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2013 | Buch

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Reactive Kripke Semantics

verfasst von: Dov M. Gabbay

Verlag: Springer Berlin Heidelberg

Buchreihe : Cognitive Technologies

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This text offers an extension to the traditional Kripke semantics for non-classical logics by adding the notion of reactivity. Reactive Kripke models change their accessibility relation as we progress in the evaluation process of formulas in the model. This feature makes the reactive Kripke semantics strictly stronger and more applicable than the traditional one. Here we investigate the properties and axiomatisations of this new and most effective semantics, and we offer a wide landscape of applications of the idea of reactivity. Applied topics include reactive automata, reactive grammars, reactive products, reactive deontic logic and reactive preferential structures.

Reactive Kripke semantics is the next step in the evolution of possible world semantics for non-classical logics, and this book, written by one of the leading authorities in the field, is essential reading for graduate students and researchers in applied logic, and it offers many research opportunities for PhD students.

Inhaltsverzeichnis

Frontmatter

1. A Theory of Hypermodal Logics:Mode Shifting in Modal Logic

Abstract

In this chapter the author introduces hypermodal logics, showing that such modalities cannot always be characterised by a class of Kripke frames.

Dov M. Gabbay

2. Introducing Reactive Kripke Semantics and Arc Accessibility

Abstract

In this chapter the author introduces reactive Kripke semantics and arc accessibility. He describes the reactive paradigm in general and then gives some specific examples from different application areas. The author then discusses several options for defining reactivity. And finally the chapter gives some technical results in the area of modal logic.

Dov M. Gabbay

3. Introducing Reactive Modal Tableaux

Abstract

In this chapter the author introduces reactive semantics and reactive Beth tableaux.

Dov M. Gabbay

4. Reactive Intuitionistic Tableaux

Abstract

In this chapter the author introduces reactive intuitionistic frames (Kripke frames and Beth frames) and studies their expressive power and properties.

Dov M. Gabbay

5. Completeness Theorems for Reactive Modal Logics

Abstract

In this chapter the author gives completeness theorems for some basic reactive Kripke models and semantics.

Dov M. Gabbay

6. Modal Logics of Reactive Frames

Abstract

In this chapter the author explains modal logics of reactive frames. He shows the correspondence between logic, reactive frame property, and the shattered frame unfolded property. He uses classical tools to study completeness in a new interpretation of modal logic, successfully studying some of its subsystems.

Dov M. Gabbay

7. Global View on Reactivity: Switch Graphs and their Logics

Abstract

In this chapter the author offers a global view on reactivity, using the term to refer to reactive systems which are history-dependent relational structures where the accessibility relation is determined not only by the point where one is but also by the previous transitions.

Dov M. Gabbay

8. Reactive Automata

Abstract

In this chapter the author offers a reactive point of view for automata and shows that using reactive arrows can reduce the number of states of an automaton. Within the framework of reactive automata he provides some examples where such a reduction is striking.

Dov M. Gabbay

9. Reactivity and Grammars: An Exploration

Abstract

In this chapter the author considers the relationship between grammars and formal languages, exploring more dynamic approaches to language generation called "reactive grammars", and examining a range of such reactivities.

Dov M. Gabbay

10. Reactive Flow Products

Abstract

In this chapter the author offers a generalisation of flow products, namely reactive flow products.

Dov M. Gabbay

11. Reactive Standard Deontic Logic

Abstract

In this chapter the author shows the power of the reactive idea in solving the Chisholm paradox, a difficult problem of SDL (Standard Deontic Logic).

Dov M. Gabbay

12. Reactive Preferential Structures and Non-monotonic Consequence

Abstract

In this chapter the author gives an abstract and also a very concrete semantics to IBRSs (information-bearing relation systems), and shows that a special case of IBRS generalizes preferential semantics in a very natural way and solves open representation problems for weak logical systems.

Dov M. Gabbay

Backmatter

Titel: Reactive Kripke Semantics
verfasst von: Dov M. Gabbay
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-41389-6
Print ISBN: 978-3-642-41388-9
DOI: https://doi.org/10.1007/978-3-642-41389-6