Abstract argumentation allows to determine in an easy, formal way which claims survive in a con icting dispute. It works by considering claims as abstract entities, and expressing attack relationships among them. However, this level of expressiveness prevents abstract argumentation systems from being directly applied to reasoning processes where the context is relevant. An outstanding example is when a claim is supported by appealing to authority, so that the audience assigns reliability to the claim's justi cation based on the authority's renowned experience in the domain. To handle this, we propose to enrich the classical representation used in abstract argumentation by associating arguments with weights that express their degree of authority. The weights' values de ne their strength in the given domain, which in turn should a ect the evaluation of their degree of justi cation. This paper de nes a strategy to combine these weights in order to determine which arguments withstand in a dispute concerning a given domain. Such a strategy was implemented in the ARCA system, that allows to comfortably set up argumentation problems and solve them using both traditional extension-based semantics and the proposed evaluation approach. ARCA is used to illustrate the proposed strategy by means of sample use cases.
Argumentation is a major component of our everyday lives, in that we are continuously faced with con icting information and associated inconsistencies. Roughly, an argument is a bunch of information (i.e., a set of assumptions ) from which conclusions can be drawn, based on a number of reasoning steps. The assumptions used are called the premises of the argument, while its conclusion (chosen from many possible ones) is called the claim of the argument. The support of an argument provides the reason (or, equivalently, a justi cation) for the claim of the argument. This structure simpli es understanding of the opinions of other people and helps in the identi cation of fallacies in their reasoning. People usually argue in turns, by providing arguments and counterarguments to initial arguments. The winner of the argumentation is the arguer with the last unchallenged argument.
Many strategies can be found in the literature for the identi cation of the successful arguments in an argumentation dispute context. Some such strategies are based on the so-called Abstract Argumentation Framework, that will be presented in the next sections. This model of argumentation takes a set of abstract arguments, i.e., arguments whose internal structure or speci c interpretation is ignored. The abstract nature of the arguments, and the relationship with nonmonotonic reasoning formalisms, yield a very simple and quite general model that allows to easily understand which sets of arguments are mutually compatible. Unfortunately, abstract system representations are not always suitable to depict real situations. This is because abstract systems lack of elements which can empower the representation setting so that con icts can be automatically identi ed or the strength of a con ict can be determined. For example, the abstract argumentation framework does not allow to consider the weight of each argument based on the authority of the person who claims it, which may be relevant to the proper evaluation process of judging an argument.
Sometimes it may be appropriate to cite an authority to support a position.
This argumentative schema is known as argument from authority, or
\argumentum ad verecundiam" [
This work proposes a novel approach to handle these situations, that extends the abstract argumentation setting by allowing the association of arguments to weights expressing their reliability. Such weights are assigned on the basis of an authority degree which takes into account the reliability of the authority who states the argument in the argued domain. The objective is to overcome the low level of expressiveness that characterizes the standard abstract argumentation framework, and to make it able to handle di erent degrees of reliability on the arguments.
This paper is organized as follows. The next section recalls useful background information, including related works. Then, Section 3 introduces the abstract argumentation framework along with the standard evaluation strategies used in the process of justifying an argument leading to a conclusion. Section 4 describes the proposed approach and how it is embedded in the standard abstract evaluation system, and Section 5 concludes the paper and outlines future work issues. 2
As a general, informal de nition, argumentation involves the identi cation of applicable assumptions and conclusions for a given problem under consideration. In this activity, it often faces con icting information, which results in the need to evaluate the justi cation for the available conclusions. This, in turn, may involve comparing arguments, evaluating them in some aspects, and judging a set of arguments and counterarguments to consider whether any part of them can be considered as warranted according to some standard principle. In this context, it can be also safely assumed that each argument has a proponent, who is the person putting forward the argument, and that each argument has an audience, who is the group of people reached by the argument.
