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Weighted Abstract Dialectical Frameworks

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Abstract

Dialectical Frameworks (ADFs) generalize Dung's argumentation frameworks allowing various relationships among arguments to be expressed in a systematic way. We further generalize ADFs so as to accommodate arbitrary acceptance degrees for the arguments. This makes ADFs applicable in domains where both the initial status of arguments and their relationship are only insufficiently specified by Boolean functions. We define all standard ADF semantics for the weighted case, including grounded, preferred and stable semantics. We illustrate our approach using acceptance degrees from the unit interval and show how other valuation structures can be integrated. In each case it is sufficient to specify how the generalized acceptance conditions are represented by formulas, and to specify the information ordering underlying the characteristic ADF operator. We also present complexity results for problems related to weighted ADFs.

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... In recent years, Artificial Intelligence (AI) field witnessed controversy. Decision-making represents an important arguable point due to its significance in multiple areas of use to solve many problems [1,2,3]. It has been adopted in various fields of AI, for example, it has been applied to the legal field to identify the legitimacy of arguments, nonmonotonic thinking, and multi-specialist frameworks [3]. ...
... The last option will then, at that point, be presented given explicit properties/sorts of the connections in the diagrams. Acknowledgment conditions enable us to present different hub and connection types [1]. ...
... More formally, a convincing hypothetical construction is an organized outline whose center points address the arguments, explanations, or positions, which can be recognized or not. The guideline ADF adds a dedicated affirmation condition to all the contentions individually [1]. ...
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... After that, the latter will be introduced based on the properties/types of the links in our graphs. Acceptance requirements enable the addition of new node and link types [7]. This article proposes a novel method for determining the preference argumentation. ...
... More officially, a theoretical persuasive structure is a coordinated chart whose hubs address arguments, statements, or positions which can be acknowledged or not. All in all, the principle thought to the ADF is adding to every argument a particular acknowledgment condition [7]. control argumentation frameworks (CAFs), provides a dynamic model, it can change over time reflecting the dynamics of the environment, it sums up the strategies, in particular the typical augmentation requirement, by obliging the chance of vulnerability in unique situations. ...
... The latter will subsequently be introduced based on the links' unique attributes kinds. Acceptance requirements enable the addition of new node and link types [7]. The recommended core of the arguments and attacks affects their acceptability or not. ...
... The argumentation is a significant focal point in Artificial Intelligence (AI), especially in recent years. It has become a very important component in this field [1,2,3]. It is associated with and helpful to other AI subfields, specifically information portrayal, nonmonotonic thinking, and multi-specialist frameworks. ...
... More officially, a theoretical persuasive structure is a coordinated chart whose hubs address arguments, the statements or positions which can be acknowledged or not. All in all, the principle thought to the ADF it adding to every argument a particular acknowledgment condition [1]. Definition 6. ...
... The Bipolar Argumentation Framework (BAF) gives to set of relationship defeat relation and support relation [38]. Abstract dialectical frameworks (ADFs), add to each argument a specific acceptance condition [1]. Control Argumentation Frameworks (CAFs) provide dynamic models that can change over time reflecting the dynamics of the environment [41]. ...
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... In the current section, we briefly restate some of the key concepts of abstract dialectical frameworks that are derived from those given in (Brewka et al. 2018;Brewka et al. 2017b;Brewka et al. 2013;Brewka and Woltran 2010). An ADF can be represented by a graph in which nodes indicate arguments and links show the relation among arguments. ...
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... As we use = for both an identity and an equation in numerical systems, we use ≡ for both an equivalence relation and an equation.6 In this section, we use X to denote a variable on the left side of an equation. ...
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... However, subsequent work has been done to study ADFs in several other directions. We can cite [20] in which the semantics of ADFs are inspired by approximation fixpoint theory, [76] which proposes a probabilistic version of ADFs, [54] which shows how to represent an ADF with only attack relations and [48] which investigates sub-classes of ADFs. A main notion in argumentation approaches with structured arguments, is that of sub-argument [17,57,66,81,87]. ...
