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In [[mathematics]], a '''translation plane''' is a particular kind of [[projective plane]], as considered as a combinatorial object.<ref>Projective Planes [http://www.maths.qmul.ac.uk/~pjc/pps/pps2.pdf On projective planes]</ref>
In [[artificial intelligence]], '''model-based reasoning''' refers to an [[inference]] method used in [[expert systems]] based on a [[model (abstract)|model]] of the physical world.  With this approach, the main focus of application development is developing the model.  Then at run time, an "engine" combines this model knowledge with observed data to derive conclusions such as a diagnosis or a prediction.


In a projective plane, <math>\scriptstyle p</math> represents a point, and <math>\scriptstyle L</math> represents a line. A central [[collineation]] with center <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> is a collineation fixing every point on <math>\scriptstyle L</math> and every line through <math>\scriptstyle p</math>. It is called an "elation" if <math>\scriptstyle p</math> is on <math>\scriptstyle L</math>, otherwise it is called a "homology". The central collineations with centre <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> form a group.<ref>Geometry [http://www.math.uni-kiel.de/geometrie/klein/math/geometry/translation.html Translation Plane] Retrieved on June 13, 2007</ref>
== Knowledge representation ==


A projective plane <math>\scriptstyle \Pi</math> is called a translation plane if there exists a line <math>\scriptstyle L</math> such that the group of elations with axis <math>\scriptstyle L</math> is transitive on the affine plane Π<sub>l</sub> (the [[Affine geometry|affine]] derivative of Π).
In a model-based reasoning system [[knowledge]] can be [[knowledge representation|represented]] using '''[[causal rules]]'''. For example, in a [[medical diagnosis system]] the [[knowledge base]] may contain the following rule:
: <math>\forall</math> patients : Stroke(patient) <math>\rightarrow</math> Confused(patient) <math>\land</math> Unequal(Pupils(patient))
In contrast in a [[diagnostic reasoning]] system knowledge would be represented through [[diagnostic rules]] such as:
: <math>\forall</math> patients : Confused(patient) <math>\rightarrow</math> Stroke(patient)
: <math>\forall</math> patients : Unequal(Pupils(patient)) <math>\rightarrow</math> Stroke(patient)


== Relationship to spreads ==
There are many other forms of models that may be used.  Models might be quantitative (for instance, based on mathematical equations) or qualitative (for instance, based on cause/effect models.)  They may include representation of uncertainty.  They might represent behavior over time.  They might represent "normal" behavior, or might only represent abnormal behavior, as in the case  of the examples above. Model types and usage for model-based reasoning are discussed in.<ref>[http://gregstanleyandassociates.com/whitepapers/FaultDiagnosis/Model-Based-Reasoning/model-based-reasoning.htm Model Based Reasoning for Fault Detection and Diagnosis]</ref>
Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction.<ref>{{cite web|url=http://www-ma4.upc.es/~simeon/bblpsympspread.pdf|title=Symplectice Spreads|last=Ball|first=Simeon|author2=John Bamberg |author3=Michel Lavrauw |author4=Tim Penttila |date=2003-09-15|publisher=[[Polytechnic University of Catalonia]]|accessdate=2008-10-08}}</ref> A spread of <math>\scriptstyle PG(3, q) </math> is a set of ''q''<sup>2</sup>&nbsp;+&nbsp;1 lines, with no two intersecting. Equivalently, it is a partition of the points of <math>\scriptstyle PG(3, q) </math> into lines.


Given a spread <math>\scriptstyle S</math> of <math>\scriptstyle PG(3, q) </math>, the André/Bruck-Bose construction<sup>1</sup> produces a translation plane <math>\scriptstyle \pi(S)</math> of order ''q''<sup>2</sup> as follows:  Embed <math>\scriptstyle PG(3, q) </math> as a hyperplane of <math>\scriptstyle PG(4, q) </math>. Define an incidence structure <math>\scriptstyle A(S)</math> with "points," the points of <math>\scriptstyle PG(4, q) </math> not on <math>\scriptstyle PG(3, q) </math> and "lines" the planes of <math>\scriptstyle PG(4, q) </math> meeting <math>\scriptstyle PG(3, q) </math> in a line of <math>\scriptstyle S</math>. Then <math>\scriptstyle A(S)</math> is a translation affine plane of order ''q''<sup>2</sup>. Let <math>\scriptstyle \pi(S)</math> be the projective completion of <math>\scriptstyle A(S)</math>.<ref>{{cite book
== See also ==
  | last =André  | first =Johannes  | authorlink =  | title = Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe  | publisher = | year =1954  | location = | pages =156–186  | url =  | doi =  | id =  }}</ref><ref>{{cite book
* [[Diagnosis (Artificial intelligence)|Diagnosis]]
  | last =Bruck  | first = R. H. | authorlink = Richard Bruck|author2=R. C. Bose  | title = The Construction of Translation Planes from Projective Spaces  | publisher =  | year =1964  | location =  | pages = 85–102  | url =  | doi =  | id =  }}</ref>


==References==
== References ==
{{Reflist}}
{{reflist}}
* {{Russell Norvig 2003|pages=260}}


==Further reading==
== External links ==
* Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) ''Foundations of Translation Planes'', [[Marcel Dekker]] ISBN 0-8247-0609-9 .
* [http://www.cs.uu.nl/docs/vakken/mbr Model-based reasoning at Utrecht University]
* [http://ti.arc.nasa.gov/ NASA Intelligent Systems Division]


==External links==
[[Category:Artificial intelligence]]
*[http://www.library.tuiasi.ro/ipm/vol13no34/pure.html  Foundations_of_Translation_Planes]
[[Category:Decision theory]]
*[http://www-math.ucdenver.edu/~wcherowi/courses/m6221/pglc3a.html Lecture Notes on Projective Geometry]
[[Category:Reasoning]]
*[http://mellinger.umwblogs.org/publications/ Publications of Keith Mellinger]


{{DEFAULTSORT:Translation Plane}}
 
[[Category:Projective geometry]]
{{compu-AI-stub}}

Revision as of 14:25, 14 August 2014

In artificial intelligence, model-based reasoning refers to an inference method used in expert systems based on a model of the physical world. With this approach, the main focus of application development is developing the model. Then at run time, an "engine" combines this model knowledge with observed data to derive conclusions such as a diagnosis or a prediction.

Knowledge representation

In a model-based reasoning system knowledge can be represented using causal rules. For example, in a medical diagnosis system the knowledge base may contain the following rule:

patients : Stroke(patient) Confused(patient) Unequal(Pupils(patient))

In contrast in a diagnostic reasoning system knowledge would be represented through diagnostic rules such as:

patients : Confused(patient) Stroke(patient)
patients : Unequal(Pupils(patient)) Stroke(patient)

There are many other forms of models that may be used. Models might be quantitative (for instance, based on mathematical equations) or qualitative (for instance, based on cause/effect models.) They may include representation of uncertainty. They might represent behavior over time. They might represent "normal" behavior, or might only represent abnormal behavior, as in the case of the examples above. Model types and usage for model-based reasoning are discussed in.[1]

See also

References

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External links


Template:Compu-AI-stub

  1. ↑ Model Based Reasoning for Fault Detection and Diagnosis
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