Probably the foundational and most important philosophical work for the
development of argumentation was made by Toulmin [
In order to handle arguments systematically, a \formalization" of
argumentation is needed. Many professions implicitly or explicitly explore these issues
and, in facts, put the systematic use of arguments at the heart of their work.
Outstanding examples can be found in the legal, medical, and journalistic
professions. The study of formal argumentation started among critical thinking and
practical reasoning philosophers [
However, the need to go beyond the systematic handling of arguments
motivates the search for techniques that are able to scale up and deal with
substantial and complex problems. Classical logic is appealing as a starting point
for argumentation: it provides a rich representation formalism and powerful
reasoning mechanism. Unfortunately, inconsistency causes problems in reasoning
with classical logic [
As a partial response to the issue of inconsistency arising in argumentation,
three main approaches to formalization for argumentation have been proposed
in the literature, namely: abstract systems [
In particular, abstract systems build on the seminal proposal by Dung [
The argumentation literature emphasized the importance of considering
additional criteria, namely preferences, when evaluating arguments in a framework.
Preferences are expressed between arguments and re ect their relative strengths.
In [
A rst introduction to weighted attack relations in an argumentation
framework can be found in [
Another early extension of Dung's proposal with weights is Value-based
Argumentation Frameworks (VAFs) [
A more general approach to extending Dung's proposal is that of Bipolar
Argumentation Frameworks (BAFs), which take into account two kinds of
interaction between arguments: a positive interaction (by which an argument can help
or support another argument) and a negative interaction (by which an argument
can attack another argument) [
An abstract argument system or Argumentation Framework (AF for short), as introduced by Dung, is a pair hA; Ri consisting of a set A, whose elements are called arguments, and a binary relation R A A on A, called attack relation. Given two arguments ; 2 A, the relation R represents an attack from against . In general, arguments and are in con ict if argument refutes argument or if is attacking premises supporting . More precisely, we talk about: { Rebutting, when there is an explicit contradiction between conclusions ; or { Undercutting, when argument attacks the applicability of a rule that supports , without necessarily denying it.
An AF has a typical representation as a directed graph where nodes are arguments and edges are drawn from attacking to attacked arguments. Representing the structure and meaning of arguments at so high a level of abstraction allows to better focus on properties that are independent from any speci c context, and makes it applicable to a wide variety of domains. On the other hand, this formalism lacks of expressiveness, which prevents its direct application in any speci c domain. Indeed, in order to set up an AF one rst needs to build an underlying knowledge base, along with mechanisms to generate the set of arguments from it and determine in which ways these arguments attack each other. Then, once the AF has been set up, a second issue is how to determine a justi cation state for the involved arguments and, in particular, how to identify which are the justi ed ones. Informally, an argument is considered to be justi ed if it survives to attack relations. Therefore, the next step is to understand which argument is not defeated from the confrontation with the others. This process, called argument evaluation, aims at determining the justi cation state of the arguments in an abstract argumentation system.
An argumentation semantics is the formal way of determining which arguments or statements can be considered as justi ed in the argument evaluation process. Two main approaches to the de nition of argumentation semantics are available in the literature: the labelling-based one and the extension-based one. In the former, the idea is to de ne a mapping that associates each argument to one of a set of labels corresponding to the possible states of argumentsin the given context. A sensible choice for the set of labels is: { in for the accepted arguments { out for the rejected arguments { undec for unde ned (not accepted or refused) arguments.
The labeling operation can be seen as the result of the reasoning carried out by an agent which analyzes the arguments and marks them as justi ed, rejected or temporarily undecided. One of the bene ts derived from the use of labellingbased semantics is the possibility of de ning a more re ned defeat-status by introducing di erent levels of justi cation and rejection (e.g., `very acceptable', `quite acceptable', `not acceptable').
In the extension-based approach the idea is to derive, from an AF, an `extension' E, that is a subset of A representing a set of argument which are considered as acceptable. These semantics can assign each node to a single status (unique-status ) or multiple statuses (multiple-status ). The di erence is in the management of the temporarily undecided state. A multiple-status semantics can resolve a mutual attack issue by generating two hypothetical solutions in which the con icting arguments can be alternately assumed as acceptable.
