it is possible to apply to $z$ the process we applied to $x_0$ until one of the following two scenarios occurs:

$\bullet$ $S^{\chi }(t)\models \nexists y,p_z\wedge a_y\wedge c{A}_{y,z}$ . This leads to trigger the exogenous event $makesAc{c}_z$ which is not possible as $S^{\chi }(t)$ is an argumentative state according to Lemma 1.

$\bullet$ $\mathrm{\forall }z,S^{\chi }(t)\models a_y\wedge p_z\wedge c{A}_{z,y}\Rightarrow S^{\chi }(t)\models \mathrm{\neg }{a}_z$ where $p_z\wedge c{A}_{z,y}$ is false. Then in $AF$ , $At{t}_y=\mathrm{\varnothing }$ . Therefore, $y$ , that attacks $x_0$ , 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 $A$ .

If $At{t}_{x_0}=\mathrm{\varnothing }$ , the result is proved.

Otherwise, let us prove that $\mathrm{\forall }y\in At{t}_{x_0}$ , $y$ is not acceptable by $A$ . If $y_0\in At{t}_{x_0}$ is acceptable by $A$ , according to $[\Rightarrow ]$ , $S^{\chi }(t)\models a_{y_0}$ . However, it also holds that $S^{\chi }(t)\models a_{x_0}\wedge p_{y_0}\wedge c{A}_{y_0,x_0}$ while implies that $S^{\chi }(t)\models \mathrm{\neg }{a}_{y_0}$ using condition (i) in Definition 9.

So, as $\mathrm{\forall }y\in At{t}_{x_0}$ , $y$ is not acceptable by $A$ , $x_0$ is acceptable by $A$ . □

Proposition 5 establishes that there is an equivalence between the acceptability status in an argumentative state and in its associated graph. Now, from a dialog 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 $\mathrm{\forall }x\in A$ , $\mathrm{\exists }{t}^{\prime }\in \mathbb{T}$ such that $t^{\prime }<t$ and $enunciat{e}_x\in E^{\chi }(t^{\prime })$ . This state exists and is unique. Indeed, as $S(0)$ is an argumentative state, using Proposition 4 a new argumentative state is reached after a new argument is enunciated. Therefore, when all the dialog is over, i.e. when all the arguments have been enunciated, the final argumentative state is reached corresponding to the last element of the state trace.

Theorem 1 (Soundness and Completeness)

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

Proof.

As $A{F}^{\prime }$ is associated to a final argumentative state, $\mathrm{\forall }x\in A$ , $\mathrm{\exists }{t}^{\prime }\in \mathbb{T}$ such that $t^{\prime }<t$ and $enunciat{e}_x\in E^{\chi }(t^{\prime })$ . Now, $e\mathit{\text{ff}}(enunciat{e}_x)=p_x\wedge a_x$ . So, $A^{\prime }=A$ .

Moreover, by construction of $c{A}_{y,x}$ and $R^{\prime }$ , $R=R^{\prime }$ . So $AF=A{F}^{\prime }$ .

Finally, from Proposition 5, as $S^{\chi }(t)$ is argumentative, $\mathrm{\forall }x\in A=A^{\prime },S^{\chi }(t)\models a_x\iff x$ is acceptable by $A$ . □

5.3. On Temporality and Causality

Proposition 3 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 uniqueness property does not hold. This section shows that this temporality does not impact the final argumentative state but impacts the causal relations.

Proposition 6

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 $(t,t^{\prime })\in \mathbb{T}\times {\mathbb{T}}^{\prime }$ , then $S^{\varsigma ,\kappa }(t)=S^{{\varsigma }^{\prime },\kappa }(t^{\prime })$ .

Proof.

Let us denote by $AF=(A,R)$ and $A{F}^{\prime }=(A^{\prime },R^{\prime })$ , the associated graphs of the final argumentative states $S^{\varsigma ,\kappa }(t)$ and $S^{{\varsigma }^{\prime },\kappa }(t^{\prime })$ induced by their respective state traces. Given that they are based on the same actions in the sequence, we have $A=A^{\prime }$ . They also share the same context, so $R=R^{\prime }$ . Therefore $AF=A{F}^{\prime }$ .

Now, according to Proposition 5, $x\in A$ acceptable by $A\iff S^{\varsigma ,\kappa }(t)\models a_x$ . So $\mathrm{\forall }x,S^{\varsigma ,\kappa }(t)\models a_x\iff \mathrm{\forall }x,S^{{\varsigma }^{\prime },\kappa }(t^{\prime })\models a_x$ . □

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 the same. This immediately leads to the following uniqueness corollary:

Corollary 1

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

Proposition 6 and its corollary are in accordance with the classical AAF. Indeed, in an acyclic argumentation graph there is a single maximal admissible set and therefore a single possible acceptability status for each argument. 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 7

Causal relations, i.e. NESS and actual causes, depend on the sequence $\varsigma$ .

