{"type":"ScholarlyArticle","title":"Bipolar Abstract Dialectical Frameworks Are Covered by Kleene’s Three-valued Logic","authors":[{"name":"Ringo Baumann","slug":"ringo-baumann","given":"Ringo","family":"Baumann","affiliation":"Computer Science Institute , Leipzig University , Germany; Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI) Dresden / Leipzig , Germany"},{"name":"Maximilian Heinrich","slug":"maximilian-heinrich","given":"Maximilian","family":"Heinrich","affiliation":"Intelligent Information Systems , Bauhaus-Universität Weimar , Germany"}],"abstract":"Abstract dialectical frameworks (ADFs) are one of the most powerful generalizations of classical Dung-style argumentation frameworks (AFs).\r\nThe additional expressive power comes with an increase in computational complexity, namely one level up in the polynomial hierarchy in comparison to\r\ntheir AF counterparts. However, there is one important subclass, so-called bipolar ADFs (BADFs) which are as complex as classical AFs while offering strictly more modeling capacities. This property makes BADFs very attractive from a knowledge representation point of view and is the main reason why this class has received much attention recently. The semantics of ADFs rely on the Gamma-operator which takes as an input a three-valued interpretation and returns a new one. However, in order to obtain the output the original definition requires to consider any two-valued completion of a given three-valued interpretation. In this paper we formally prove that in case of BADFs we may bypass the computationally intensive procedure via applying Kleene's three-valued logic K. We therefore introduce the so-called bipolar disjunctive normal form which is simply a disjunctive normal form where any used atom possesses either a positive or a negative polarity. We then show that: First, this normal form is expressive enough to represent any BADF and secondly, the computation can be done via Kleene's K instead of dealing with two-valued completions. Inspired by the main correspondence result we present some first experiments showing the computational benefit of using Kleene.","doi":"10.24963/ijcai.2023/348","url":"https://ijcai.org/proceedings/2023/348","pdf":"https://ijcai.org/proceedings/2023/0348.pdf","markdown":"https://ijcai.org/proceedings/2023/348.md","bibtex":"https://ijcai.org/proceedings/2023/bibtex/348","volume":{"slug":"2023","title":"Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2023","url":"https://ijcai.org/proceedings/2023/","year":2023,"location":"Macao, SAR","dates":"19-25 August 2023","editors":"Edith Elkind","isbn":"978-1-956792-03-4","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2023"},"section":"Knowledge Representation and Reasoning","track":"Main Track","number":348,"pages":{"first":3123,"last":3131},"keywords":["Knowledge Representation and Reasoning: KRR: Argumentation","Knowledge Representation and Reasoning: KRR: Non-monotonic reasoning"],"videos":[],"references":[{"position":1,"raw":"[Alviano et al., 2016] Mario Alviano, Wolfgang Faber, and Hannes Strass. 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