Abstract
This chapter aims at providing logical encodings for translating interactions in an argumentation graph themselves into propositional knowledge bases. This translation will be used for identifying or redefining some properties of argumentation graphs. The graphs we consider are used to formalize abstract argumentation with at least two different kinds of interaction (attack and support) and also recursive interactions.
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- 1.
- 2.
An extension of this work is presented in (Nouioua 2013) but it is not discussed here.
- 3.
Related to a given semantics, whatever this semantics.
- 4.
The notation \({\texttt {n+attack}}\) means that necessary supports (n) are combined with (+) attacks (attack).
- 5.
In a context of preferences, it is a way for choosing between two conflicting positions.
- 6.
This assumption is done only for a simplification of the presentation. Nevertheless, it can be easily removed and the definition extended in order to take into account several supports and attacks to a given interaction by replacing \((c \wedge A\beta )\) (resp. \((\lnot d \vee \lnot A\delta )\)) by \(\bigwedge _i(c_i \wedge A\beta _i)\) (resp. \(\bigwedge _j(\lnot d_j \vee \lnot A\delta _j)\)).
- 7.
In case of a Case 2-\({\texttt {n+attack}}\) from a to b, such as \(a {\mathbf{R}}_{\text {att}}c\) and \(c {\mathbf{R}}_{\text {sup}}b\), \(\varSigma \) contains the formulas \(a \rightarrow {\mathtt{Att}}c\), \({\mathtt{Att}}c \rightarrow \lnot c\), \(b \rightarrow {\mathtt{Sup}}c\) and \({\mathtt{Sup}}c \rightarrow c\). So \(\varSigma \) infers \(a \rightarrow {\mathtt{Att}}c\) and \(b \rightarrow \lnot {\mathtt{Att}}c\), but \(\varSigma \) does not infer \(a \rightarrow {\mathtt{Att}}b\) since the propositional variable \({\mathtt{Att}}b\) does not appear in \(\varSigma \).
- 8.
Indeed, Case 2-\({\texttt {n+attacks}}\) cannot be detected using our language (see the footnote on Proposition 18.2).
- 9.
Transitivity: the relation \({\mathtt{support}}\) is transitive.
- 10.
Closure: if \(c {\mathbf{R}}_{\text {sup}}b\), then “the acceptance of b implies the acceptance of c”.
- 11.
Conflicting sets: from a logical point of view, from “the acceptance of a implies the non-acceptance of b” using contrapositives, we obtain a symmetric conflict involving a and b; however this kind of conflict is not equivalent to the existence of an attack.
- 12.
Addition of new attacks: a stronger constraint that CFS as it enables to add new attacks which are \({\texttt {n+attacks}}\).
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Cayrol, C., Fariñas del Cerro, L., Lagasquie-Schiex, MC. (2018). A Propositional Logical Encoding of Enriched Interactions in Abstract Argumentation Graphs. In: Golińska-Pilarek, J., Zawidzki, M. (eds) Ewa Orłowska on Relational Methods in Logic and Computer Science. Outstanding Contributions to Logic, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-97879-6_18
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