Leila Amgoud

Florence Dupin De Saint Cyr

Measures for persuasion dialogs: A preliminary investigation

Keywords: Argumentation, Persuasion dialogs, Quality measures

Persuasion is one of the main types of dialogs encountered in everyday life. The basic idea behind a persuasion is that two (or more) agents disagree on a state of affairs, and each one tries to persuade the other to change his mind. For that purpose, agents exchange arguments of different strengths.

Several systems, grounded on argumentation theory, have been proposed in the literature for modeling persuasion dialogs. These systems have studied more or less deeply the different protocols required for this kind of dialogs, and have investigated different termination criteria. However, nothing is said about the properties of the generated dialogs, nor on the behavior of the interacting agents. Besides, analyzing dialogs is a usual task in everyday life. For instance, political debates are generally deeply dissected.

In this paper we define measures for analyzing dialogs from the point of view of an external agent. In particular, three kinds of measures are proposed: i) measures of the quality of the exchanged arguments in terms of their strengths, ii) measures of the behavior of each participating agent in terms of its coherence, its aggressiveness in the dialog, and finally in terms of the novelty of its arguments, iii) measures of the quality of the dialog itself in terms of the relevance and usefulness of its moves.

Introduction

Since the seminal work by Walton and Krabbe [START_REF] Walton | Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning[END_REF] on the role of argumentation in dialog, and on the classification of different types of dialogs, there is an increasing interest on modeling those dialog types using argumentation techniques. Indeed, several dialog systems have been proposed in the literature for modeling information seeking dialogs (e.g. [START_REF] Parsons | Properties and complexity of some formal inter-agent dialogues[END_REF]), inquiry dialogs (e.g. [START_REF] Black | A generative inquiry dialogue system[END_REF]), negotiation (e.g. [START_REF] Sycara | Persuasive argumentation in negotiation[END_REF]), and finally persuasion dialogs (e.g. [START_REF] Amgoud | Modelling dialogues using argumentation[END_REF][START_REF] Gordon | The pleadings game[END_REF][START_REF] Prakken | Coherence and flexibility in dialogue games for argumentation[END_REF][START_REF] Zabala | Beliefs, reasons and moves in a model for argumentative dialogues[END_REF]). Persuasion dialogs are initiated from a position of conflict in which one agent believes p and the other believes ¬p, and both try to persuade the other to change its mind by presenting arguments in support of their thesis.

It is worth noticing that in all these disparate works, a dialog system is built around three main components: i) a communication language specifying the locutions that will be used by agents during a dialog for exchanging information, arguments, offers, etc., ii) a protocol specifying the set of rules governing the well-definition of dialogs, and iii) agents' strategies which are the different tactics used by agents for selecting their moves at each step in a dialog.

All the above systems allow agents to engage in dialogs that obey of course to the rules of the protocol. Thus, the only properties that are guaranteed for a generated dialog are those related to the protocol. For instance, one can show that a dialog terminates, the turn shifts equally between agents in that dialog (if such rule is specified by the protocol), agents can backtrack to an early move in the dialog, etc. Note that the properties inherited from a protocol concern the way the dialog is generated. However, they don't say anything about the properties of that dialog. Judging the properties of a dialog may be seen as a subjective issue. Two people listening to the same political debate may disagree, for instance, on the "winner" of the debate, and more generally on their feeling about the dialog itself. Nevertheless, it is possible to define more objective criteria, for instance, the aggressiveness of each participant, the way agents may borrow ideas from each others, the self-contradiction of agents, the relevance of the exchanged information, etc.

Focusing only on persuasion dialogs, in this paper, we are concerned by analyzing already generated dialogs whatever the protocol used is and whatever the strategies of the agents are. We place ourselves in the role of an external observer that tries to evaluate the dialog. For this purpose, three kinds of measures are proposed: 1) Measures of the quality of the exchanged arguments in terms of their weights, 2) Measures of the behavior of each participating agent in terms of its coherence, its aggressiveness in the dialog, and finally in terms of the source of its arguments, 3) Measures of the properties of the dialog itself in terms of the relevance and usefulness of its moves. These measures are of great importance since they can be used as guidelines for a protocol in order to generate the "best" dialogs. They can also serve as a basis for analyzing dialogs that hold between agents.

