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See detailModelling defeasible and prioritized support in bipolar argumentation
Villata, Serena; Boella, Guido; Gabbay, Dov M. UL et al

in Annals of Mathematics & Artificial Intelligence (2012), 66(1-4), 163-197

Cayrol and Lagasquie-Schiex introduce bipolar argumentation frameworks by introducing a second relation on the arguments for representing the support among them. The main drawback of their approach is ... [more ▼]

Cayrol and Lagasquie-Schiex introduce bipolar argumentation frameworks by introducing a second relation on the arguments for representing the support among them. The main drawback of their approach is that they cannot encode defeasible support, for instance they cannot model an attack towards a support relation. In this paper, we introduce a way to model defeasible support in bipolar argumentation frameworks. We use the methodology of meta-argumentation in which Dung’s theory is used to reason about itself. Dung’s well-known admissibility semantics can be used on this meta-argumentation framework to compute the acceptable arguments, and all properties of Dung’s classical theory are preserved. Moreover, we show how different contexts can lead to the alternative strengthening of the support relation over the attack relation, and converse. Finally, we present two applications of our methodology for modeling support, the case of arguments provided with an internal structure and the case of abstract dialectical frameworks. [less ▲]

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See detailA logic of argumentation for specification and verification of abstract argumentation frameworks
Villata, Serena; Boella, Guido; Gabbay, Dov M. UL et al

in Annals of Mathematics & Artificial Intelligence (2012), 66(1-4), 199-230

In this paper, we propose a logic of argumentation for the specification and verification (LA4SV) of requirements on Dung’s abstract argumentation frameworks. We distinguish three kinds of decision ... [more ▼]

In this paper, we propose a logic of argumentation for the specification and verification (LA4SV) of requirements on Dung’s abstract argumentation frameworks. We distinguish three kinds of decision problems for argumentation verification, called extension verification, framework verification, and specification verification respectively. For example, given a political requirement like “if the argument to increase taxes is accepted, then the argument to increase services must be accepted too,” we can either verify an extension of acceptable arguments, or all extensions of an argumentation framework, or all extensions of all argumentation frameworks satisfying a framework specification. We introduce the logic of argumentation verification to specify such requirements, and we represent the three verification problems of argumentation as model checking and theorem proving properties of the logic. Moreover, we recast the logic of argumentation verification in a modal framework, in order to express multiple extensions, and properties like transitivity and reflexivity of the attack relation. Finally, we introduce a logic of meta-argumentation where abstract argumentation is used to reason about abstract argumentation itself. We define the logic of meta-argumentation using the fibring methodology in such a way to represent attack relations not only among arguments but also among attacks. [less ▲]

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See detailOverview on the connection between reactive Kripke models and argumentation networks
Gabbay, Dov M. UL

in Annals of Mathematics & Artificial Intelligence (2012), 66(1-4), 1-5

The collection in this special issue contains mainly papers on reactive Kripke semantics and on argumentation. This overview says a few words about how these papers fit in the general picture.

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See detailIntroducing reactive modal tableaux
Gabbay, Dov M. UL

in Annals of Mathematics & Artificial Intelligence (2012), 66(1-4), 55-79

This paper introduces the idea of reactive semantics and reactive Beth tableaux for modal logic and quotes some of its applications. The reactive idea is very simple. Given a system with states and the ... [more ▼]

This paper introduces the idea of reactive semantics and reactive Beth tableaux for modal logic and quotes some of its applications. The reactive idea is very simple. Given a system with states and the possibility of transitions moving from one state to another, we can naturally imagine a path beginning at an initial state and moving along the path following allowed transitions. If our starting point is s0, and the path is s0,s1,...,sn, then the system is ordinary non-reactive system if the options available at sn (i.e., which states t we can go to from sn) do not depend on the path s0,...,sn (i.e., do not depend on how we got to sn). Otherwise if there is such dependence then the system is reactive. It seems that the simple idea of taking existing systems and turning them reactive in certain ways, has many new applications. The purpose of this paper is to introduce reactive tableaux in particular and illustrate and present some of the applications of reactivity in general. Mathematically one can take a reactive system and turn it into an ordinary system by taking the paths as our new states. This is true but from the point of view of applications there is serious loss of information here as the applicability of the reactive system comes from the way the change occurs along the path. In any specific application, the states have meaning, the transitions have meaning and the paths have meaning. Therefore the changes in the system as we go along a path can have very important meaning in the context, which enhances the usability of the model. [less ▲]