By considering the expressiveness of the two approaches to semantics, it can be observed that any extension-based semantics can be equivalently translated in a labelling-based one by adopting a set of two labels in and out that correspond to extension membership. Vice versa, in general an arbitrary assignment of labels cannot be translated in terms of extensions. This is because labellings always include a label that corresponds to the extension membership, while other labels are derivable from extension membership and the attack relation. Consequently, equivalent extension-based de nitions of labelling-based semantics are in general applicable. This is the reason why extension-based semantics are more widely exploited in the literature.
A basic requirement for any extension E is derived from its interpretation as a set of arguments which can survive together. In other words, if an argument attacks another argument , one reasonably does not expect to have them together in the same extension. This corresponds to the concept of con ict-free that is at the basis of all extension-based semantics.
De nition 3.1 (con ict-free) Given an Argumentation Framework AF = hA; Ri, a set S A is con ict-free i @ ; 2 S s.t. R ( attacks ).
A second requirement corresponds to the need of a set of arguments to resist the attacks it receives from other arguments by counterattacking them. This feature is based on the notions of acceptable argument and admissible set. De nition 3.2 (acceptability) Given an Argumentation Framework AF = hA; Ri, an argument 2 A is acceptable wrt a set S A i 8 2 A : R ) 9 2 S s.t. R ( is defended by S).
De nition 3.3 (admissibility) Given an Argumentation Framework AF = hA; Ri, a set S A is admissible i S is con ict-free and 8 2 S is acceptable wrt S.
Now suppose that the attackers of an argument are all attacked by an extension E. Then the attack su ered by is canceled because E is `defending' , and is reinstated because it should belong to E. This property takes the name of reinstatement and leads to the following principle:
work AF = hA; Ri, a semantics satis es the reinstatement principle i for all extensions E A it holds that if is acceptable w.r.t E then 2 E. Since semantics provide the basis for evaluating the justi cation state of arguments, one may rst require that the evaluation basis of an AF is not empty. Some (labelling- or extension-based) semantics may allow many alternative justi cation states for the arguments. Two main alternatives may be considered for the notion of justi cation state: { skeptical justi cation requires that an argument is accepted in all semantics; { credulous justi cation requires that an argument is accepted in at least one semantics.
Of course in a unique-status approach credulous and skeptical justi cations coincide, but in multiple-status approaches typically the credulous justi cation includes the skeptical justi cation.
Let us now consider some approaches to determine argumentation semantics proposed in the literature.
Complete semantics The notion of complete extension is based on the principles of admissibility and reinstatement. It is a set which is able to defend itself and includes all arguments it defends.
AF = hA; Ri, a set S A is complete extension i S is admissible and @ 2 A such that: { { is acceptable wrt S 2= S
The following semantics build their own extensions referring to the complete extensions.
Ground semantics For each AF there exists only one ground extension which corresponds to the set of arguments that satis es the conditions of admissibility and that is minimal with respect to the inclusion relation between the admissible sets of AF. Compared to complete extensions, the ground is the complete minimal one with respect to set inclusion.
De nition 3.6 (ground extension) Given an Argumentation Framework AF = hA; Ri, a set S A is a ground extension i S is admissible and S is a minimal subset of A.
Preferred semantics A preferred extension S of an AF is the admissible set of AF which is maximal with respect to set inclusion. For each admissible set E of AF there exists at least one preferred extension S such that E S (it can be also the empty set). Compared to complete extensions, the preferred extension is the complete maximal one with respect to set inclusion.
AF = hA; Ri, a set S A is a preferred extension i S is admissible and S is a -maximal subset of A.
Stable semantics A stable extension of an AF is a complete extension which attacks all arguments that are not its members. Any stable extension is also a maximal con ict-free set of AF.