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... The semantics of ADFs have been defined using a framework called approximation fixpoint theory [Denecker et al., 2000, Brewka et al., 2013, Strass, 2013. There, knowledge bases of knowledge representation formalisms are mapped to operators (their so-called characteristic consequence operators) on an order-theoretic structure (for example a lattice, a meet-complete semi-lattice, or a complete partial order [Davey and Priestley, 2002]). ...
... There, knowledge bases of knowledge representation formalisms are mapped to operators (their so-called characteristic consequence operators) on an order-theoretic structure (for example a lattice, a meet-complete semi-lattice, or a complete partial order [Davey and Priestley, 2002]). Certain points of these operators (for example fixpoints, least fixpoints, postfixpoints, or maximal postfixpoints [Davey and Priestley, 2002]) then correspond to models of the knowledge base according to various semantics [Denecker et al., 2000, Brewka et al., 2013, Strass, 2013]. ...
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... The additional expressive power is achieved by adding acceptance conditions to the arguments which allow for the specification of more complex relationships between them. Of particular interest might be the subclass of bipolar ADFs (BADFs) which are as complex as AFs while arguably offering more modelling capabilities (Brewka, Ellmauthaler, Strass, Wallner, & Woltran, 2017;Straß & Wallner, 2015;Baumann & Heinrich, 2023). It is one highly relevant future task to investigate notions of forgetting in these more expressive argumentation formalisms. ...
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... The principle-based analysis of argumentation semantics was initiated by Baroni and Giacomin (2007) to choose among the many extension-based argumentation semantics that have been proposed in the formal argumentation literature. The handbook chapter of van der Torre and Vesic (2018) semantics (Amgoud and Ben-Naim, 2013), and for extended argumentation frameworks, for example, for abstract dialectical frameworks (Brewka et al., 2018). ...
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... The calculation of grounded semantics is considerably dierent. Since Γ D is ≤ i -monotonic and the grounded interpretation is dened as ≤ i -least xpoint we calculate it via iteratively applying the gamma operator starting from ≤ i -least interpretation u [2]. for ...
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... The denition of ADFs [14] was motivated by the eort to obtain more expressive power than classical AFs. This is achieved by equipping each argument with a so-called acceptance condition which can be given as a logical formula [15]. Denition 1. ...
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dialectical frameworks (ADFs) are one of the most powerful generalization of classical Dung-style AFs. In this paper we show how to use ADFs if we want to deal with acceptance conditions changing over time. We therefore introduce so-called timed abstract di-alectical frameworks (tADFs) which are essentially ADFs equipped with time states. Beside a precise formal denition of tADFs and an illustrating example we prove that Kleene's three-valued logic K 3 facilitate the evaluation of acceptance functions if we do not allow multiple occurrences of atoms.
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Graphical models are widely used in argumentation to visualize relationships among propositions or arguments. The intuitive meaning of the links in the graphs is typically expressed using labels of various kinds. In this paper we introduce a general semantical framework for assigning a precise meaning to labelled argument graphs which makes them suitable for automatic evaluation. Our approach rests on the notion of explicit acceptance conditions, as first studied in Abstract Dialectical Frameworks (ADFs). The acceptance conditions used here are functions from multisets of labels to truth values. We define various Dung style semantics for argument graphs. We also introduce a pattern language for specifying acceptance functions. Moreover, we show how argument graphs can be compiled to ADFs, thus providing an automatic evaluation tool via existing ADF implementations. Finally, we also discuss complexity issues.
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Textual Entailment (TE) systems aim at recognizing the relations of entailment or non entailment holding between two text fragments (i.e. a pair). The identified TE pairs are considered as independent one from the others. However, in the latest years TE systems have been challenged against a number of real world application scenarios like analyzing costumers interactions about a service, or analyzing online debates. These applications have underlined the need to move from TE pairs to TE graphs where pairs are no more independent. Moving from single pairs to graphs has the advantage of providing an overall view of the topic discussed in the text. The challenge here is to define ways to exploit such graph-based representation for text exploration. In the literature, some approaches apply abstract argumentation theory to compute the accepted arguments of a debate, but they present a number of drawbacks, e.g., the non entailment relation and the attack relation in abstract argumentation are assumed to be equivalent, but this is not always the case. In this paper, we define bipolar entailment graphs, i.e., graphs whose nodes are text fragments and the edges represent the entailment or non entailment relations. We adopt abstract dialectical frameworks to define acceptance conditions for the nodes such that the resulting framework returns us relevant information for our text exploration task. Experimental evaluation shows the feasibility of our approach.
