An introduction to argumentation semantics

The second author has been supported by the National Research Fund, Luxembourg (LAAMI project).

Please notice that terms like ‘preferred semantics’ or ‘ideal semantics’ correspond to existing terminology in the literature and do not imply any value judgments.

In Dung's theory, attack is a one-to-one relationship, which deviates from the earlier work of, for instance, Vreeswijk (1993), which is centered around the notion of collective attack, meaning that a set of arguments is collectively attacking another argument.

The original terminology in Dung (1995) was that an argumentA is acceptable with respect to a set of arguments \[--><$>{\cal A}rgs<$><!--\]. However, we find it more intuitive to say that an argumentA is defended by a set of arguments\[--><$>{\cal A}rgs<$><!--\].

If a set \[--><$>{\cal A}rgs<$><!--\] of arguments is not conflict-free \[--><$>{\cal A}rgs\: \cap \:{\cal A}rg{{s}^ + } <$><!--\] is not empty, that is, some argument would be labelled both in and out according to Ext2Lab\[--><$>({\cal A}rgs)<$><!--\].

We use the Definition of Caminada (2011). Note that clause 2 is needed for defining stage labellings (see Section 2.9).

Definition 21 is not literally the same as the one originally given by Dung (1995). We provide this equivalent version as more coherent with our presentation line.

An argumentation framework is finitary if every argument receives a finite number of attacks.

As a counterexample, consider an argumentation framework AF = ({A,B}, {A, B}, {B,A}). Let \[--><$>{\cal L}a{{b}_1}\: = \:(\{ A\} ,\{ B\},\,\emptyset), {\cal L}a{{b} _2}\: = \:(\emptyset ,\emptyset ,\{ A,B\} ), and {\cal L}a{{b}_3}\: = \:(\{ B\} ,\{ A\},\,\emptyset)<$><!--\]. It holds that \[--><$>{\cal L}a{{b}_1}\: \approx \:{\cal L}a{{b}_2}<$><!--\] and \[--><$>{\cal L}a{{b}_2}\: \approx \:{\cal L}a{{b}_3} but {\cal L}a{{b}_1}\:\not\approx \:{\cal L}a{{b}_3}<$><!--\].

This is because \[--><$>{\cal L}a{{b}_1}\: \approx \:{\cal L}a{{b}_2}<$><!--\] iff in\[--><$>({\cal L}a{{b}_1})\: \subseteq \: in({\cal L}a{{b}_2})\ \cup \: undec({\cal L}a{{b}_2}) and out({\cal L}a{{b}_1})\: \subseteq \: out({\cal L}a{{b} _2})\ \cup \:<$><!--\] undec\[--><$>({\cal L}a{{b}_2})<$><!--\].

The idea is to perform the skeptical judgment aggregation procedure of Caminada and Pigozzi (2011) on all preferred labellings.

Pollock (2001) discusses odd-length attack cycles in the context of a set of ‘reference’ inference graphs for testing the intuitive validity of justification status assignments. Actually, the paper where the problem is raised (Pollock, 2001) is mainly focused on an approach to reasoning with variable degrees of justification and does not provide an explicit ‘solution’ to this problematic example.

It can be remarked that all Dung's original semantics can be equivalently characterized using SCC-recursive definitions similar to Definition 33, as proved in Baroni et al. (2005).

This observation is immediate for all the considered semantics but ideal. The proof that an ideal labelling is also complete is given in Caminada (2011).

This is commonly called skeptical acceptance as it will be better discussed in Section 4.

Recall that the rejection property is defined only in the context of labelling-based approaches and that directionality implies non-interference, which in turn implies crash resistance.

A similar table is given in Baroni and Giacomin (2007); here, we add the treatment of stage semantics and the properties of cardinality, rejection, allowing abstention, crash resistance, and non-interference, while excluding prudent semantics and some variants of admissibility and reinstatement properties.

Note in particular that a partial order can be defined among different justification statuses both labelling-based and extension-based, for example as specified in Wu and Caminada (2010) and Baroniet al. (2004).

As recalled at the beginning of Section 4.1, we assume that an empty set of extensions/labellings does not support any justification status evaluation and therefore cannot be involved in skepticism comparison.

The set of stable extensions is empty in this case.

The skepticism relations described in the following have been analyzed in Baroni and Giacomin (2009b) for the extension-based approach. Due to the one-to-one correspondence between extensions and labellings holding for all the semantics involved in the comparison, it is possible to prove that the skepticism relations also hold in the labelling-based approach.

Since the empty set is an ideal set and the union of two ideal sets is an ideal set, as proved in Dung et al. (2007), it follows that the set of ideal sets is a non-empty partially ordered (w.r.t. inclusion) set whose totally ordered subsets have an upper bound (their union). Then, by Zorn's lemma, the set of ideal sets contains at least one maximal element. The maximal element is unique, since supposing that there are two distinct maximal ideal sets would contradict the fact that their union is an ideal set too.

Stable model semantics (Gelfond & Lifschitz, 1988,1991) has originally been stated in native logic programming terms. However, as it has been shown in Dung (1995), it is also possible to describe this approach using argumentation under stable semantics.

Restricted rebutting basically means that conclusion-based attacks can only be done against a conclusion that is the consequent of a defeasible reasoning step. Thus, in our example, A₈ attacks A₅ but A₅ does not attack A₈. The reader may refer to Caminada and Amgoud (2007) for more details.

This counter example was presented at COMMA 2010 and is available at: http://www.ing.unibs.it/comma2010/presentations/P15-Caminada.pdf