De nition 3.8 (stable extension) Given an Argumentation Framework AF = hA; Ri, a set S A is a stable extension i { S is a complete extension { S [ Sdefeated = A where Sdefeated = f 2 A j 2 E ^
R g
Semi-stable semantics A semi-stable extension [
AF = hA; Ri, a set S A is a stable extension i { S is a complete extension { S [ Sdefeated is maximal wrt A where Sdefeated = f 2 A j 2 E ^
R g Ideal semantics An extension of an AF is called ideal if it corresponds to the largest admissible set that is a subset of each preferred extension. De nition 3.10 (ideal extension) Given an Argumentation Framework AF = hA; Ri, a set S A is an ideal extension i S is the admissible -maximal subset of A such that 8Spreferred : S Spreferred.
Eager semantics An eager extension of an AF corresponds to the largest admissible set that is a subset of each semi-stable extension. It relies on a concept that is similar to the ideal semantics, with the restriction that the admissible set must be in the intersection of semi-stable extensions.
De nition 3.11 (eager extension) Given an Argumentation Framework AF = hA; Ri, a set S A is an eager extension i S is the admissible -maximal subset of A such that 8Ssemi stable : S Ssemi stable.
An ordering relationship exists among the semantics described above [
The semantics shown above neglect information that, in some cases, may turn out to be of crucial importance to the argumentation. For instance, the Abstract Argumentation Framework does not distinguish between rebuttal and undercutting attacks, in order to provide more e cient computation of extensions. Also, in some cases, it would be advisable to evaluate the set of reliable arguments by taking into account the context in which the sentences are claimed, and speci cally the trustworthiness of those who claim them. Adding quantitative information becomes of crucial importance when arguments have di erent levels of strength. Hence, adding a weight to arguments allows to give them the right strength, so as to represent real dialogues.
A rst re nement to deal with this scenario could be to distinguish utterances made by domain experts from those made by novices or by outsiders of the domain of the argumentation. By domain we mean a context in which a person is skilled. The more con dent a person within a domain, the higher his authority in that domain. For example, in a wine and food context, the opinion or contradiction of a mathematician has a minor signi cance compared to that of a winemaker of unquestionable professionalism. Conversely, in a mathematical context the winemaker level of reliability should be less than that of the mathematician. This degree of reliability might be captured in an Argumentation Framework by introducing an authority degree associated to nodes, such that two nodes reporting utterances made by experts in di erent domains will have di erent weights into attacking the same node. Thus, arguments are partitioned in domains which re ect the area of expertise of each arguer.
A second re nement might be to consider the number of attackers and defenders for a node in the graph. In abstract argumentation terms, the larger the number of attackers of a node, the more likely it is that it should be defeated, and, conversely, the larger number of defenders (i.e., of reinstatements) it has, the more likely it is that it may be admissible.
In general, our proposal is to associate a `social ' weight to attack relations,
preferring large admissible sets which attack external nodes and at the same time
defend their members, rather smaller sets of isolated admissible nodes. In this
setting, the domain-based authority degree works as `normalizer ': it rebalances
weights with the aim of avoiding over-defended conclusions. Another novelty of
our approach is that the authority degree may di er depending on the intended
domain. This is di erent than in PAFs [
The setting we propose is de ned through a number of functions to be used in the evaluation strategy of justi cation. In the following, hA; Ri will indicate an argumentation framework and the authority degree for an argument 2 A in its domain . A domain function allows us to focus on the most represented domains in the framework rather than on niche domains:
( ) =
N A j j with N number of arguments having domain .
The domain function acts as a moderator to balance the weights of the attacking nodes within the argued context. In fact, this ensures that the arguments in support of the discussed domain have more relevance.
In the following, we propose three functions which allow to consider the strength of a group of persons involved in the same domain.
