Rich preference-based argumentation frameworks

An argumentation framework is seen as a directed graph whose nodes are arguments and arcs are attacks between the arguments. Acceptable sets of arguments, called extensions, are computed using a semantics. Existing semantics are solely based on the attacks and do not take into account other important criteria like the intrinsic strengths of arguments.The contribution of this paper is three fold. First, we study how preferences issued from differences in strengths of arguments can help in argumentation frameworks. We show that they play two distinct and complementary roles: (i) to repair the attack relation between arguments, (ii) to refine the evaluation of arguments. Despite the importance of both roles, only the first one is tackled in existing literature. In a second part of this paper, we start by showing that existing models that repair the attack relation with preferences do not perform well in certain situations and may return counter-intuitive results. We then propose a new abstract and general framework which treats properly both roles of preferences. The third part of this work is devoted to defining a bridge between the argumentation-based and the coherence-based approaches for handling inconsistency in knowledge bases, in particular when priorities between formulae are available. We focus on two well-known models, namely the preferred sub-theories introduced by Brewka and the demo-preferred sets defined by Cayrol, Royer and Saurel. For each of these models, we provide an instantiation of our abstract framework which is in full correspondence with it.     

Generating possible intentions with constrained argumentation systems

Practical reasoning (PR), which is concerned with the generic question of what to do, is generally seen as a two steps process: (1) deliberation, in which an agent decides what state of affairs it wants to reach - that is, its desires; and (2) means-ends reasoning, in which the agent looks for plans for achieving these desires. The agent’s intentions are a consistent set of desires that are achievable together.This paper proposes the first argumentation system for PR that computes in one step the possible intentions of an agent, avoiding thus the drawbacks of the existing systems. The proposed system is grounded on a recent work on constrained argumentation systems, and satisfies the rationality postulates identified in argumentation literature, namely the consistency and the completeness of the results.     

Postulates for logic-based argumentation systems

Logic-based argumentation systems are developed for reasoning with inconsistent information. Starting from a knowledge base encoded in a logical language, they define arguments and attacks between them using the consequence operator associated with the language. Finally, a semantics is used for evaluating the arguments.In this paper, we focus on systems that are based on deductive logics and that use Dung's semantics. We investigate rationality postulates that such systems should satisfy. We define five intuitive postulates: consistency and closure under the consequence operator of the underlying logic of the set of conclusions of arguments of each extension, closure under sub-arguments and exhaustiveness of the extensions, and a free precedence postulate ensuring that the free formulas of the knowledge base (i.e., the ones that are not involved in inconsistency) are conclusions of arguments in every extension. We study the links between the postulates and explore conditions under which they are guaranteed or violated.     

Looking-ahead in backtracking algorithms for abstract argumentation

We refine implemented backtracking algorithms for a number of problems related to Dung's argumentation frameworks. Under admissible, preferred, complete, stable, semi stable, and ideal semantics we add enhancements, what are so-called global looking-ahead pruning strategies, to the-state-of-the-art implementations of two problems. First, we tackle the extension enumeration problem: constructing some/all set(s) of acceptable arguments of a given argumentation framework. Second, we address the acceptance decision problem: deciding whether an argument is in some/all set(s) of accepted arguments of a given argumentation framework. The experiments that we report show that the speedup gain of the new enhancements is quite significant.     

Eliminating counterexamples: A Grade 8 student’s learning trajectory for contrapositive proving

This study uses a teaching experiment and retrospective analysis to develop a learning trajectory for improving a Grade 8 student’s ability to construct, critique, and validate contrapositive arguments. The study is predicated on the hypothesis that adolescents perform poorly on contrapositive reasoning tasks because they lack sufficient ways of justifying contrapositive argumentation as a viable mode of argumentation. By studying a student’s actions and comments as she develops, critiques, and validates not-the-conclusion-implies-the-conditions-are-impossible arguments for conditional claims, a promising learning trajectory for contrapositive argumentation is developed. The student’s learning trajectory demonstrates how a conception of contrapositive proving as eliminating counterexamples can be useful in developing, critiquing, and validating contrapositive arguments.     

Classroom mathematical practices and gesturing

The purpose of this paper is to illustrate a methodological approach for empirically investigating the function of gesturing in the collective development of knowledge. We extend the earlier work of Stephan and Rasmussen [Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior 21, 459-490] who analyzed classroom discourse and symbolizing to document the emergence of six classroom mathematical practices over the course of 22 days of instruction on first-order differential equations. We complement and extend this previous analysis by re-examining the same data for gesturing and coordinate this analysis with the evolution of the classroom mathematical practices as they developed in this particular community of learners. Our illustration of the methodology we developed suggests that (1) gestures and argumentation can function as a unit that supports the establishment of one or more taken-as-shared ideas, and (2) that a gesture/argumentation pair that develops while establishing one practice can change function to support the establishment of ideas embedded in other classroom mathematical practices.     