The rest of the paper is organized as follows: Section 2 recalls the basics of argumentation theory. In Section 3, we present the basic concepts of a persuasion dialog. Section 4 details our dialog measures as well as their properties. Section 5 is devoted to some concluding remarks and conclusions.

Basics of argumentation systems

Argumentation is a reasoning model based on the construction and the comparison of arguments whose definition will be given in Section 3. In [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], an argumentation system is defined as follows:

Definition 1 (Argumentation system) An argumentation system (AS) is a pair T = A, R , where A is a set of arguments and R ⊆ A × A is an attack relation. We say that an argument α i attacks an argument α j iff (α i , α j ) ∈ R (or α i R α j ).

Note that to each argumentation system is associated an oriented graph whose nodes are the different arguments, and the edges represent the attack relation between them. Let G T denote the graph associated to the argumentation system T = A, R .

Since arguments are conflicting, it is important to know which arguments are acceptable. For that purpose, in [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], different acceptability semantics have been proposed. Let us recall them here.

Definition 2 (Conflict-free, Defence) Let B ⊆ A. • B is conflict-free iff ∄α i , α j ∈ B such that (α i , α j ) ∈ R. • B defends an argument α i iff for each argument α j ∈ A, if (α j , α i ) ∈ R, then ∃α k ∈ B such that (α k , α j ) ∈ R.
Definition 3 (Acceptability semantics) Let B be a conflict-free set of arguments of A. • α is skeptically accepted iff α ∈ E i , ∀E i = ∅ with i = 1, . . . , n;

• α is credulously accepted iff α is in some extensions and not in others.

• α is rejected iff ∄E j such that α ∈ E j .

Persuasion dialogs

Let L be a logical language from which arguments may be built. In our application, arguments are reasons of believing something. Throughout the paper, the structure and the origin of such arguments are supposed to be unknown. However, an argument is assumed to have at least two parts: a support (representing the set of premises or formulas used to build the argument) and a conclusion (representing the belief one wants to justify through the argument). Arguments will be denoted by lowercase Greek letters. In what follows, arg denotes a function that returns for a given set S ⊆ L all the arguments that may be built from formulas of S. Thus, arg(L) is the set of arguments that may be built from the whole logical language L.

As well established in the literature and already said, arguments may be conflicting since, for instance, they may support contradictory conclusions. In what follows, R L is a binary relation that captures all the conflicts that may exist among arguments of arg(L).

Thus, R L ⊆ arg(L) × arg(L). For two arguments α, β ∈ arg(L), the pair (α, β) ∈ R L means that the argument α attacks the argument β.

Let Ag = {a 1 , . . . , a n } be a set of symbols representing agents that may be involved in a persuasion dialog. Each agent is supposed to be able to recognize each argument of arg(L) and each conflict in R L . Note that this does not mean at all that an agent is aware of all the arguments. This assumption means that agents use the same logical language and the same definition of arguments.

A persuasion dialog consists mainly of an exchange of arguments. Of course other kinds of moves can be exchanged like questions and assertions. However, arguments play the key role in determining the outcome of the dialog. Thus, throughout the paper, we are only interested by the arguments exchanged in a dialog. The subject of such a dialog is an argument, and its aim is to compute the status of that argument. If at the end of the dialog, the argument is "skeptically accepted" or "rejected", then we say that the dialog has succeeded, otherwise the dialog has failed.

Definition 6 (Moves)

A move m ∈ M is a triple S, H, x such that:

• S ∈ Ag is the agent that utters the move, Speaker(m) = S • H ⊆ Ag is the set of agents to which the move is addressed,

Hearer(m) = H • x ∈ arg(L) is the content of the move, Content(m) = x.
During a dialog several moves may be uttered. Those moves constitute a sequence denoted by m 0 , . . . , m n , where m 0 is the initial move whereas m n is the final one. The empty sequence is denoted by . For any integer n, the set of sequences of length n is denoted by M n . These sequences are built under a given protocol. A protocol amounts to define a function that associates to each sequence of moves, a set of valid moves. Several protocols have been proposed in the literature, like for instance [START_REF] Amgoud | Modelling dialogues using argumentation[END_REF][START_REF] Prakken | Coherence and flexibility in dialogue games for argumentation[END_REF]. In what follows, we don't focus on particular protocols.

