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Logic Journal of IGPL Advance Access originally published online on November 14, 2008
Logic Journal of IGPL 2009 17(1):1-54; doi:10.1093/jigpal/jzn021
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Defeasible inheritance systems and reactive diagrams*

Dov M Gabbay {dagger}

King's College, London {ddagger}

Karl Schlechta §

Laboratoire d'Informatique Fondamentale de Marseille ¶


  Abstract

Inheritance diagrams are directed acyclic graphs with two types of connections between nodes: x -> y (read x is a y) and x nrarr y (read as x is not a y). Given a diagram D, one can ask the formal question of "is there a valid (winning) path between node x and node y?" Depending on the existence of a valid path we can answer the question "x is a y" or "x is not a y".

The answer to the above question is determined through a complex inductive algorithm on paths between arbitrary pairs of points in the graph.

This paper aims to simplify and interpret such diagrams and their algorithms. We approach the area on two fronts.

(1)

Suggest reactive arrows to simplify the algorithms for the winning paths.

(2)

We give a conceptual analysis of (defeasible or nonmonotonic) inheritance diagrams, and compare our analysis to the "small" and "big sets" of preferential and related reasoning.

In our analysis, we consider nodes as information sources and truth values, direct links as information, and valid paths as information channels and comparisons of truth values. This results in an upward chaining, split validity, off-path preclusion inheritance formalism of a particularly simple type.

We show that the small and big sets of preferential reasoning have to be relativized if we want them to conform to inheritance theory, resulting in a more cautious approach, perhaps closer to actual human reasoning.

We will also interpret inheritance diagrams as theories of prototypical reasoning, based on two distances: set difference, and information difference.

We will see that some of the major distinctions between inheritance formalisms are consequences of deeper and more general problems of treating conflicting information.

It is easily seen that inheritance diagrams can also be analysed in terms of reactive diagrams - as can all argumentation systems.

AMS Classification: 68T27, 68T30

Received for publication 15 March 2007.

*Paper 326

{dagger}Dov.Gabbay{at}kcl.ac.uk, www.dcs.kcl.ac.uk/staff/dg

{ddagger}Department of Computer Science, King's College London, Strand, London WC2R 2LS, UK

§ks{at}cmi.univ-mrs.fr, karl.schlechta{at}web.de, http://www.cmi.univ-mrs.fr/~ks

UMR 6166, CNRS and Université de Provence, Address: CMI, 39, rue Joliot-Curie, F-13453 Marseille Cedex 13, France


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