Optimization of dialectical outcomes in dialogical argumentation

When informal arguments are presented, there may be imprecision in the language used, and so the audience may be uncertain as to the structure of the argument graph as intended by the presenter of the arguments. For a presenter of arguments, it is useful to know the audience's argument graph, but the presenter may be uncertain as to the structure of it. To model the uncertainty as to the structure of the argument graph in situations such as these, we can use probabilistic argument graphs. The set of subgraphs of an argument graph is a sample space. A probability value is assigned to each subgraph such that the sum is 1, thereby reflecting the uncertainty over which is the actual subgraph. We can then determine the probability that a particular set of arguments is included or excluded from an extension according to a particular Dung semantics. We represent and reason with extensions from a graph and from its subgraphs, using a logic of dialectical outcomes that we present. We harness this to define the notion of an argumentation lottery, which can be used by the audience to determine the expected utility of a debate, and can be used by the presenter to decide which arguments to present by choosing those that maximize expected utility. We investigate some of the options for using argumentation lotteries, and provide a computational evaluation.     

Modelling Inference in Argumentation through Labelled Deduction: Formalization and Logical Properties

Artificial Intelligence (AI) has long dealt with the issue of finding a suitable formalization for commonsense reasoning. Defeasible argumentation has proven to be a successful approach in many respects, proving to be a confluence point for many alternative logical frameworks. Different formalisms have been developed, most of them sharing the common notions of argument and warrant. In defeasible argumentation, an argument is a tentative (defeasible) proof for reaching a conclusion. An argument is warranted when it ultimately prevails over other conflicting arguments. In this context, defeasible consequence relationships for modelling argument and warrant as well as their logical properties have gained particular attention. This article analyzes two non-monotonic inference operators C_arg and C_war intended for modelling argument construction and dialectical analysis (warrant), respectively. As a basis for such analysis we will use the LDS_ar framework, a unifying approach to computational models of argument using Labelled Deductive Systems (LDS). In the context of this logical framework, we show how labels can be used to represent arguments as well as argument trees, facilitating the definition and study of non-monotonic inference operators, whose associated logical properties are studied and contrasted. We contend that this analysis provides useful comparison criteria that can be extended and applied to other argumentation frameworks. Mathematics Subject Classification (2000): Primary 03B22; Secondary 03B42.     

Argumentation update in YALLA (Yet Another Logic Language for Argumentation)

This article proposes a complete framework for handling the dynamics of an abstract argumentation system. This frame can encompass several belief bases under the form of several argumentation systems, more precisely it is possible to express and study how an agent who has her own argumentation system can interact on a target argumentation system (that may represent a state of knowledge at a given stage of a debate). The two argumentation systems are defined inside a reference argumentation system called the universe which constitutes a kind of “common language”. This paper establishes three main results. First, we show that change in argumentation in such a framework can be seen as a particular case of belief update. Second, we have introduced a new logical language called YALLA in which the structure of an argumentation system can be encoded, enabling to express all the basic notions of argumentation theory (defense, conflict-freeness, extensions) by formulae of YALLA. Third, due to previous works about dynamics in argumentation we have been in position to provide a set of new properties that are specific for argumentation update.     

Theory of Semi-Instantiation in Abstract Argumentation

We study instantiated abstract argumentation frames of the form (S, R, I), where (S, R) is an abstract argumentation frame and where the arguments x of S are instantiated by I(x) as well formed formulas of a well known logic, for example as Boolean formulas or as predicate logic formulas or as modal logic formulas. We use the method of conceptual analysis to derive the properties of our proposed system. We seek to define the notion of complete extensions for such systems and provide algorithms for finding such extensions. We further develop a theory of instantiation in the abstract, using the framework of Boolean attack formations and of conjunctive and disjunctive attacks. We discuss applications and compare critically with the existing related literature.     

Valued preference-based instantiation of argumentation frameworks with varied strength defeats

A Dung-style argumentation framework aims at representing conflicts among elements called arguments. The basic ingredients of this framework is a set of arguments and a Boolean abstract (i.e., its origin is not known) binary defeat relation. Preference-based argumentation frameworks are instantiations of Dung's framework in which the defeat relation is derived from an attack relation and a preference relation over the arguments. Recently, Dung's framework has been extended in order to consider the strength of the defeat relation, i.e., to quantify the degree to which an argument defeats another argument. In this paper, we instantiate this extended framework by a preference-based argumentation framework with a valued preference relation. As particular cases, the latter can be derived from a weight function over the arguments or a Boolean preference relation. We show under some reasonable conditions that there are “less situations” in which a defense between arguments holds with a valued preference relation compared to a Boolean preference relation. Finally, we provide some conditions that the valued preference relation shall satisfy when it is derived from a weight function.     