Definition 7 (Persuasion dialog)

A persuasion dialog D is a non-empty and finite sequence of moves m 0 , . . ., m n . The subject of D is Subject(D) = Content(m 0 ), and the length of D, denoted |D|, is the number of moves n+1. Each sub-sequence m 0 , . . ., m i (with i < n) is a sub-dialog D i of D. We will write also D i < D.

It is worth noticing that to each persuasion dialog D, one may associate an argumentation system that will be used to evaluate the status of each argument uttered during the dialog. This argumentation system is also used to compute the output of the dialog.

Definition 8 (AS of a persuasion dialog)

Let D = m 0 , . . . , m n be a persuasion dialog. The argumentation system of D is the pair AS D = Args(D), Confs(D) s.t:

1. Args(D) = {Content(m i ) | i = 0, . . . , n} 2. Confs(D) = {(α, β) such that α, β ∈ Args(D) and (α, β) ∈ R L }
In other words, Args(D) and Confs(D) return respectively, the set of arguments exchanged during the dialog and the different conflicts among those arguments.

Example 1 Let D be the following persuasion dialog between two agents a 1 and a 2 .

D = a 1 , {a 2 }, α 1 , a 2 , {a 1 }, α 2 , a 1 , {a 2 }, α 3 , a 1 , {a 2 }, α 4 , a 2 , {a 1 }, α 1 .
Let us assume that there exist conflicts in R L among some of these arguments. Those conflicts are summarized in the figure below.

α 1 α 2 α 3 α 4
In this case,

Args(D) = {α 1 , α 2 , α 3 , α 4 } and Confs(D) = {(α 2 , α 1 ), (α 3 , α 2 ), (α 4 , α 2 )}.
Property 1 Let D be a persuasion dialog. ∀D j such that D j < D, Args(D j ) ⊆ Args(D), and Confs(D j ) ⊆ Confs(D).

Any dialog has an output. In case of a persuasion, the output of a dialog is either the status of the argument under discussion (i.e. the subject) when that status is "skeptically accepted" or "rejected", or failure in case the status of the subject is "credulously accepted". The idea is that a dialog succeeds as soon as the status of the subject is determined, and thus a winner agent is known. However, when an argument is credulously accepted, this means that each agent keeps its position w.r.t. the subject, and the dialog fails to meet its objective.

Definition 9 (Output of a persuasion dialog) Let D be a persuasion dialog. The output of D, denoted by

Output(D) is    A iff Subject(D) is skeptically accepted in AS D R iff Subject(D) is rejected in AS D Fail iff Subject(D) is credulously accepted in AS D
It may be the case that from the set of formulas involved in a set E of arguments, it is possible to build new arguments that do not belong to E. Let

Formulas(E) = ∪ α∈E Support(α)
be that set of formulas. Due to the monotonic construction of arguments, E ⊆ arg(Formulas(E)) but the reverse is not necessarily true. Indeed, an argument remains always an argument even when new attackers are received. However, its status is nonmonotonic, and may change. As a consequence of the previous inclusion, new conflicts may also appear among those new arguments, and even between new arguments and elements of E. This shows clearly that the argumentation system associated with a dialog is not necessarily "complete". In what follows, we define the complete version of an argumentation system associated with a given dialog D.

Definition 10 (Complete AS) Let D be a persuasion dialog, and

AS D = Args(D), Confs(D) its associated AS. The complete AS is CAS D = arg(Formulas(Args(D))), R c where R c = {(α, β) s.t. α, β ∈ arg(Formulas(Args(D))) and (α, β) ∈ R L }.
Recall that Args(D) ⊆ arg(Formulas(Args(D))) and Confs(D) ⊆ R c ⊆ R L . Note that the status of an argument α in the system AS D is not necessarily the same as in the complete system CAS D .

Measuring persuasion dialogs

In this section we discuss different measures of persuasion dialogs. Three aspects can be analyzed: 1) the quality of the exchanged arguments during a persuasion dialog, 2) the agent's behavior, and 3) the properties of the dialog itself.

Measuring the quality of arguments

During a dialog, agents utter arguments that may have different weights. A weight may highlight the quality of information involved in the argument in terms for instance of its certainty degree. It may also be related to the cost of revealing that information. [START_REF] Amgoud | Inferring from inconsistency in preference-based argumentation frameworks[END_REF], several definitions of such arguments' weights have been proposed, and their use for comparing arguments has been made explicit. It is worth noticing that the same argument may not have the same weight from one agent to another. In what follows, a weight in terms of a numerical value is associated to each argument. The greater this value is, the better the argument.

weight : arg(L) -→ IN *
The function weight is given by the agent which wants to analyze the dialog. Thus, it may be given by an agent that is involved in the dialog, or by an external one. On the basis of arguments' weights, it is possible to compute the weight of a dialog as follows: This measure allows to compare pairs of persuasion dialogs only on the basis of the exchanged arguments. It is even more relevant when the two dialogs have the same subject and got the same output.

