Description

Modal logics, originally conceived in philosophy, have recently found many applications in computer science, artificial intelligence, the foundations of mathematics, linguistics and other disciplines. Celebrated for their good computational behaviour, modal logics are used as effective formalisms for talking about time, space, knowledge, beliefs, actions, obligations, provability, etc. However, the nice computational properties can drastically change if we combine some of these formalisms into a many-dimensional system, say, to reason about knowledge bases developing in time or moving objects.

To study the computational behaviour of many-dimensional modal logics is the main aim of this book. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many-dimensional systems (e.g., multi-agent systems, description logics with epistemic, temporal and dynamic operators, spatio-temporal logics, etc.) fit in perfectly with this theoretical framework, and so their computational behaviour can be analyzed using the developed machinery.

We start with concrete examples of applied one- and many-dimensional modal logics such as temporal, epistemic, dynamic, description, spatial logics, and various combinations of these. Then we develop a mathematical theory for handling a spectrum of 'abstract' combinations of modal logics - fusions and products of modal logics, fragments of first-order modal and temporal logics - focusing on three major problems: decidability, axiomatizability, and computational complexity. Besides the standard methods of modal logic, the technical toolkit includes the method of quasimodels, mosaics, tilings, reductions to monadic second-order logic, algebraic logic techniques. Finally, we apply the developed machinery and obtained results to three case studies from the field of knowledge repres

Readership

Logicians. Logicians in Computer Science (reseachers and students). Researchers and students in knowledge representation and reasoning.

Table of Contents

I Introduction

1 Modal logic basics
1.1 Modal axiomatic systems
1.2 Possible world semantics
1.3 Classical first-order logic and the standard translation
1.4 Multimodal logics
1.5 Algebraic semantics
1.6 Decision, complexity and axiomatizability problems

2 Applied modal logic
2.1 Temporal logic
2.2 Interval temporal logic
2.3 Epistemic logic
2.4 Dynamic logic
2.5 Description logic
2.6 Spatial logic
2.7 Intuitionistic logic
2.8 'Model level' reductions between logics

3 Many-dimensional modal logics
3.1 Fusions
3.2 Spatio-temporal logics
3.3 Products
3.4 Temporal epistemic logics
3.5 Classical first-order logic as a propositional multimodal logic
3.6 First-order modal logics
3.7 First-order temporal logics
3.8 Description logics with modal operators
3.9 HS as a two-dimensional logic
3.10 Modal transition logics
3.11 Intuitionistic modal logics

II Fusions and products

4 Fusions of modal logics
4.1 Preserving Kripke completeness and the finite model property
4.2 Algebraic preliminaries
4.3 Preserving decidability of global consequence
4.4 Preserving decidability<

Details

No. of pages:
766
Language:
English
Copyright:
© 2003
Published:
Imprint:
North Holland
Electronic ISBN:
9780080535784
Print ISBN:
9780444508263
Print ISBN:
9780444551832

About the authors

Reviews

"This book will be a valuable reference for the modal logic researcher. It can serve as a brief but useful introduction (....) for the suitably qualified newcomer. And it contributes a careful and rewarding comprehensive account of some of the latest foundational results in the area of combining modal logics."
Mark Reynolds, The University of Western Australia. Studia Logica, 2004.