Logic

 

Information Flow Foundation

A local logic L = áinst(L), typ(L), ⊨L, ⊢L, NLñ, which is an inclusive idea combining the notions of classification and theory into a (not necessarily sound) whole, consists of

1.      a theory th(L) = átyp(L), ⊢Lñ of types and constraints,

2.      a classification cla(L) = áinst(L), typ(L), ⊨Lñ of instances,

3.      a subset NL Í inst(L) of normal instances which satisfy all the constraints.

A logic is sound when every instance of inst(L) is normal. For any local logic L, the sound part of L is obtained by throwing away all abnormal instances and restricting the classification relation to normal instances. A (sound) logic L = áinst(L), typ(L), ⊨L, ⊢Lñ consists of

1.      a theory th(L) = átyp(L), ⊢Lñ of types and constraints, and

2.      a classification cla(L) = áinst(L), typ(L), ⊨Lñ of instances which satisfy all the constraints.

 

Overview

The IFF representation for the theory component of an Information Flow logic has been dealt with in a separate discussion. Here we are mainly concerned with the representation for the classification component. Clearly, the specification form for a classification needs to contain the following information: name (or id) of classification; the instance set; the type set; and the classification relation. There are two specification forms for Information Flow classifications.

1.      An individual incidence in the classification relation is represented as an implicit has relationship. For example, to make the claims that the movie Toy_Story_(1995) is an Animation, and the movie Casablanca_(1942) is a Romance, we use the following specification forms.

<Animation obj="Toy_Story_(1995)"/>

<Romance obj="Casablanca_(1942)"/>

Note that these declarations do not assert the existence of the objects (instances) involve. In IFF the hallmark of an assertion of existence is use of the id attribute instead of the obj attribute. The use of these declarations is only to make a claim that the objects are of certain types. This specification form for Information Flow logics is called the has relationship form. As illustrated below, any binary relation of IFF can be "entified" (reified) by being defined in terms of the primitive has incidence relation and the genus type for a reifying theory.

Binary conceptual relation

Entified binary relation

2.      When a large number of incidence relationships must be asserted, an aggregate form may come in handy. This aggregate form, called the type set form, uses the Information Flow notion of the type set of an instance in a classification. The type set of an instance is represented by the typ element. For example, the following specification makes a claim for three incidence relationships.

<typ obj="Wrong_Trousers,_The_(1993)">

  <Animation/><Comedy/><Thriller/>

</typ>

 

 

Movie Genre Logic

As an example of the “type set” specification form for logics, consider the genre of movies. Movie genre provides an interesting example of a binary ontological relation that can be directly used (direct conceptual scaling) as the classification component of an Information Flow logic. In a movie ontology movie instances can be of several genres, and genre information is represented as a binary ontological relation

genre Í Movie´Genre.

Since the binary genre relation is between the extent (set of instances) of an object type Movie and a controlled vocabulary Genre, it can be considered to be an Information Flow classification with instances being movies, types being genres, and incidence being the genre relation.

o        The theory of the logic is the controlled Genre vocabulary. This controlled vocabulary is specified as an Information Flow theory associated with the ontology being used.

o        The classification of the logic needs only an instance set and an incidence relation

-             The Movie extent forms the instance set. This instance metadata may be contained in a instance component of the entity classification of an IFF model. The only information needed from this model is id and display information.

-             The binary genre relation is transformed by direct conceptual scaling into the incidence relation for the classification component of the logic. This relational instance metadata is contained in an (possibly the same) IFF model as the model containing the instance set information.

The fact that the controlled vocabulary is a theory has implications for the classification relation. For example, if the theory is a partial order, then the classification relation is most likely closed (on the right) with respect to this order.

To illustrate this, consider the following classification of 10 top-rated movies.

 Here is the representation as an IFF logic using the typ element.