Examining the role of examples in proving processes through a cognitive lens: the case of triangular numbers

In this paper, we analyze the role of examples in the proving process. The context chosen for this study was finding a general rule for triangular numbers. The aim of this paper is to show that examples are effective for the construction of a proof when they allow cognitive unity and structural continuity between argumentation and proof. Continuity in the structure is possible if the inductive argumentation is based on process pattern generalization (PPG), but this is not the case if a generalization is made on the results. Moreover, the PPG favors the development of generic examples that support cognitive unity and structural continuity between the argumentation and proof. The cognitive analysis presented in this paper is performed through Toulmin??s model.     

Argumentation and algebraic proof

This paper concerns a study analysing cognitive continuities and distances between argumentation supporting a conjecture and its algebraic proof, when solving open problems involving properties of numbers. The aim of this paper is to show that, unlike the geometrical case, the structural distance between argumentation and proof (from an abductive argumentation to a deductive proof) is not one of the possible difficulties met by students in solving such problems. On the contrary, since algebraic proof is characterized by a strong deductive structure, abductive steps in the argumentation activity can be useful in linking the meaning of the letters used in the algebraic proof with numbers used in the argumentation. The analysis of continuities and distances between argumentation and proof is based on the use of Toulmin’s model combined with ck¢ model.     

The FE-closure operator in countable-valued logic

The FE-closure operator on set P _N of functions of countable-valued logic is considered. It is proved that any general recursive operator (defined in the Herbrand-Gödel formalism) can be realized in the language of FE-closure. It is established that the class of relations definable in the language of FE-closure coincides with class Σ ₁ ¹ in the Kleene analytic hierarchy. It is shown that the membership in an FE-closed class of any nontrivial homogeneous function implies that the entire set of homogeneous functions is contained in this class.     

Bipolarity in argumentation graphs: Towards a better understanding

Different abstract argumentation frameworks have been used for various applications within multi-agents systems. Among them, bipolar frameworks make use of both attack and support relations between arguments. However, there is no single interpretation of the support, and the handling of bipolarity cannot avoid a deeper analysis of the notion of support.In this paper we consider three recent proposals for specializing the support relation in abstract argumentation: the deductive support, the necessary support and the evidential support. These proposals have been developed independently within different frameworks. We restate these proposals in a common setting, which enables us to undertake a comparative study of the modellings obtained for the three variants of the support. We highlight relationships and differences between these variants, namely a kind of duality between the deductive and the necessary interpretations of the support.     

Pre-recursive categories

How can we obtain in a natural way the primitive recursive functions in categories? In this paper, we study the free ‘cartesian closed category with a natural numbers object (in the sense of the Peano-Lawvere axiom)’ generated by the empty category. In this category, every morphism 1 → N represents a natural number and every morphism N → N represents a function. Furthermore, the set of functions represented by the morphisms of this category contains strictly the set of primitive recursive functions and is strictly contained in the set of recursive functions. Then, we see that this category is a categorical version of Grzegorczyk's recursive functionals of finite type, with the addition of product types.     

Recursive solution to a quorum queueing system

In this paper, a single-channel bulk service queueing system is considered. Given an accumulation level r, the server stops processing new customers whenever the queue falls below r and resumes service when the queue reaches level r. We have developed a recursive numerical procedure to find the steady-state system size probabilities. We assess accuracy and efficacy of the recursive procedure by comparing it to the exact solution procedure. Effectiveness and computation time of the recursive procedure are further assessed through comparison with other approximate solution procedures on a sample of 323 experimental problems.     

Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions

The main assumption of this article is that learning mathematics depends on the student's participation in processes of collective argumentation. On the empirical level, such processes will be analyzed with Toulmin's theory of argumentation and Goffman's idea of decomposition of the speaker's role. On the theoretical level, different statuses of participation in processes of argumentation will be considered. By means of the method of comparative analysis, different grades of autonomy according to the interactional contribution of a student can be reconstructed. The paper finishes with remarks about consequences for improving mathematics teaching in schools and mathematics teacher education at university level.     

The Validity of the Multifractal Formalism: Results and Examples

By obtaining a new sufficient condition for a valid multifractal formalism, we improve in this paper a result developed by L. Olsen (1995, Adv. Math.116, 82-196). In particular, we describe a large class of measures satisfying the multifractal formalism and for which the construction of Gibbs measures is not possible. Some of these measures are not unidimensional but have a nontrivial multifractal spectrum, giving a negative answer to a question asked by S. J. Taylor (1995, J. Fourier Anal. Appl., special issue). We also describe a necessary condition of validity for the formalism which is very close to the sufficient one. This necessary condition allows us to describe a measure μ for which the multifractal packing dimension function B_μ(q) is a nontrivial real analytic function but the multifractal formalism is nowhere satisfied. This example gives also a solution to a problem posed by Taylor (cited above).