It is also possible to compute the weight of arguments uttered by each agent in a given dialog. For that purpose, one needs to know what has been said by each agent. This can be computed by a simple projection on the dialog given that agent.

Definition 12 (Dialog projection) Let D = m 0 , . . . , m n be a persuasion dialog, and

a i ∈ Ag. The projection of D on agent a i is D ai = m i1 , . . . , m i k such that 0 ≤ i 1 ≤ . . . ≤ i k ≤ n and ∀l ∈ [1, k], m i l ∈ D and Speaker(m i l ) = a i .
The contribution of each agent is defined as follows: It is easy to check that when the protocol under which a dialog is generated does not allow an agent to repeat an argument already given by another agent, then the sum of the contributions of the different agents is equal to 1.

Proposition 1 Let D = m 0 , . . . , m n be a persuasion dialog and a 1 , . . . , a n the agents involved in it. i=1,...,n Contr(a i , D) = 1 iff ∄m i , m j with 0 ≤ i, j ≤ n, such that Speaker(m i ) = Speaker(m j ), and Content(m i ) = Content(m j ).

It is worth noticing that the measure Contr is not monotonic. Indeed, the contribution of an agent may change during the dialog. It may increase then decreases in the dialog. However, at a given step of a dialog, the contribution of the agent that will present the next move will for sure increase, whereas the contributions of the other agents may decrease. Formally: Proposition 2 Let D = m 0 , . . . , m n be a persuasion dialog and a i ∈ Ag. Let m ∈ M such that Speaker(m) = a i . Then, Contr(a i , D ⊕ m) ≥ Contr(a i , D), and ∀a j = a i , Contr(a j , D ⊕ m) ≤ Contr(a j , D), with D ⊕ m = m 0 , . . . , m n , m .

Analyzing the behavior of agents

The behavior of an agent in a given persuasion dialog may be analyzed on the basis of three main criteria: i) its degree of aggressiveness in the dialog, ii) the source of its arguments, i.e. whether it builds arguments using its own formulas, or rather the ones revealed by other agents, and finally iii) its degree of coherence in the dialog.

The first criterion, i.e. the aggressiveness of an agent in a dialog, amounts to compute to what extent an agent was attacking arguments sent by other agents. Such agents prefer to destroy arguments presented by other parties rather than presenting arguments supporting their own point of view. Formally, the aggressiveness degree of an agent a i towards an agent a j during a persuasion dialog is equal to the number of its arguments that attack the other agent's arguments over the number of arguments it has uttered in that dialog.

Definition 14 (Measure of aggressiveness) Let D = m 0 , . . . , m n be a persuasion dialog, and a i , a j ∈ Ag. The aggressiveness degree of agent a i towards a j in D is

Agr(a i , a j , D) = |{α∈Args(D a i ) s.t. ∃β∈Args(D a j ) and (α,β)∈Confs(D)}| |Args(D a i )| 2 .
Example 4 Let D be a persuasion dialog between two agents a 1 and a 2 . Let us assume that

Args(D) = {α 1 , α 2 , β 1 , β 2 }, D a1 = {α 1 , α 2 } and D a2 = {β 1 , β 2 }.
The conflicts among the four arguments are depicted in the figure below.

α 1 α 2 β 1 β 2
The aggressiveness degrees of the two agents are respectively Agr(a 1 , a 2 , D) = 0/2 = 0, and Agr(a 2 , a 1 , D) = 1/2.

The aggressiveness degree of an agent changes as soon as a new argument is uttered by that agent. It decreases when that argument does not attack any argument of the other agent, and increases otherwise. Formally: Proposition 3 Let D = m 0 , . . . , m n be a persuasion dialog and a i , a j ∈ Ag. Let m ∈ M such that Speaker(m) = a i , and let D ⊕ m = m 0 , . . . , m n , m . Agr(a i , a j , D⊕m) ≥ Agr(a i , a j , D) iff ∃α ∈ Args(D aj ) such that (Content(m), α) ∈ R L , and Agr(a i , a j , D ⊕ m) < Agr(a i , a j , D) otherwise.