IFF

Ontology

(located at http://www.movie.org/ontology)

 

<IFF>

  <Ontology id="Movie" version="1.0">

    …

    <Object name="Movie"/>

    <Object name="Genre"/>

    …

    <Relation name="genre" source="Movie" target="Genre"/>

    …

  </Ontology>

</IFF>

 

Model

(located at http://www.movie.org/model)

 

<IFF>

  <Model ontology="http://www.movie.org/ontology">

    <Classification kind="ent">

      …

      <Movie id="Star_Wars_(1977)"/>

      <Movie id="Wrong_Trousers,_The_(1993)"/>

      <Movie id="Shawshank_Redemption,_The_(1994)"/>

      <Movie id="Pulp_Fiction_(1994)"/>

      <Movie id="Usual_Suspects,_The_(1995)"/>

      <Movie id="Toy_Story_(1995)"/>

      <Movie id="Schindler's_List_(1993)"/>

      <Movie id="Casablanca_(1942)"/>

      <Movie id="Blade_Runner_(1982)"/>

      <Movie id="Citizen_Kane_(1941)"/>

      …

    </Classification>

    <Classification kind="rel">

      <genre src="Star_Wars_(1977)" tgt="Action"/>

      <genre src="Star_Wars_(1977)" tgt="Adventure"/>

      ...

      <genre src="Citizen_Kane_(1941)" tgt="Drama"/>

    </Classification>

  </Model>

</IFF>

 

Logic – (type set form)

 

<IFF>

  <Logic

    theory="http://www.movie.org/theory/Genre">

    <Classification kind="inc">

      <typ obj="Star_Wars_(1977)">

        <Action/><Adventure/><Drama/><Sci-Fi/><War/></typ>

      <typ obj="Wrong_Trousers,_The_(1993)">

        <Animation/><Comedy/><Thriller/></typ>

      <typ obj="Shawshank_Redemption,_The_(1994)">

        <Drama/><Mystery/><Thriller/></typ>

      <typ obj="Pulp_Fiction_(1994)"><Comedy/><Crime/><Drama/></typ>

      <typ obj="Usual_Suspects,_The_(1995)"><Crime/><Film-Noir/><Thriller/></typ>

      <typ obj="Toy_Story_(1995)"><Animation/><Comedy/></typ>

      <typ obj="Schindler's_List_(1993)"><Drama/><War/></typ>

      <typ obj="Casablanca_(1942)"><Drama/><Romance/><War/></typ>

      <typ obj="Blade_Runner_(1982)"><Action/><Drama/><Film-Noir/><Sci-Fi/></typ>

      <typ obj="Citizen_Kane_(1941)"><Drama/></typ>

    </Classification>

  </Logic>

</IFF>

 

 

 

Automobile Color Logic

As an example of the “has relationship” specification form for logics, consider the following natural language description for the color of automobiles.

“There are seven automobiles - a old Chevrolet convertible, a Dodge truck, a Ford sedan, a Honda Civic, a Jeep Wrangler, a Lotus Esprit sports car and a Volvo luxury sedan. The Chevrolet has been repainted yellow. The Dodge is a two-tone green and blue truck. The Ford is a two-tone sedan with a blue top and a black bottom. The Honda Civic is violet colored, and the Jeep is painted green. The Lotus is blue with a white top, and the Volvo is indigo.”

The color of automobiles is naturally modeled with the binary relation

<Relation name="color" source="Automobile" target="RGB"/>

where the data type RGB represents the Red-Green-Blue values for the color. For example, a red automobile has RGB value "FF0000". Then the above description could be represented as an Information Flow model with the entity classification representing the set of automobiles and the relation classification representing the color relation. Below this is the representation as an IFF logic using the typ element.

