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Structural constraints for dynamic operators in abstract argumentation

Johannes P. Wallner

2019
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Argument & Computation
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Many recent studies of dynamics in formal argumentation within AI focus on the well-known formalism of Dung's argumentation frameworks (AFs). Despite the usefulness of AFs in many areas of argumentation, their abstract notion of arguments creates a barrier for operators that modify a given AF, e.g., in the case that dependencies between arguments have been abstracted away that are important for subsequent modifications. In this paper we aim to support development of dynamic operators on formal
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... odels in abstract argumentation by providing constraints imposed on the modification of the structure that can be used to incorporate information that has been abstracted away. Towards a broad reach, we base our results on the general formalism of abstract dialectical frameworks (ADFs) in abstract argumentation. To show applicability, we present two cases studies that adapt an existing extension enforcement operator that modifies AFs: in the first case study, we show how to utilize constraints in order to obtain an enforcement operator on ADFs that is allowed to only add support relations between arguments, and in the second case study we show how an enforcement operator on AFs can be defined that respects dependencies between arguments. We show feasibility of our approach by studying the complexity of the proposed structural constraints and the operators arising from the case studies, and by an experimental evaluation of an answer set programming (ASP) implementation of the enforcement operator based on supports. Keywords: Abstract argumentation, dynamics of argumentation, argumentation frameworks, abstract dialectical frameworks, structured argumentation, computational complexity, constraints, answer set programming This article is published online with Open Access and distributed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC 4.0). Example 3. Consider the AF F in Example 2. For the considered semantics σ we show the corresponding extensions in Table 1 . Abstract dialectical frameworks We recall ADFs from [30] , which are based on earlier works [32] . We begin with basics from propositional logic and three-valued interpretations. Let A be a finite set of arguments (statements). An interpretation is a function I mapping arguments to one of the three truth values I : A → {t, f, u}. That is, an interpretation maps each argument to either true (t), false (f), or undefined (u). An interpretation I is two-valued if I (a) ∈ {t, f} for all a ∈ A, and trivial, denoted as I u , if I (a) = u for all a ∈ A. Further, let I t (I f ) be the interpretation assigning all arguments to t (f). For Boolean formulas ϕ we consider the classical connectives of logical conjunction "∧", logical disjunction "∨", logical negation "¬", and material implication "→". A two-valued interpretation I extends to the evaluation of a formula ϕ under I as usual, denoted by I (ϕ). For a formula ϕ and a three-valued interpretation I let ϕ[I ] be the formula obtained from ϕ with each argument that I assigns to either true or false being replaced by the corresponding truth constant, i.e., f]; arguments assigned to undefined are not modified. An interpretation I is equally or more informative than J , denoted by J i I , if J (a) ∈ {t, f} implies J (a) = I (a) for all a ∈ A. We denote by < i the strict version of i , i.e., J < i I if J i I and ∃a ∈ A s.t. J (a) = u and I (a) ∈ {t, f}. Definition 4. An ADF is a tuple D = (A, L, C) where A is a set of arguments, L ⊆ A × A is a set of links, and C = {ϕ a } a∈A is a collection of acceptance conditions, each given by a formula over the parents of an argument: par D (a) = {b ∈ A | (b, a) ∈ L}. Example 4. Fig. 2(b) shows an ADF D = ({a, b, c, d}, L, C) with L = {(a, b), (b, a), (a, c), (b, c), (c, d)}. The acceptance conditions are shown close to the arguments. The semantics of ADFs are based on the characteristic function D mapping interpretations to updated interpretations. Definition 5. Let D = (A, L, C) be an ADF. The characteristic function D is defined by D (I ) = J with J (a) = ⎧ ⎪ ⎨ ⎪ ⎩ t if ϕ a [I ] is a tautology, f if ϕ a [I ] is unsatisfiable, and u otherwise. Semantics of ADFs are defined in a similar fashion as for AFs, based on the characteristic function. Definition 6. Given an ADF D, an interpretation I • is admissible in D iff I i D (I ); • is complete in D iff I = D (I ); • is grounded in D iff I is the least fixed-point of D ; and • is preferred in D iff I is i -maximal admissible in D. We refer to the set of all admissible, complete, grounded, and preferred interpretations of an ADF D by adm(D), com(D), grd(D), and prf(D), respectively. In any ADF D, it holds that prf(D) ⊆ com(D) ⊆ adm (D). Further, by definition, it holds that the grounded interpretation is a complete interpretation.

doi:10.3233/aac-190471
fatcat:onyuqrzn6zaynnxupq2izsdaey