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See detailIntroducing reactive Kripke semantics and arc accessibility
Gabbay, Dov M. UL

in Annals of Mathematics & Artificial Intelligence (2012), 66(1-4), 7-53

Ordinary Kripke models are not reactive. When we evaluate (test/ measure) a formula A at a model m, the model does not react, respond or change while we evaluate. The model is static and unchanged. This ... [more ▼]

Ordinary Kripke models are not reactive. When we evaluate (test/ measure) a formula A at a model m, the model does not react, respond or change while we evaluate. The model is static and unchanged. This paper studies Kripke models which react to the evaluation process and change themselves during the process. The additional device we add to Kripke semantics to make it reactive is to allow the accessibility relation to access itself. Thus the accessibility relation R of a reactive Kripke model contains not only pairs (a,b)∈R of possible worlds (b is accessible to a, i.e., there is an accessibility arc from a to b) but also pairs of the form (t ,(a,b))∈R, meaning that the arc (a,b) is accessible to t, or even connections of the form((a,b),(c,d))∈R. This new kind of Kripke semantics allows us to characterise more axiomatic modal logics (with one modality) by a class of reactive frames. There are logics which cannot be characterised by ordinary frames but which can be characterised by reactive frames. We also discuss the manifestation of the ‘reactive’ idea in the context of automata theory, where we allow the automaton to react and change it’s own definition as it responds to input, and in graph theory, where the graph can change under us as we manipulate it. [less ▲]

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See detailGlobal view on reactivity: switch graphs and their logics
Gabbay, Dov M. UL; Marcelino, Sergio

in Annals of Mathematics & Artificial Intelligence (2012), 66(1-4), 131-162

The notion of reactive graph generalises the one of graph by allowing the base accessibility relation to change when its edges are traversed. Can we represent these more general structures using points ... [more ▼]

The notion of reactive graph generalises the one of graph by allowing the base accessibility relation to change when its edges are traversed. Can we represent these more general structures using points and arrows? We prove this can be done by introducing higher order arrows: the switches. The possibility of expressing the dependency of the future states of the accessibility relation on individual transitions by the use of higher-order relations, that is, coding meta-relational concepts by means of relations, strongly suggests the use of modal languages to reason directly about these structures. We introduce a hybrid modal logic for this purpose and prove its completeness. [less ▲]

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See detailEmbedding and automating conditional logics in classical higher-order logic
Benzmuller, Christoph; Gabbay, Dov M. UL; Genovese, Valerio UL et al

in Annals of Mathematics & Artificial Intelligence (2012), 66(1-4), 257-271

A sound and complete embedding of conditional logics into classical higher-order logic is presented. This embedding enables the application of off-the-shelf higher-order automated theorem provers and ... [more ▼]

A sound and complete embedding of conditional logics into classical higher-order logic is presented. This embedding enables the application of off-the-shelf higher-order automated theorem provers and model finders for reasoning within and about conditional logics. [less ▲]

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See detailCompleteness theorems for reactive modal logics
Gabbay, Dov M. UL

in Annals of Mathematics & Artificial Intelligence (2012), 66(1-4), 81-129

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See detailCausal dynamic inference
Bochman, Alexander; Gabbay, Dov M. UL

in Annals of Mathematics & Artificial Intelligence (2012), 66(1-4), 231-256

We suggest a general logical framework for causal dynamic reasoning. As a first step, we introduce a uniform structural formalism and assign it two kinds of semantics, abstract dynamic models and ... [more ▼]

We suggest a general logical framework for causal dynamic reasoning. As a first step, we introduce a uniform structural formalism and assign it two kinds of semantics, abstract dynamic models and relational models. The corresponding completeness results are proved. As a second step, we extend the structural formalism to a two-sorted state-transition calculus, and prove its completeness with respect to the associated relational semantics. [less ▲]

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