Dickson Lukose,  Rob Kremer

KNOWLEDGE ENGINEERING, PART A: Knowledge Representation

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Lecture 7. Reasoning and Computation

7.1. Schemata and Prototype

• The basic structure for representing background knowledge for human-like inference is called the schema.
  
  
• Schema is a pattern derived from past experience that is used for interpreting, planning, and imagining other experiences.
  
  
• Examples of schemata are:
• Constellations (Ceccato, 1961)
• Frames (Minsky, 1975)
• Scripts (Schank and Abelson, 1977)
  
  
• In terms of complexities of conceptual graphs, schemata form the third level of complexity:
  
  
• Arbitrary conceptual graphs impose no constraints on permissible combinations.
  
  
• Canonical graphs enforce selectional constraints. They correspond to the case frames in linguistics and the category restrictions in philosophy.
  
  
• Schemata incorporate domain-specific knowledge about the typical constellations of entities, attributes, and events in the real world.
  
  
• By enforcing selectional constraints, canonical graphs rule out anomalies like green ideas sleeping, but they allow such unlikely combinations as purple cows:

[SLEEP]->(AGNT)->[IDEA]->(COLR)->[GREEN]

[SLEEP]->(AGNT)->[COW]->[COLR)->[PURPLE]

• Canonical graphs represent everything that is conceivable, and schemata represents everything that is plausible.
  
  
• Schemata are similar in structure to type definition.
  
  
• A concept type may have at most one definition, but arbitrarily many schemata.
  
  
• Type definition presents the narrow notion of a concept, and schemata present the broad notion.
  
  
• Type definitions are obligatory conditions that state only the essential properties, but schemata are optional defaults that state the commonly associated accidental properties.
  
  
• Type definition contains obligatory or essential properties that must hold for the type.
  
  
• Type definitions are appropriate for some of the formal concepts of science, law, or accounting.
  
  
• Schemata are necessary for the loosely structured concepts of everyday life.
  
  
• Each schemata presents a perspective on one way a concept type may be used.
  
  
• The collection of all the perspectives for a type is called its schematic cluster.
  
  
• 4.1.1 Definition. A schematic cluster for a type t is a set of monadic abstractions {la₁u₁, ......., la_nu_n} where each formal parameter a_i is of type t. Each abstraction la₁u₁ in the set is called a schema for the type t.
  
  
• Example:

• 4.1.2 Definition. Any schema for a supertype of a type t is also a schema for type t. If lau is a schema in the schematic cluster for t, then it is called an immediate schema for t. If a schema occurs in a schematic cluster of the supertype of t, it is called an indirect schema of t.
  
  
• The concepts and relations of a schema serve both as conditions for determining whether the schema is applicable and as defaults that may be joined to a graph as long as they are consistent with it.
  
  
• 4.1.3 Definition. Let v be a canonical graph containing a concept b, and let lau be a schema for type(b). Then a schematic join of lau to v is a maximal join of u to v with the concept b joined to the formal parameter a.
  
  
• Schemata show the typical ways in which a concept may be used but they do not describe a typical instance of a concept.
  
  
• A prototype is a typical instance.
  
  
• 4.2.4 Definition. A prototype p for a type t is a monadic abstraction lau with the following properties:
• The formal parameter a is of type t.
• The prototype p is derived by a schematic join of one or more schemata in the schematic cluster for t, with some or all of the concepts in p restricted from generic to individual.
  
  
• Example:

prototype for ELEPHANT(x) is
  [ELEPHANT: *x]-
    (CHRC)->[HEIGHT: @ 3.3 m]
    (CHRC)->[WEIGHT: @ 5400 kg]
    (COLR)->[DARK-GRAY]
    (PART)->[NOSE]-
      (ATTR)->[PREHENSILE]
      (IDNT)->[TRUNK],
    (PART)->[EAR: {*}]-
      (QTY)->[NUMBER: 2]
      (ATTR)->[FLOPPY],
    (PART)->[TUSK: {*}]-
      (QTY)->[NUMBER: 2]
      (MATR)->[IVORY],
    (PART)->[LEG: {*}]->(QTY)->[NUMBER: 4]
    (STAT)->[LIVE]-
      (LOC)->[CONTINENT: {Africa | Asia}]
      (DUR)->[TIME: @ 50 years].

• Summary:
• A type definition introduces a new type defined in terms of a graph called the differentia.
• A aggregation specialises concepts in the differentia of a basis type in order to define a composite individual of that type.
• A schema shows concepts and relations that are commonly associated with a particular concept type. Unlike type definition, the relationships in a schema are not necessary and sufficient conditions for that type.
• A prototype specialises concepts in one or more schemata to show the form of a typical individual. Unlike aggregations, a prototype specifies defaults that are true of a typical case, but necessary for any particular case.

Dickson Lukose &  Rob Kremer.  KNOWLEDGE ENGINEERING, PART A: Knowledge Representation.  July 1996.

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