The second criterion concerns the source of arguments. An agent can build its arguments either from its own knowledge base, thus using its own formulas, or using formulas revealed by other agents in the dialog. In [START_REF] Amgoud | Strategical considerations for argumentative agents (preliminary report)[END_REF], this idea of borrowing formulas from other agents has been presented as one of the tactics used by agents for selecting the argument to utter at a given step of a dialog. The authors argue that by doing so, an agent may minimize the risk of being attacked subsequently. Let us now check to what extent an agent borrows information from other agents. Before that, let us first determine which formulas are owned by each agent according to what has been said in a dialog. Informally, a formula is owned by an agent, if this formula is revealed for the first time by that agent. Note that a formula revealed for the first time by agent a i may also pertain to the base of another agent a j . Here, we are interested by who reveals first that formula.

Definition 15 (Agent's formulas) Let D = m 0 , . . . , m n be a persuasion dialog, and a i ∈ Ag. The formulas owned by agent a i are: OwnF(a i , D) = {x ∈ L such that ∃m j with Speaker(m j ) = a i and x ∈ Support(Content(m j )) and ∄m k s.t k < j and Speaker(m k ) = a i and x ∈ Support(Content(m k )}. Now that we know which formulas are owned by each agent, we can compute the degree of loan for each participating agent. It may be tactically useful to turn an agents' arguments against him since they should be immune from challenge. This loan degree can thus help for evaluating the strategical behavior of an agent.

Definition 16 (Measure of loan) Let D = m 0 , . . . , m n be a persuasion dialog, and a i , a j ∈ Ag. The loan degree of agent a i from agent a j in D is:

Loan(a i , a j , D) = |Formulas(Args(D ai )) ∩ OwnF(D, a j )| |Formulas(Args(D ai ))|
The third criterion concerns the coherence of an agent. Indeed, in a persuasion dialog where an agent a i defends its point of view, it is important that this agent does not contradict itself. In fact, there are two kinds of self contradiction:

1. an explicit contradiction in which an agent presents at a given step of a dialog an argument, and later it attacks that argument. Such conflicts already appear in the argumentation system AS D a i = Args(D ai ), Confs(D ai ) associated to the moves uttered by agent a i . In other words, the set Confs(D ai ) is not empty. 2. an implicit contradiction that appears only in the complete version of that system, i.e. in CAS D a i .

In what follows, we will define a measure, a degree of incoherence, for evaluating to what extent an agent was incoherent in a dialog.

Definition 17 (Measure of incoherence) Let D be a persuasion dialog, a i ∈ Ag, and CAS D a i = A ai c , R ai c its complete system. The incoherence degree of agent a i in D is

Inc(a i , D) = |R ai c | |A ai c × A ai c |
Example 5 Let D be a persuasion dialog in which agent a 1 has uttered two arguments α 1 and α 2 . Let us assume that from the formulas of those arguments a third argument, say α 3 , is built. The figure below depicts the conflicts among the three arguments. The incoherence degree of agent a 1 is equal to 2/9.

α 1 α 2 α 3
Note that, the above definition is general enough to capture both explicit and implicit contradictions. Moreover, this measure is more precise than the one defined on the basis of attacked arguments, i.e. Inc(a i , D)

= |{β∈A a i c such that ∃(α,β)∈R a i c }| |A a i c |
. Using this measure, the incoherence degree of the agent in the above example is 1/3. Indeed, even if the argument α 1 is attacked by two arguments, only one conflict is considered.

It is easy to check that if an agent is aggressive towards itself, then it is incoherent. Property 3 Let D be a persuasion dialog, and a i ∈ Ag. If Agr(a i , a i , D) > 0 then Inc(a i , D) > 0.

The following example shows that the reverse is not always true. Similarly, it can be shown that if all the formulas of an agent a i are borrowed from another agent a j and that a i is aggressive towards a j , then a j is for sure incoherent.

Proposition 4 Let D be a persuasion dialog, and a i , a j ∈ Ag. If Loan(a i , a j , D) = 1 and Agr(a i , a j , D) > 0, then Inc(a j , D) > 0.