An attacking function returns the number of attacks launched by an argument towards other nodes.
fa( ) = jU ( )j; where U ( ) = f 2 A j R ^ ( ) g
A defeating function returns the number of attacks an argument su ers from other nodes.
fd( ) = jE( )j; where E( ) = f 2 A j R ^ < ( )g
A defending function returns the number of attacks su ered by an argument that are not defended by other su ciently reliable arguments.
fr( ) = jD( )j; where D( ) = f 2 A j R ^ 9 2 A s:t: R ^ ( ) g
In the last three functions described above, the domain function serves to support the attacking node in order to contextualize the weight of the attack compared to the weight of the attacked node.
Now, let call us each argument 2 A as authority-node wrt an argument 2 A, that argument which launches attacks towards other nodes 2 A such that ( ) . An authority function measures the degree of an argument 2 A based on the number of attacks launched as authority-node wrt 2 A and the number of attacks su ered by other arguments being authority-nodes against it.
fauthority( ) = fa( ) fd( ) + fr( ) The more attacks launched by 2 A as an authority-node wrt 2 A, the higher its fauthority; the more attacks it su ers by arguments which are authority-nodes against it, the lower its fauthority. Intuitively, if the attacks su ered by a node are defended by other authority-nodes, then its authority degree will depend only on its successful attacks (i.e. attacks towards less reliable nodes). Indeed, the su ered attacks, which decrease the value of its authority, are balanced by the number of attacks from which it is defended by authority-nodes. Hence, the authority function is used to select the `stronger' admissible set: namely the most reliable set will be the one with the lower number of nodes such that their authority function value is maximum. In this setting, none of the classical Dung's extensions are considered, but only the collection of admissible sets, ordered according to the value of their fauthority. Then, the smallest admissible set with highest value of the authority function is chosen as more reliable justi ed set. Applying a more skeptical semantics may limit the aim of this paper because we would lose the sense of the domain's weight associated to nodes. In general, an attack is considered valid if the attacker's authority (decreased by a domaindependent factor) is higher than the attacked authority.
Fig. 1 shows an example of a graph depicting an AF. Node labels indicate the level of authority. Nodes in gray belong to domain 0 and those in white to domain 00. Due to space constraints, we will determine in the following the authority degree for node only, using the above functions.
8 3
7 4
9
Let consider the node , it's domain value for domain 0 is ( ) = 52 = 0:4; fa( ) = 0 because 3 0.4 = 1.2 s.t. 1.2 8 and 1.2 9; fd( ) = 1 because 3 < 8 0.6 = 4.8; fr( ) = 0 because node su ers no attack; fauthority( ) = 0 - 1 + 0 = -1.
The authority functions for the remaining nodes are calculated in the same way. Then, the smallest admissible set, having the maximum sum of authority functions values for its members, is chosen as the most reliable justi ed subset of arguments. 4.2
The proposed strategy was implemented in the ARCA system (acronym for Abstract Resolution of Con icts in Argumentation ). ARCA includes a logic program core which can compute both the classical extension based semantics and the proposed authority degree-based evaluation of arguments. On top of it, ARCA provides a graphic tool (see Fig. 2) to enter and display arguments and attacks in an AF, and to associate to each claim a domain of origin and a degree of experience in that domain. It also allows to associate and display utterances associated to arguments and to show nodes with di erent colors denoting their acceptability according to the di erent semantics. This allows one to easily see con icts and observe the di erences between di erent semantic extensions. Let us now show the various features of ARCA using a sample scenario in which some professionals olive growers are arguing about the most useful criterion of olive trees pruning. In mature trees, pruning is mainly required to renew the fruiting surface of the tree and achieve high yields, maintain vegetative growth of sprouts, maintain the skeleton structure, contain tree size, favor light penetration and air circulation inside the canopy, permit control of pests and diseases, prevent aging of the canopy, and eliminate dead wood.
Three novice croppers, Albert, John and Jack, are expressing their point of view, according their own (limited) experience: 1. Albert: \When in doubt, less pruning is better." 2. John: \Not all trees in a grove need to be pruned every year." 3. Jack: \Pruning should be rapid and simple."
