# Knowledge space

In mathematical psychology, a knowledge space is a combinatorial structure describing the possible states of knowledge of a human learner. [citation|title=Knowledge Spaces|last1=Doignon|first1=J.-P.|last2=Falmagne|first2=J.-Cl.|publisher=Springer-Verlag|year=1999|isbn = 3-540-64501-2.] To form a knowledge space, one models a domain of knowledge as a set of concepts, and a feasible state of knowledge as a subset of that set containing the concepts known or knowable by some individual. Typically, not all subsets are feasible, due to prerequisite relations among the concepts. The knowledge space is the family of all the feasible subsets. Knowledge spaces were introduced in 1985 by Jean-Paul Doignon and Jean-Claude Falmagne [citation|last1=Doignon|first1=J.-P.|last2=Falmagne|first2=J.-Cl.|year=1985|title=Spaces for the assessment of knowledge|journal=International Journal of Man-Machine Studies|volume=23|pages=175–196.] and have since been studied by many other researchers. [A [http://wundt.uni-graz.at/kst.php bibliography on knowledge spaces] maintained by Cord Hockemeyer contains over 300 publications on the subject.] They also form the basis for two computerized tutoring systems, [http://wundt.kfunigraz.ac.at/rath/ RATH] and [http://www.aleks.com/ ALEKS] . [ [http://wundt.uni-graz.at/MathPsych/cda/overview_sokrates.htm Introduction to Knowledge Spaces: Theory and Applications] , Christof Körner, Gudrun Wesiak, and Cord Hockemeyer, 1999 and 2001.]

It is possible to interpret a knowledge space as a special form of a restricted latent class model [citation|title=About the connection between knowledge structures and latent class models. |last1=Schrepp |first1=M. |publisher=Methodology, 1 (3) ,92-102. |year=2005 .] .

Knowledge space is also a term used with a different meaning in philosophy by Pierre Lévy in his 1997 book "Collective Intelligence". [Jillian Dellit, "Education (and EdNA) in the knowledge age", available at [http://www.csu.edu.au/special/online99/proceedings99/206b.htm] accessed 19 July 2007]

Definitions

Some basic definitions used in the knowledge space approach:
A tuple $\left(Q, K\right)$ consisting of a non-empty set $Q$ and a set $K$ of subsets from $Q$ is called a "knowledge structure" if $K$ contains the empty set and $Q$.
A knowledge structure is called a "knowledge space" if it is closed under union, i.e. if $S, T in Q$ implies $Scup T$.
A knowledge space is called a "quasi-ordinal knowledge space" if it is in addition closed under intersection, i.e. if $S, T in Q$ implies $Scap T$.
The well-known Birkhoff Theorem shows that there is a one-to-one correspondence between the set of all quasiorders on Q and the set of all quasi-ordinal knowledge spaces on Q, i.e. each quasi-ordinal knowledge space can be represented by a quasi-order and vice versa.

An important subclass of knowledge spaces, the "well-graded knowledge spaces" or "learning spaces", can be defined as satisfying two mathematical axioms:
# If $S$ and $T$ are both feasible subsets of concepts, then $Scup T$ is also feasible. In educational terms: if it is possible for someone to know all the concepts in "S", and someone else to know all the concepts in "T", then we can posit the potential existence of a third person who combines the knowledge of both people.
# If $S$ is a nonempty feasible subset of concepts, then there is some concept "x" in "S" such that $Ssetminus\left\{x\right\}$ is also feasible. In educational terms: any feasible state of knowledge can be reached by learning one concept at a time.A set family satisfying these two axioms forms a mathematical structure known as an antimatroid.

Construction

In practice there exist several methods to construct knowledge spaces. The most frequently used method is querying experts. There exists several querying algorithms which allow one or several experts to construct a knowledge space by answering a sequence of simple questions [citation|title=Extracting human expertise for constructing knowledge spaces: An algorithm. |last1=Koppen |first1=M. |publisher=Journal of Mathematical Psychology, 37 ,1-20 |year=1993 .] [citation|title=How to build a knowledge space by querying an expert. |last1=Koppen |first1=M. |last2=Doignon |first2=J.-P. |publisher=Journal of Mathematical Psychology, 34 ,311-331 |year=1990 .] .Another method is to construct the knowledge space by explorative data analysis (for example by Item tree analysis) from data [citation|title=Extracting knowledge structures from observed data. |last1=Schrepp |first1=M. |publisher=British Journal of Mathematical and Statistical Psychology, 52 ,213-224 |year=1999 .] [citation|title=A method for the analysis of hierarchical dependencies between items of a questionnaire. |last1=Schrepp |first1=M. |publisher= Methods of Psychological Research --- Online, 19 ,43-79 |year=2003 .] .

A third method is to derive the knowledge space from an analysis of the problem solving processes in the corresponding domain [citation|title=Knowledge Spaces: Theories, Empirical Research, Applications.|last1=Albert|first1=D.|last2=Lukas|first2=J.|publisher=Lawrence Erlbaum Associates, Mahwah, NJ|year=1999.] .

References

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