Measuring the dialog itself

In the previous sections, we have defined measures for evaluating the quality of arguments uttered in a persuasion dialog, and others for analyzing the behavior of agents involved in such a dialog. In this section, we define two other measures for evaluating the quality of the dialog itself. It is clear that the greater this degree is, the better the dialog. When the relevance degree of a dialog is equal to 1, this means that agents did not deviate from the subject of the dialog. However, this does not mean at all that all the moves have an impact on the result of the dialog, i.e. on the status of the subject. Another measure is then needed to compute the percentage of useful moves. Before introducing this measure, let us first define what is a useful move. The following definition is similar to the one used in [START_REF] Prakken | Coherence and flexibility in dialogue games for argumentation[END_REF]. It can be checked that the two notions of usefulness and relevance are closely related.

Proposition 5 Let D be a persuasion dialog, and m ∈ M. If m is useful in D, then m is relevant to the subject of D.

Note that the converse is not true as shown in the following example.

Example 7 Let us assume a dialog D whose subject is α 1 , and whose graph is the one presented below.

α 1 α 3 α 2 α 4
The set {α 1 , α 3 , α 4 } is the only preferred extension of AS D . It is clear that the argument α 4 is relevant to α 1 , but it is not useful for D. Indeed, the removal of α 4 will not change the status of α 1 which is skeptically accepted.

On the basis of the above notion of usefulness of moves, it is possible to compute the degree of usefulness of the dialog as a whole. It is worth noticing that according to the structure of the graph associated to the argumentation system of a dialog, it is possible to know whether the degrees of relevance and usefulness of that dialog are less that 1 or not. Formally:

Proposition 6 Let D be a persuasion dialog, and G be the graph associated with AS D . If G is not connected then Usefulness(D) < 1 and Relevance(D) < 1.

Conclusion

Several systems have been proposed in the literature for allowing agents to engage in persuasion dialogs. Different dialog protocols have then been discussed. These latter are the high level rules that govern a dialog. Examples of such rules are "how the turn shifts between agents", and "how moves are chained in a dialog". All these rules should ensure "correct" dialogs, i.e. dialogs that terminate and reach their goals. However, they 3 D \ m i =    m 0 , . . . m i-1 , m i+1 , . . . , mn if i = 0 and i = n m 1 , . . . , mn if i = 0 m 0 , . . . , m n-1 if i = n don't say anything on the quality of the dialogs. One even wonders whether there are criteria for measuring the quality of a dialog. In this paper we argue that the answer to this question is yes. Indeed, under the same protocol, different dialogs on the same subject may be generated, and some of them may be judged better than others. There are three kinds of reasons, each of them is translated into quality measures: i) the arguments exchanged are stronger, ii) the generated dialogs are more concise (i.e. all the uttered arguments have an impact on the result of the dialog), iii) the behavior of agents was "ideal". In the paper, the behavior of an agent is analyzed on the basis of three main criteria: its degree of aggressiveness, its degree of loan, and its degree of coherence. In sum, different measures have been proposed in this paper for the quality of dialogs. To the best of our knowledge, this is the first work on such measures in dialogs. Exceptions may be the works by Hunter [START_REF] Hunter | Towards higher impact argumentation[END_REF] and by Yuan et col. [START_REF] Yuan | An assessment of dialogue strategies for a human computer debating system, via computational agents[END_REF] on defining dialog strategies. For instance, Hunter has defined a strategy for selecting arguments in a dialog. The basic idea is that an agent selects the ones that will satisfy the goals of the audience. The agent is thus assumed to maintain two bases: a base containing its own beliefs, and another base containing what the agent thinks are the goals of the audience. These works are thus more concerned with proposing dialog strategies than with analyzing dialogs. An extension of this work would be the study of the general properties of protocols generating good dialogs w.r.t. the measures presented in this paper. Another future work consists of applying these measures to other types of dialogs, especially negotiation.

Definition 5 (

 5 Argument) An argument α is a pair α = H, h where h ∈ L and H ⊆ L.H is the support of the argument returned by the function H = Support(α), and h is its conclusion returned by the function h = Conc(α).

Definition 11 (

 11 Measure of dialog weight) Let D be a persuasion dialog. The weight of D is Weight(D) = αi∈Args(D) weight(α i ) It is clear that this measure is monotonic. Formally: Property 2 Let D be a persuasion dialog. ∀D j < D then Weight(D j ) ≤ Weight(D).