IFF

Ontology

(located at http://www.automobile.org/ontology)

 

<IFF>

  <Ontology>

    <extends href="http://www.ontologos.org/IFF/ontology"/>

    <extends theory="RGB" href="http://www.color.org/theory/RGB/"/>

    <Theory kind="ent" id="Car-Model" genus="Automobile">

      <language>

        <Language>

          <Object name="Chevrolet"/>

          <Object name="Dodge"/>

          <Object name="Ford"/>

          <Object name="Honda"/>

          <Object name="Jeep"/>

          <Object name="Lotus"/>

          <Object name="Volvo"/>

        </Language>

      </language>

      <partition>

        <Chevrolet/><Dodge/><Ford/><Honda/><Jeep/><Lotus/><Volvo/>

      </partition>

    </Theory>

    <Relation name="color">

      <sign pos="1" name="automobile" target="Automobile"/>

      <sign pos="2" name="rgb" target="Color:RGB"/>

    </Relation>

    …

    <subtype specific="Chevrolet" generic="Automobile"/>

    …

  </Ontology>

</IFF>

 

Model

 

 

<IFF>

  <Model

    ontology="http://www.automobile.org/ontology">

    <Classification kind="ent" type="Automobile">

      <Chevrolet id="c"/>

      <Dodge     id="d"/>

      <Ford      id="f"/>

      <Honda     id="h"/>

      <Jeep      id="j"/>

      <Lotus     id="l"/>

      <Volvo     id="v"/>

    </Classification>

    <Classification kind="rel" type="color">

      <color automobile="c" rgb="FFFF00"/>

      <color automobile="d" rgb="00FF00"/>

      <color automobile="d" rgb="0000FF"/>

      <color automobile="f" rgb="0000FF"/>

      <color automobile="f" rgb="000000"/>

      <color automobile="h" rgb="4F2F4F"/>

      <color automobile="j" rgb="00FF00"/>

      <color automobile="l" rgb="0000FF"/>

      <color automobile="l" rgb="FFFFFF"/>

      <color automobile="v" rgb="6B238E"/>

  </Model>

</IFF>

 

But it is natural to define a simple color theory for this situation, where the common colors are given names. This theory represents a controlled vocabulary of colors. Normally one would interpret the colors with simple conceptual scaling according to their RGB values, but for simplicity we will assume the color binary relation has target type Color and use direct conceptual scaling here.

<Relation name="color">

  <sign pos="1" name="automobile" target="Automobile"/>

  <sign pos="2" name="color" target="Color:Color"/>

</Relation>

Here is the new color relation.

IFF

Model

The has relation is represented by a classification using the obj attribute.

 

<IFF>

  <Model>

    <extends ontology="http://www.example.org/ontology/">

    <Classification kind="ent">

      <Chevrolet id="c"/>

      <Dodge     id="d"/>

      <Ford      id="f"/>

      <Honda     id="h"/>

      <Jeep      id="j"/>

      <Lotus     id="l"/>

      <Volvo     id="v"/>

    </Classification>

    <Classification kind="inc" relation="color" id="automobile-color">

      <color automobile="c" color="Color.yellow"/>

      <color automobile="d" color="Color.green"/>

      <color automobile="d" color="Color.blue"/>

      <color automobile="f" color="Color.blue"/>

      <color automobile="f" color="Color.black"/>

      <color automobile="h" color="Color.violet"/>

      <color automobile="j" color="Color.green"/>

      <color automobile="l" color="Color.blue"/>

      <color automobile="l" color="Color.white"/>

      <color automobile="v" color="Color.indigo"/>

    </Classification>

  </Model>

</IFF>

 

Logic – (has relationship form)

 

<IFF>

  <Logic

    theory="http://www.color.org/theory/Color/">

    <Classification kind="inc">

      <Color.yellow obj="c"/>

      <Color.green  obj="d"/>

      <Color.blue   obj="d"/>

      <Color.blue   obj="f"/>

      <Color.black  obj="f"/>

      <Color.violet obj="h"/>

      <Color.green  obj="j"/>

      <Color.blue   obj="l"/>

      <Color.white  obj="l"/>

      <Color.indigo obj="v"/>

    </Classification>

  </Logic>

</IFF>

 

 

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Please send questions, comments and suggestions about this page to: Robert E. Kent rekent@ontologos.org

Copyright © 2000 TOC (The Ontology Consortium). All rights reserved. Revised: July 2000