Gödel semantics of fuzzy argumentation frameworks with consistency degrees

Argumentation frameworks (AF) play important roles in artificial intelligence. This paper is an exploration in establishing semantics of fuzzy AFs by fuzzy sets. There are many ways to characterize the semantics of fuzzy AFs. In this paper, our work is based on the assumption that some inconsistency of the system is permitted. Firstly, we formalize the conflict-freeness with a consistency degree x and the acceptability with a consistency degree y. Various types of extensions are then defined in a way similar to Dung's approach. The conflict-freeness and acceptability can be seen as an interpretation of the corresponding notion in Janssen's work. Formally, we add the conflict-freeness into the admissible extensions and the preferred extensions. We also introduce the complete extensions and the grounded extensions. Moreover, some basic properties are proven, such as the Fundamental Lemma, the algorithm of the grounded extension, etc. At last, it is proven to be consistent with Dung's original semantics in crisp AFs. Keywords: argumentation frameworks; fuzzy arguments; fundamental lemma; characteristic function; fuzzy argumentation frameworks Mathematics Subject Classification: 03E72, 03E75