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Abstract dialectical frameworks (in short, ADFs) are one of the most general and unifying approaches to formal argumentation. As the semantics of ADFs are based on three-valued interpretations, the question poses itself as to whether some and which monotonic three-valued logic underlies ADFs, in the sense that it allows to capture the main semantic concepts underlying ADFs. As an entry-point for such an investigation, we take the concept of model of an ADF, which was originally formulated on the basis of Kleene’s three-valued logic. We show that an optimal concept of a model arises when instead of Kleene’s three-valued logic, possibilistic logic is used. We then show that in fact, possibilistic logic is the most conservative three-valued logic that fulfils this property, and that possibilistic logic can faithfully encode all other semantical concepts for ADFs. Based on this result, we also make some observations on strong equivalence and introduce possibilistic ADFs.

@misc{422,
  author = {Jesse Heyninck, Matthias Thimm, Gabriele Kern-Isberner, Tjitze Rienstra, Kenneth Skiba},
  title = {On the Relation between Possibilistic Logic and Abstract Dialectical Frameworks},
  abstract = {Abstract dialectical frameworks (in short, ADFs) are one of the most general and unifying approaches to formal argumentation. As the semantics of ADFs are based on three-valued interpretations, the question poses itself as to whether some and which monotonic three-valued logic underlies ADFs, in the sense that it allows to capture the main semantic concepts underlying ADFs. As an entry-point for such an investigation, we take the concept of model of an ADF, which was originally formulated on the basis of Kleene’s three-valued logic. We show that an optimal concept of a model arises when instead of Kleene’s three-valued logic, possibilistic logic is used. We then show that in fact, possibilistic logic is the most conservative three-valued logic that fulfils this property, and that possibilistic logic can faithfully encode all other semantical concepts for ADFs. Based on this result, we also make some observations on strong equivalence and introduce possibilistic ADFs.},
  year = {2021},
  url = {https://sites.google.com/view/nmr2021/home?authuser=0)},
}

Abstract dialectical frameworks (in short, ADFs) are a unifying model of formal argumentation, where argumentative relations between arguments are represented by assigning acceptance conditions to atomic arguments. This idea is generalized by letting acceptance conditions being assigned to complex formulas, resulting in conditional abstract dialectical frameworks (in short, cADFs). We define the semantics of cADFs in terms of a non-truth-functional four-valued
logic, and study the semantics in-depth, by showing existence results and proving that all semantics are generalizations of the corresponding semantics for ADFs.

@misc{421,
  author = {Jesse Heyninck, Matthias Thimm, Gabriele Kern-Isberner, Tjitze Rienstra, Kenneth Skiba},
  title = {Arguing about Complex Formulas: Generalizing Abstract Dialectical Frameworks},
  abstract = {Abstract dialectical frameworks (in short, ADFs) are a unifying model of formal argumentation, where argumentative relations between arguments are represented by assigning acceptance conditions to atomic arguments. This idea is generalized by letting acceptance conditions being assigned to complex formulas, resulting in conditional abstract dialectical frameworks (in short, cADFs). We define the semantics of cADFs in terms of a non-truth-functional four-valued
logic, and study the semantics in-depth, by showing existence results and proving that all semantics are generalizations of the corresponding semantics for ADFs.},
  year = {2021},
  url = {https://sites.google.com/view/nmr2021/home?authuser=0},
}

Approximation fixpoint theory (AFT) constitutes an abstract and general algebraic framework for studying the semantics of nonmonotonic logics. It provides a unifying study of the semantics of different formalisms for nonmonotonic reasoning, such as logic programming, default logic and autoepistemic logic. In this paper we extend AFT to non-deterministic constructs such as disjunctive information. This is done by generalizing the main constructions and corresponding results to non-deterministic operators, whose ranges are sets of elements rather than single elements. The applicability and usefulness of this generalization is illustrated in the context of disjunctive logic programming.

@{420,
  author = {Jesse Heyninck, Ofer Arieli},
  title = {Approximation Fixpoint Theory for Non-Deterministic Operators and Its Application in Disjunctive Logic Programming},
  abstract = {Approximation fixpoint theory (AFT) constitutes an abstract and general algebraic framework for studying the semantics of nonmonotonic logics. It provides a unifying study of the semantics of different formalisms for nonmonotonic reasoning, such as logic programming, default logic and autoepistemic logic. In this paper we extend AFT to non-deterministic constructs such as disjunctive information. This is done by generalizing the main constructions and corresponding results to non-deterministic operators, whose ranges are sets of elements rather than single elements. The applicability and usefulness of this generalization is illustrated in the context of disjunctive logic programming.},
  year = {2021},
  journal = {18th International Conference on Principles of Knowledge Representation and Reasoning},
  pages = {334-344},
  month = {03/11-12/11},
  publisher = {IJCAI Organization},
  address = {Online},
  isbn = {978-1-956792-99-7},
  url = {https://proceedings.kr.org/2021/32/},
  doi = {10.24963/kr.2021/32},
}

For propositional beliefs, there are well-established connections between belief revision, defeasible conditionals and
nonmonotonic inference. In argumentative contexts, such connections have not yet been investigated. On the one hand, the exact relationship between formal argumentation and nonmonotonic inference relations is a research topic that keeps on eluding researchers despite recently intensified efforts, whereas argumentative revision has been studied in numerous works during recent years. In this paper, we show that similar relationships between belief revision, defeasible conditionals and nonmonotonic inference hold in argumentative contexts as well. We first define revision operators for abstract dialectical frameworks, and use such revision operators to define dynamic conditionals by means of the Ramsey test. We show that such conditionals can be equivalently defined using a total preorder over three-valued interpretations, and study the inferential behaviour of the resulting conditional inference relations.

@{418,
  author = {Jesse Heyninck, Gabriele Kern-Isberner, Tjitze Rienstra, Kenneth Skiba, Matthias Thimm},
  title = {Revision and Conditional Inference for Abstract Dialectical Frameworks},
  abstract = {For propositional beliefs, there are well-established connections between belief revision, defeasible conditionals and
nonmonotonic inference. In argumentative contexts, such connections have not yet been investigated. On the one hand, the exact relationship between formal argumentation and nonmonotonic inference relations is a research topic that keeps on eluding researchers despite recently intensified efforts, whereas argumentative revision has been studied in numerous works during recent years. In this paper, we show that similar relationships between belief revision, defeasible conditionals and nonmonotonic inference hold in argumentative contexts as well. We first define revision operators for abstract dialectical frameworks, and use such revision operators to define dynamic conditionals by means of the Ramsey test. We show that such conditionals can be equivalently defined using a total preorder over three-valued interpretations, and study the inferential behaviour of the resulting conditional inference relations.},
  year = {2021},
  journal = {18th International Conference on Principles of Knowledge Representation and Reasoning},
  pages = {345-355},
  month = {03/11-12/11},
  publisher = {IJCAI Organization},
  address = {Online},
  isbn = {978-1-956792-99-7},
  url = {https://proceedings.kr.org/2021/33/},
  doi = {10.24963/kr.2021/33},
}