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We analyse the expressiveness of Brewka and Woltran's abstract dialectical frameworks for two-valued semantics. By expressiveness we mean the ability to encode a desired set of two-valued interpretations over a given propositional vocabulary A using only atoms from A. We also compare ADFs' expressiveness with that of (the two-valued semantics of) abstract argumentation frameworks, normal logic programs and propositional logic. While the computational complexity of the two-valued model existence problem for all these languages is (almost) the same, we show that the languages form a neat hierarchy with respect to their expressiveness. We then demonstrate that this hierarchy collapses once we allow to introduce a linear number of new vocabulary elements. We finally also analyse and compare the representational succinctness of ADFs (for two-valued model semantics), that is, their capability to represent two-valued interpretation sets in a space-efficient manner.
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This paper provides equational semantics for Dung's argumentation networks. The network nodes get numerical values in [0,1], and are supposed to satisfy certain equations. The solutions to these equations correspond to the “extensions” of the network. This approach is very general and includes the Caminada labelling as a special case, as well as many other so-called network extensions, support systems, higher level attacks, Boolean networks, dependence on time, and much more. The equational approach has its conceptual roots in the nineteenth century following the algebraic equational approach to logic by George Boole, Louis Couturat, and Ernst Schroeder.
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Probabilistic abstract argumentation combines Dung's abstract argumentation framework with probability theory in order to model uncertainty in argumentation. In this setting, we address the fundamental problem of computing the probability that a set of arguments is an extension according to a given semantics. We focus on the most popular semantics (i.e., admissible, stable, complete, grounded, preferred, ideal-set, ideal, stage, and semi-stable), and show the following dichotomy result: computing the probability that a set of arguments is an extension is either FP or FP^\#P-complete depending on the semantics adopted. Our polynomial time results are particularly interesting, as they hold for some semantics for which no polynomial-time technique was known so far.
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We present various new concepts and results related to abstract dialectical frameworks (ADFs), a powerful generalization of Dung's argumentation frameworks (AFs). In particular, we show how the existing definitions of stable and preferred semantics which are restricted to the subcase of so-called bipolar ADFs can be improved and generalized to arbitrary frameworks. Furthermore, we introduce preference handling methods for ADFs, allowing for both reasoning with and about preferences. Finally, we present an implementation based on an encoding in answer set programming.
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We provide a systematic in-depth study of the semantics of abstract dialectical frameworks (ADFs), a recent generalisation of Dungʼs abstract argumentation frameworks. This is done by associating with an ADF its characteristic one-step consequence operator and defining various semantics for ADFs as different fixpoints of this operator. We first show that several existing semantical notions are faithfully captured by our definition, then proceed to define new ADF semantics and show that they are proper generalisations of existing argumentation semantics from the literature. Most remarkably, this operator-based approach allows us to compare ADFs to related nonmonotonic formalisms like Dung argumentation frameworks and propositional logic programs. We use polynomial, faithful and modular translations to relate the formalisms, and our results show that both abstract argumentation frameworks and abstract dialectical frameworks are at most as expressive as propositional normal logic programs.
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We revisit the issue of epistemological and semantic foundations for autoepistemic and default logics, two leading formalisms in nonmonotonic reasoning. We develop a general semantic approach to autoepistemic and default logics that is based on the notion of a belief pair and that exploits the lattice structure of the collection of all belief pairs. For each logic, we introduce a monotone operator on the lattice of belief pairs. We then show that a whole family of semantics can be defined in a systematic and principled way in terms of fixpoints of this operator (or as fixpoints of certain closely related operators). Our approach elucidates fundamental constructive principles in which agents form their belief sets, and leads to approximation semantics for autoepistemic and default logics. It also allows us to establish a precise one-to-one correspondence between the family of semantics for default logic and the family of semantics for autoepistemic logic. The correspondence exploits the modal interpretation of a default proposed by Konolige. Our results establish conclusively that default logic can be viewed as a fragment of autoepistemic logic, a result that has been long anticipated. At the same time, they explain the source of the difficulty to formally relate the semantics of default extensions by Reiter and autoepistemic expansions by Moore. These two semantics occupy different locations in the corresponding families of semantics for default and autoepistemic logics.