Samuel, a renowned expert olive grower, counterargues all three statements according to his large experience: \The type of pruning must be adjusted in relation to plant age, training system, crop load, product use, environmental conditions, soil fertility, and farm structure." Thus, Samuel's opinion has more relevance than those of the three novice growers. Hence, his authority degree is such that the attacks he is su ering have no e ect in the argumentation. In fact, Fig. 3 shows that, among all admissible subsets, the one with higher rank is precisely the set containing Samuel's (winning) argument.
Now, suppose Julian, an agronomist, takes part in the discussion. As a person with special knowledge in soil management and eld-crop production, he explains in which direction new techniques in this domain are going and, therefore, which strategy is better to accomplish all aims: \Current tendency is to prune olive trees as little as possible, so as to reduce costs substantially and simplify pruning management." This sentence attacks Samuel's claim and generates circular con icts between Julian's claim and the novice growers' statements. Since Julian has less practical experience in pruning olive trees, his claim has less weight in the argumentation. In this situation, the evaluation of extension-based semantics and the ARCA solution are quite di erent. In skeptical extension-based semantics such as Semi-Stable and Stable extensions (Fig. 4 (a)), the admissible undefeated set of arguments includes both the claims of novice growers, and the agronomist's one. In the ARCA solution (see Fig. 4 (b)), the weight of the agronomist in the argumentation is such that his argument is undefeated, despite his weight is lower than Samuel's one in the argued domain.
Suppose now that Samuel counterattacks also Julian's claim with another argument: \Pruning should be more severe on old trees and trees of low vigor than on young plants, or on trees growing in irrigated conditions and in fertile soils. The authority weight of Samuel's last argument determines the winning arguments in the ARCA solution. Indeed, in Fig. 5 the two arguments expressed by Samuel are a subset of admissible elements which has a higher rank in the ARCA solution.
This sample scenario clearly shows how the weighted evaluation strategy of justi ed arguments may lead to more reliable and sensible results when the context domain is relevant. 5
Abstract Argumentation is a formal approach to de ne which claims withstand in a dispute, in which the only expressed property is a binary `attack' relation representing the rebuttal of an argument to another. The aim is determining an evaluation strategy that allows to justify con icting arguments. While several such strategies have been de ned, not always they are useful or sensible. This is due to the low level of expressiveness of abstract systems which doesn't allow to represent all relevant contextual situations. E.g., using an appeal to authority, claims may have di erent weights in the discourse, and justi cations might be evaluated di erently. In turn, the weight of a claim depends on the degree of experience (i.e., the authority) of the person expressing it in a particular domain. This work proposed a novel evaluation strategy which may take into account the authority degree of arguments in a given domain in order to understand which arguments survive in a debate. It was implemented in the ARCA system, that allows easily to set up abstract argumentation frameworks and solve justi ed arguments with both classical extension-based semantics and the proposed authority-weighted approach. A sample scenario is used to illustrate how ARCA works and how the proposed strategy is useful to best handle a real-world argumentation problem in which the reliability of claims is signi cant.
As future work, we will investigate the adoption of modal logics depending on the actor's domain membership. E.g., deontic logic in legal eld or temporal logic in historical-literary eld. We would also like to implement (semi)automatic analysis of arguments expressed in natural language, so as to help the users of ARCA in setting up their argumentation frameworks. Speci cally concerning the use of weights, we would like to extend the abstract model with the use of Value-based Argumentation Frameworks in which the justi cation of an argument depends on the social values that it advances, and the process of justi cation of one argument depends on the strength of the values involved in the argumentation. Also the extension to Bipolar Argumentation Frameworks will be considered, so as to distinguish the sense of `support' to an argument from the sense of `defense' of an argument.
This work was partially funded by the Italian PON 2007-2013 project PON02 00563 3489339 `Puglia@Service'.