Proof.

This proposition is proved by example, commented in detail in the next section that illustrates the effect of considering Example 1, and a modification thereof in Example 2. Let $\mathrm{\Pi }(\chi )$ be the program obtained given $\kappa ,\varsigma$ of Example 1, as described in Section 4.3, and let $\mathrm{\Pi }({\chi }^{\prime })$ be the program similarly obtained given $\kappa ,{\varsigma }^{\prime }$ of Example 2. Given the NESS-cause definition in Sarmiento et al. (2023), it holds that $\mathrm{\Pi }(\chi )\models ness(o(enunciat{e}_d,4),h(neg(a_c),31))$ , where occurrences 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 $\nexists t,t^{\prime }\in {\mathbb{T}}^2$ , $\mathrm{\Pi }({\chi }^{\prime })\models ness(o(enunciat{e}_d,t),h(neg(a_c),t^{\prime }))$ . □

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 (see Theorem 1), it is essential to take temporality into account when dealing with notions related to causality, especially in explainability.

6. Application to the Example and Discussion

The formalization of acyclic abstract argumentation graphs into an action language as proposed in Section 4 allows us to dispose of complete knowledge of the evolution of the dialog. Indeed, it allows generating two traces: One tracing the evolution of world states, the other tracing the occurrence of events. This section proposes to explore at three levels how the use of action languages can benefit AAFs to generate explanations. In particular, it considers successively two classes of argumentation explanations in the taxonomy proposed by Čyras et al. (2021): it first shows how it can lead to graphical representations of the processes of accepting/rejecting arguments, it then discusses causal explanations that provide information on the reason for the acceptability status of arguments. Example 1 is used throughout this section to illustrate the discussion. The state trace corresponding to the example contains thirty one states and the event trace fourteen actions and twenty exogenous events.

6.1. Graphical Representation and Explanation

According to Čyras et al. (2021), 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 Section 4 makes it possible to derive graphical representations of the argumentative process.

Indeed, transition systems such as action languages offer a way to model the world that is suitable for visual representation: the representation of two successive states and the events that led to this evolution gives an accurate representation of how the evolution of the world is modeled. For that, in the case of the considered action language, we use the traces that record the evolution of the world. Therefore, displaying them in chronological order provides a narration of the dialog.

Following this idea of using the narration of the dialog provided by traces, Figure 3 proposes a possible display of the event and state traces obtained from Example 1 by showing fluents as hexagons and triggered events as rectangles. Since the acceptability of arguments is what mainly matters in a dialog, we represent only the fluents $a_x$ , using only the argument names for the sake of legibility. Moreover, we do not show fluents when their negation holds 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 (see $d$ at $t=8$ and $t=9$ ). The events $enunciat{e}_x$ , $makesUnac{c}_{y,x}$ , and $makesAc{c}_x$ are shortened as $en{u}_x$ , $un{a}_{y,x}$ , and $ac{c}_x$ , respectively.

Figure 3.

Partial graphical representation associated with Example 1. Hexagons represent fluents and rectangles events.

Example 1

(continued) – Figure 3 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 Definition 9, which allows for the enunciation of the next argument: since all arguments preceding $e$ have already been enunciated, the action $enunciat{e}_e$ can be performed. The occurrence of this event is the transition to the next state $S(7)$ where, as shown in Figure 3, argument $e$ is acceptable. Unlike $S(6)$ , $S(7)$ is not an argumentative state: condition (i) of Definition 9 is not satisfied because $(a_d\wedge c{A}_{e,d})$ and $a_e\in S(7)$ . Therefore, the next argument cannot be enunciated. However, since the triggering conditions of $makesUnac{c}_{e,d}$ 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 Definition 9 is now satisfied. However, condition (ii) is not satisfied by $S(8)$ , preventing the next argument from being enunciated. Instead, $makesAc{c}_c$ is triggered, leading to the following state $S(9)$ . As this new state is argumentative, the next argument, $f$ , can be stated. The dialog continues step by step and ends at state $S(31)$ . At this state, all arguments have been enunciated and the argumentative updates have been performed making $S(31)$ the final argumentative state. In particular, fluents $a_a$ and $a_c$ are false and fluent $a_l$ is true: the only acceptable decision variable is $l$ (do an MRI today), which is the same result as would be obtained in a classical AAF.