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We explore a framework for argumentation (based on classical logic) in which an argument is a pair where the first item in the pair is a minimal consistent set of formulae that proves the second item (which is a formula). We provide some basic definitions for arguments, and various kinds of counter-arguments (defeaters). This leads us to the definition of canonical undercuts which we argue are the only defeaters that we need to take into account. We then motivate and formalise the notion of argument trees and argument structures which provide a way of exhaustively collating arguments and counter-arguments. We use argument structures as the basis of our general proposal for argument aggregation.There are a number of frameworks for modelling argumentation in logic. They incorporate formal representation of individual arguments and techniques for comparing conflicting arguments. In these frameworks, if there are a number of arguments for and against a particular conclusion, an aggregation function determines whether the conclusion is taken to hold. We propose a generalisation of these frameworks. In particular, our new framework makes it possible to define aggregation functions that are sensitive to the number of arguments for or against. We compare our framework with a number of other types of argument systems, and finally discuss an application in reasoning with structured news reports.
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In this paper we study fixpoints of operators on lattices and bilattices in a systematic and principled way. The key concept is that of an approximating operator, a monotone operator on the product bilattice, which gives approximate information on the original operator in an intuitive and well-defined way. With any given approximating operator our theory associates several different types of fixpoints, including the Kripke–Kleene fixpoint, stable fixpoints, and the well-founded fixpoint, and relates them to fixpoints of operators being approximated. Compared to our earlier work on approximation theory, the contribution of this paper is that we provide an alternative, more intuitive, and better motivated construction of the well-founded and stable fixpoints. In addition, we study the space of approximating operators by means of a precision ordering and show that each lattice operator O has a unique most precise—we call it ultimate—approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We then discuss applications of these results in logic programming.
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We introduce and investigate a natural extension of Dung's well-known model of argument systems in which attacks are associated with a weight, indicating the relative strength of the attack. A key concept in our framework is the notion of an inconsistency budget, which characterises how much inconsistency we are prepared to tolerate: given an inconsistency budget β, we would be prepared to disregard attacks up to a total weight of β. The key advantage of this approach is that it permits a much finer grained level of analysis of argument systems than unweighted systems, and gives useful solutions when conventional (unweighted) argument systems have none. We begin by reviewing Dung's abstract argument systems, and motivating weights on attacks (as opposed to the alternative possibility, which is to attach weights to arguments). We then present the framework of weighted argument systems. We investigate solutions for weighted argument systems and the complexity of computing such solutions, focussing in particular on weighted variations of grounded extensions. Finally, we relate our work to the most relevant examples of argumentation frameworks that incorporate strengths.
Conference Paper
In this paper we introduce dialectical frameworks, a powerful generalization of Dung-style argumentation frameworks where each node comes with an associated acceptance condition. This allows us to model different types of dependencies, e.g. support and attack, as well as different types of nodes within a single framework. We show that Dung's standard semantics can be generalized to dialectical frameworks, in case of stable and preferred semantics to a slightly restricted class which we call bipolar frameworks. We show how acceptance conditions can be conveniently represented using weights respectively priorities on the links and demonstrate how some of the legal proof standards can be modeled based on this idea. Copyright © 2010, Association for the Advancement of Artificial Intelligence.
Abstract dialectical frameworks for legal reasoning
  • L Al-Abdulkarim
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Al-Abdulkarim, L.; Atkinson, K.; and Bench-Capon, T. J. M. 2014. Abstract dialectical frameworks for legal reasoning. In Proc. JU-RIX'14, volume 271 of FAIA, 61-70. IOS Press.
On the graded acceptability of arguments
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A Uniform Account of Realizability in Abstract Argumentation
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Analyzing the Computational Complexity of Abstract Dialectical Frameworks via Approximation Fixpoint Theory
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Pührer, J. 2017. ArgueApply: A mobile app for argumentation. In Proc. LPNMR'17, volume 10377 of LNCS, 250-262. Springer. Strass, H., and Wallner, J. P. 2015. Analyzing the Computational Complexity of Abstract Dialectical Frameworks via Approximation Fixpoint Theory. Artif. Intell. 226:34-74.