Definition 13 (Example 2 Example 3

 1323 Measure of agent's contribution) Let D = m 0 , . . . , m n be a persuasion dialog, and a i ∈ Ag. The contribution of agent a i in D is Contr(a i , D) = weight(αi) s.t. αi∈Args(D a i ) Weight(D) . Let us consider the persuasion dialog D presented in Example 1. Recall that Args(D) = {α 1 , α 2 , α 3 , α 4 }, D a1 = {α 1 , α , α 4 } and D a2 = {α 1 , α 2 }. Suppose now that an external agent wants to analyze this dialog. The function "weight" of this agent is as follow: weight(α 1 ) = 1, weight(α 2 ) = 4, weight(α 3 ) = 2 and weight(α 4 ) = 3. It is then clear from the definitions that the overall weight of the dialog is Weight(D) = 10. The contributions of the two agents are respectively Contr(a 1 , D) = 6/10 and Contr(a 2 , D) = 5/10. Consider now an example in which an agent sends several times the same argument. Let us consider a persuasion dialog D such that Args(D) = {α, β}. D a1 = {α} and D a2 = {β}. Let us assume that there are 50 moves in this dialog of which 49 moves are uttered by agent a 1 and one move uttered by agent a 2 . Suppose now that an external agent wants to analyze this dialog. The function "weight" of this agent is as follow: weight(α) = 1 and weight(β) = 30. The overall weight of the dialog is Weight(D) = 31. The contributions of the two agents are respectively Contr(a 1 , D) = 1/31 and Contr(a 2 , D) = 30/31.

Example 6

 6 Let D be a persuasion dialog, a i ∈ Ag. Let us assume that Args(D ai ) = {α 1 , α 2 }, and Confs(D ai ) = ∅. This means that Agr(a i , a i , D) = 0. Let CAS D a i = A ai c , R ai c be the complete version of the previous system with A ai c = {α 1 , α 2 , α 3 } and R ai c = {(α 3 , α 1 ), (α 3 , α 2 )}. It is clear that Inc(a i , D) = 2/9.

  The first measure checks to what extent moves uttered in a given dialog are in relation with the subject of that dialog. It is very common in everyday life, that agents deviate from the subject of the dialog. Before introducing the measure, let us first define formally the notion of relevance. Definition 18 (Relevant move) Let D = m 0 , . . . , m n be a persuasion dialog. A move m i=0,...,n is relevant to the subject of D iff there exists a path (not necessarily directed) from Subject(D) to Content(m i ) in the directed graph associated with AS D . Example 4 (continued): Let us consider the persuasion dialog given in Example 4. Suppose that Subject(D) = α 1 . It is clear that the arguments α 2 , β 1 are relevant, whereas the argument β 2 is not. On the basis of this notion of relevance, one can define a measure for knowing the percentage of moves that are relevant in a dialog. Definition 19 (Measure of relevance) Let D = m 0 , . . . , m n be a persuasion dialog. The relevance degree of D is Relevance(D) = |{m i=0,...,n such that m i is relevant }| |D| Example 4 (continued): In the previous example, Relevance(D) = 3/4.

Definition 20 (

 20 Useful move) Let D = m 0 , . . . , m n be a persuasion dialog. A move m i=0,...,n is useful in D iff Output(D) = Output(D \ m i ) where D \ m i is the dialog obtained by removing the move m i 3 .

Definition 21 (

 21 Measure of usefulness) Let D = m 0 , . . . , m n be a persuasion dialog. The usefulness degree of D is Usefulness(D) = |{m i=0,...,n such that m i is useful }| |D|

Definition 4 (Argument status) Let

  Now that the acceptability semantics are defined, we can define the status of any argument. As we will see, an argument may have one among three possible statuses: skeptically accepted, credulously accepted and rejected. A, R be an argumentation system, and E 1 , . . . , E n its extensions under stable (resp. preferred) semantics. Let α ∈ A.

• B is an admissible extension iff B defends all its elements; • B is a preferred extension iff it is a maximal (w.r.t. set-⊆) admissible extension; • B is a stable extension iff it is a preferred extension that attacks w.r.t. the relation R all arguments in A\B. Let E 1 , . . ., E n denote the possible extensions under a given semantics.

Corresponding Author: IRIT-CNRS, 118, route de Narbonne, 31062, Toulouse Cedex, France; E-mail: amgoud@irit.fr.

The expression |E| denotes the cardinal of the set E.