This partial graphical representation shows the causal relations that can be deduced immediately, these are the direct causal relations. They can be used to derive directly basic explanations such as the fact that enunciating argument $e$ at time $t=6$ causes $e$ to be acceptable in the next state, i.e. at time $t=7$ . Now, if one wonders why argument $c$ has become acceptable at $t=9$ , such a representation shows that it is because $makesAcceptabl{e}_c$ has been triggered. In contrast to the case of $e$ , which we find acceptable due to its recent enunciation, the present causal relationship indicates $c$ ’s acceptability has now been restored, which means only for someone knowledgeable in argumentation that all of its attackers must have been rendered unacceptable. Thus, using the temporal modeling that is offered by the action language permits to give a visual display of the dialog, helping to increase the understanding of the interaction.

However, using only the temporality to derive simple causal relations between states is not enough, and more complex relations are needed. Indeed, saying that $c$ is acceptable because it has been made acceptable is not satisfactory. To solve this issue, causal reasoning is needed to find more complex and relevant causes, called actual causes.

A second, more compact, tabular visualization is proposed, illustrated in Figure 4: the arguments are represented in the first column and the order of the performed actions in the first row. For the sake of readability, $enunciat{e}_x$ is shortened as $x$ , $\bullet$ 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 cells. In contrast to the previous representation where the updating stages are shown, this second form has the advantage of being more compact and allows displaying the whole dialog. 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.

Figure 4.

Tabular representation of the entire interaction of Example 1. The first line represents the sequence and the others the acceptability status of the arguments.

Example 2

Let us consider the same dialog 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 as in the initial example.

Figure 5 displays the proposed compact visualization 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 6 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. enunciation of $n$ , with no modification until the end.

Figure 5.

Impact of the order in which arguments are enunciated (lines 1, 3) on the acceptability of the argument c (lines 2, 4).

This visualization 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.

6.2. Causal Reasoning

Beyond the graphical representations of the acceptance/rejection process, the proposed formalization of AAF into action models allows us to transfer the notion of actual causality recalled in Section 3.2.2 to the argumentation framework, providing tools for richer explanations. Indeed, the extraction of causal chains has been shown to be an important property for explanations (Miller, 2019).

Using the three causal relations described in Section 3.2.2, we propose a richer graphical representation that gives a deeper understanding of the argumentation process illustrated in Figure 6. Indeed, in the context of argumentation, the NESS causes provide information about what are the changes in the acceptability status of arguments that have led to the acceptability of a particular argument. This includes the enunciation of a new argument as it makes that argument acceptable. To find an actual cause, we look at the NESS causes of the preconditions and triggering conditions of an event. Thus, in this context, actual causes correspond to the changes in the acceptability status of arguments that have caused the changes in the acceptability status of other arguments. Those relations are inferred from the state and event traces using the program ${\pi }_{\mathbb{C}}$ of the ASP implementation and are returned as a new trace, the causal trace. They are represented using different arrows in Figure 6: (––) for the direct NESS-causes, (-----) for the NESS-cause and (—) for the actual causes.

Figure 6.

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

For example, the richer causal relations tell us that the enunciation of $n$ is a cause of $l$ being acceptable and $c$ and $m$ not being acceptable, i.e. the fact that it is an emergency cause doing the MRI today instead of in three days despite there is officially no available slot left. Moreover, by looking at the actual causal relations, we can understand, for example, that the enunciation of $n$ causes $l$ to be acceptable by making $m$ not acceptable, which is much more satisfactory than the direct causal relations illustrated in Section 6.1. More complex relationships of this kind can then be obtained by reconstructing the chain backward. This can help to explain the dialog in a simple way when the number of arguments and attack relations between them becomes large.

Example 1

(continued) – Figure 6 graphically displays the last four states in the traces 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 6 its enunciation in state $S(28)$ is a direct NESS-cause (dNc) of its acceptability in the following states, a relation we denote by $(enunciat{e}_n,28)$ dNc $(a_n,29-31)$ . Similarly, we have $(makesUnac{c}_{n,c},29)$ dNc $(\mathrm{\neg }{a}_c,30-31)$ , $(makesUnac{c}_{n,m},29)$ dNc $(\mathrm{\neg }{a}_m,30-31)$ , and $(makesAc{c}_l,30)$ dNc $(a_l,31)$ . As these examples show, the direct-NESS cause relationship looks at the actual effects of the occurrence of an event and is therefore the basic building block of causality. Yet this relationship is not enough. If we want to know why argument $l$ is acceptable at the end of the dialog (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 $(makesAc{c}_l,30)$ , i.e. because it becomes acceptable.

