Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-vkn6t Total loading time: 0.301 Render date: 2022-08-17T06:50:35.197Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

3 - A Little Bit of Model Theory

Published online by Cambridge University Press:  13 June 2017

Franz Baader
Affiliation:
Technische Universität, Dresden
Ian Horrocks
Affiliation:
University of Oxford
Carsten Lutz
Affiliation:
Universität Bremen
Uli Sattler
Affiliation:
University of Manchester
Get access

Summary

The main purpose of this chapter is to show that sets of models of ALC concepts or knowledge bases satisfy several interesting properties, which can be used to prove expressivity and decidability results. To be more precise, we will introduce the notion of bisimulation between elements of ALC interpretations, and prove that ALC concepts cannot distinguish between bisimilar elements. On the one hand, we will use this to show restrictions of the expressive power of ALC: number restrictions, inverse roles and nominals cannot be expressed within ALC. On the other hand, we will employ bisimulation invariance of ALC to show that ALC has the tree model property and satisfies closure under disjoint union of models. We will also show that ALC has the finite model property, though not as a direct consequence of bisimulation invariance. These properties will turn out to be useful in subsequent chapters and of interest to people writing knowledge bases: for example, ALC's tree model property implies that it is too weak to describe the ring structure of many chemical molecules since any ALC knowledge base trying to describe such a structure will also have acyclic models. In the present chapter, we will only use the finite model property (or rather the stronger bounded model property) to show a basic, not complexity-optimal decidability result for reasoning in ALC. For the sake of simplicity, we concentrate here on the terminological part of ALC, i.e., we consider only concepts and TBoxes, but not ABoxes.

To obtain a better intuitive view of the definitions and results introduced below, one should recall that interpretations of ALC can be viewed as graphs, with edges labelled by roles and nodes labelled by sets of concept names. More precisely, in such a graph

  1. • the nodes are the elements of the interpretation and they are labeled with all the concept names to which this element belongs in the interpretation;

  2. • an edge with label r between two nodes says that the corresponding two elements of the interpretation are related by the role r.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×