Coherent Systems

Coherent Systems

1st Edition - September 21, 2004

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  • Author: Karl Schlechta
  • eBook ISBN: 9780080502199

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Description

One aspect of common sense reasoning is reasoning about normal cases, e.g. a physician will first try to interpret symptoms by a common disease, and will take more exotic possibilities only later into account. Such "normality" can be encoded, e.g. by a relation, where case A is considered more normal than case B. This gives a standard semantics or interpretation to nonmonotonic reasoning (a branch of common sense reasoning), or, more formally, to nonmonotonic logics. We consider in this book the repercussions such normality relations and similar constructions have on the resulting nonmonotonic logics, i.e. which types of logic are adequate for which kind of relation, etc. We show in this book that some semantics correspond nicely to some logics, but also that other semantics do not correspond to any logics of the usual form.

Key Features

  • Provides a coherent picture of several formalisms of nonmonotonic logics
  • Gives completeness and incompleteness results for many variants of preferential, distance based, and other semantics
  • Gives probably the first systematic investigation of definability preservation and its consequences
  • Gives new proof techniques for completeness results
  • Is centered on semantics

Readership

Libraries and researchers in nonmonotonic and related logics

Table of Contents

  • Foreword (by David Makinson)

    Summary

    Acknowledgements



    CHAPTER 1 : INTRODUCTION

    1.1 The main topics of the book
    1.1.1 Conceptual analysis
    1.1.2 Generalized modal logic and integration
    1.1.3 Formal results
    1.1.4 The role of semantics
    1.1.5 Various remarks
    1.2 Historical remarks
    1.3 Organization of the book
    1.4 Overview of the chapters
    1.5 Specific remarks on propositional logic
    1.6 Basic definitions


    CHAPTER 2 : CONCEPTS

    2.1 Introduction
    2.2 Reasoning types
    2.2.1 Traditional nonmonotonic logics
    2.2.2 Prototypical and ideal cases
    2.2.3 Extreme cases and interpolation
    2.2.4 Clustering
    2.2.5 Certainty
    2.2.6 Quality of an answer, approximation, and complexity
    2.2.7 Useful reasoning
    2.2.8 Inheritance and argumentation
    2.2.9 Dynamic systems
    2.2.10 Theory revision
    2.2.11 Update
    2.2.12 Counterfactual conditionals
    2.3 Basic semantical concepts
    2.3.1 Preference
    2.3.2 Distance
    2.3.3 Size
    2.4 Coherence

    CHAPTER 3 : PREFERENCES

    3.1 Introduction
    3.2 General preferential structures
    3.2.1 General minimal preferential structures
    3.2.2 Transitive minimal preferential structures
    3.2.3 One copy version
    3.2.4 A very short remark on X-logics
    3.3 Smooth minimal preferential structures
    3.3.1 Smooth minimal preferential structures with arbitrarily many copies
    3.3.2 Smooth and transitive minimal preferential structures
    3.4 The logical characterization of general and smooth preferential models
    3.4.1 Simplifications of the general transitive limit case
    3.5 A counterexample to the KLM-system
    3.5.1 The formal results
    3.6 A nonsmooth model of cumulativity
    3.6.1 The formal results
    3.7 Plausibility logic - problems without closure under finite union
    3.7.1 Introduction
    3.7.2 Completeness and incompleteness results for plausibility logic
    3.8 The role of copies in preferential structures
    3.9 A new approach to preferential structures
    3.9.1 Introduction
    3.9.2 Validity in traditional and in our preferential structures
    3.9.3 The disjoint union of models and the problem of multiple copies
    3.9.4 Representation in the finite case
    3.10 Ranked preferential structures
    3.10.1 Introduction
    3.10.2 The minimal variant
    3.10.3 The limit variant without copies

    CHAPTER 4 : DISTANCES

    4.1 Introduction
    4.1.1 Theory revision
    4.1.2 Counterfactuals
    4.1.3 Summary
    4.2 Revision by distance
    4.2.1 Introduction
    4.2.2 The algebraic results
    4.2.3 The logical results
    4.2.4 There is no finite characterization
    4.2.5 The limit case
    4.3 Local and global metrics for the semantics of counterfactuals
    4.3.1 Introduction
    4.3.2 The results

    CHAPTER 5 : DEFINABILITY PRESERVATION

    5.1 Introduction
    5.1.1 The problem
    5.1.2 The remedy
    5.1.3 Basic definitions and results
    5.1.4 A remark on definability preservation and modal logic
    5.2 Preferential structures
    5.2.1 The algebraic results
    5.2.2 The logical results
    5.2.3 The general case and the limit version cannot be characterized
    5.3 Revision
    5.3.1 The algebraic result
    5.3.2 The logical result


    CHAPTER 6 : SUMS

    6.1 Introduction
    6.1.1 The general situation and the Farkas algorithm
    6.1.2 Update by minimal sums
    6.1.3 Comments on "Belief revision with unreliable observations"
    6.1.4 "Between" and "behind"
    6.1.5 Summary
    6.2 The Farkas algorithm
    6.3 Representation for update by minimal sums
    6.3.1 Introduction
    6.3.2 An abstract result
    6.3.3 Representation
    6.3.4 There is no finite representation for our type of update possible
    6.4 Comments on "Belief revision with unreliable observations"
    6.4.1 Introduction
    6.4.2 A characterization of Markov systems (in the finite case)
    6.4.3 There is no finite representation possible
    6.5 "Between" and "Behind"
    6.5.1 There is no finite representation for "between" and "behind"


    CHAPTER 7 : SIZE

    7.1 Introduction
    7.1.1 The details
    7.2 Generalized quantifiers
    7.2.1 Introduction
    7.2.2 Results
    7.3 Comparison of three abstract coherent systems based on size
    7.3.1 Introduction
    7.3.2 Presentation of the three systems
    7.3.3 Comparison of the systems of Ben-David/Ben-Eliyahu and the author
    7.3.4 Comparison of the systems of Ben-David/Ben-Eliyahu and of
    Friedman/Halpern
    7.4 Theory revision based on model size
    7.4.1 Introduction
    7.4.2 Results


    CHAPTER 8 : INTEGRATION

    8.1 Introduction
    8.1.1 Rules or object language?
    8.1.2 Various levels of reasoning
    8.2 Reasoning types and concepts
    8.3 Formal aspects
    8.3.1 Classical modal logic
    8.3.2 Classical propositional operators have no unique interpretation
    8.3.3 Combining individual completeness results


    CHAPTER 9 : CONCLUSION AND OUTLOOK

    Bibliography.
    Index.

Product details

  • No. of pages: 468
  • Language: English
  • Copyright: © Elsevier Science 2004
  • Published: September 21, 2004
  • Imprint: Elsevier Science
  • eBook ISBN: 9780080502199

About the Author

Karl Schlechta

Affiliations and Expertise

K. Schlechta Université de Provence and Laboratoire d'Informatique Fondamentale (CNRS UMR 6166), Marseille, France

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