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 leads to it. By transitivity, we get that $(makesUnac{c}_{n,m},29)$ is a cause of the fact that $makesAc{c}_l$ 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 $(makesUnac{c}_{n,m},29)$ took place, we find $(enunciat{e}_n,28)$ actual cause $(makesUnac{c}_{n,m},29)$ and therefore $(enunciat{e}_n,28)$ NESS-cause $(\mathrm{\neg }{a}_m,30-31)$ . By transitivity we can derive $(enunciat{e}_n,28)$ 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 $(\mathrm{\neg }{a}_c,31)$ , the other decision variable.

Example 2

(continued) – To illustrate the impact of temporality when dealing with causality, let us consider the modified version of the dialog, in which the order of enunciation has been modified. In the original scenario, when building the complete causal chain of why $c$ is unacceptable, the enunciation of argument $d$ appears as a NESS cause of it. However, this is not the case in this modified scenario, where the only action that causes the state of $c$ is the enunciation of $n$ .

6.3. Towards Explanations

The graphical representations proposed in the previous sections, like the causal chains, may not represent an explanation per se. Indeed, these relations, as helpful as they can be, tend to form long causal chains that may contain redundancy or useless variables. This raises the question of the appropriate way to use causal chains to derive explanations. The aim of the ongoing work is therefore to propose a method that makes it possible to extract, from this enrichment of argument graphs, explanations on the acceptability or non-acceptability of one or more arguments. A first direction can be found in the works that study the links between the notions of causality and explanation. These are summarized in Miller’s paper (2019), which establishes several interesting properties that an explanation should satisfy. We discuss in this section five of them: Proximity of the explanation to the explanandum, consideration of an agent’s volition, contrastivity, robustness, and short length. The question is how to apply these properties to the framework we propose in order to provide explanations to a final user.

To apply the first two principles, an action language is particularly well suited. In fact, the temporal proximity of an event to the consequence can be evaluated and formalized, using the inclusion of time in the framework, by comparing positions in the traces. Another interesting property is that a deliberate action is often preferred to one that is not. In the proposed formalism, this means that an action is preferred to an exogenous event. Then, given two explanations that are identical except for one element, the one with the temporally closest deliberate actions is preferred.

The next two properties, contrastivity and robustness, require to introduce new notions. First, following the principles Miller applies to Halpern’s structural causal model (Miller, 2021), to define a contrastive explanation, we need to introduce the formalism for a contrastive causal chain. Intuitively, contrastive causes of $a_l$ and $\mathrm{\neg }{a}_c$ would be, if they exist, causes of $a_l$ and $\mathrm{\neg }{a}_c$ resulting in the common elements of the two causal chains, i.e. $en{u}_n$ in this case. Regarding robustness, an explanation can be considered more robust than another if it holds in a larger number of scenarios. Finally, an explanation has to be short.

Applying these principles to Example 1, an explanation for the acceptability of $l$ and the unacceptability of $c$ (doing an MRI today instead of in two days) could be the enunciation of $n$ stating that it is an emergency. Indeed, it is a short (it cannot be shorter than one fluent) and contrastive explanation as it belongs to $a_l$ and $\mathrm{\neg }{a}_c$ causal chains. Moreover, it is a deliberate action that is temporally close to the consequence. Finally, in the scenario of Example 2, $en{u}_n$ is an explanation with regard to the all criteria except for robustness. Therefore, $en{u}_n$ is robust for at least this two scenarios and so it is an explanation satisfying the five properties.

7. Conclusion

This paper proposes a formalization of acyclic abstract argumentation systems in an action language that offers tools for identifying causal relations (Sarmiento et al., 2022), and it establishes its formal properties: the proposed formalization first allows increasing the expressiveness of AAF 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 with the action language, offering the possibility to give rich information about the argument acceptance or rejection and justifications about the latter. The paper proposes two types of graphical representations of the argumentation process that can be used as visual support, it also discusses how to extract explanations from causal chains. Thus, this work makes it possible to model the evolution of a dialog and opens the way to the generation of user-adapted causal explanations.

Future works lies in the Action Language and the Explanation Model parts of the framework presented in Figure 1. On the one hand, we are working on generalizing the transformation into the action language framework to richer argumentation cases, especially cyclic argumentation graphs. The latter raise technical issues regarding the termination and completeness of the transformation. Moreover, in contrast to acyclic graphs, there is more than one possible outcome for the dialog, and several extensions can be obtained for a given a semantic. Using temporality, we think that it will be possible to distinguish between these extensions given an argumentation semantics, which is another advantage of modeling the order of enunciation. On the other hand, ongoing works also aim at formalizing Miller’s desired explanation properties (Miller, 2019) in the action language framework. Then, the next step will aim at evaluating the method experimentally, conducting user studies to assess the intelligibility of the generated explanations, both in terms of objective understanding and subjective satisfaction.