# Knowledge space

In mathematical psychology, a knowledge space is a combinatorial structure describing the possible states of knowledge of a human learner.[1] 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[2] and have since been studied by many other researchers.[3][4] They also form the basis for two computerized tutoring systems, RATH and ALEKS.[5]

It is possible to interpret a knowledge space as a special form of a restricted latent class model.[6]

## Definitions

Some basic definitions used in the knowledge space approach -

• A tuple ${\displaystyle (Q,K)}$ consisting of a non-empty set ${\displaystyle Q}$ and a set ${\displaystyle K}$ of subsets from ${\displaystyle Q}$ is called a knowledge structure if ${\displaystyle K}$ contains the empty set and ${\displaystyle Q}$.
• A knowledge structure is called a knowledge space if it is closed under union, i.e. ${\displaystyle \cup F\in K}$ whenever ${\displaystyle F\subseteq K}$.[7]
• A knowledge space is called a quasi-ordinal knowledge space if it is in addition closed under intersection, i.e. if ${\displaystyle S,T\in K}$ implies ${\displaystyle S\cap T\in K}$. Closure under both unions and intersections gives (Q,∪,∩) the structure of a distributive lattice; Birkhoff's representation theorem for distributive lattices 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 additional mathematical axioms:

1. If ${\displaystyle S}$ and ${\displaystyle T}$ are both feasible subsets of concepts, then ${\displaystyle S\cup T}$ 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.
2. If ${\displaystyle S}$ is a nonempty feasible subset of concepts, then there is some concept x in S such that ${\displaystyle S\setminus \{x\}}$ is also feasible. In educational terms: any feasible state of knowledge can be reached by learning one concept at a time, for a finite set of concepts to be learned.

A set family satisfying these two axioms forms a mathematical structure known as an antimatroid.

## Construction of knowledge spaces

In practice, there exist several methods to construct knowledge spaces. The most frequently used method is querying experts. There exist several querying algorithms that allow one or several experts to construct a knowledge space by answering a sequence of simple questions.[8][9][10]

Another method is to construct the knowledge space by explorative data analysis (for example by Item tree analysis) from data.[11][12] A third method is to derive the knowledge space from an analysis of the problem solving processes in the corresponding domain.[13]

## References

1. ^ Doignon, J.-P.; Falmagne, J.-Cl. (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2.
2. ^ Doignon, J.-P.; Falmagne, J.-Cl. (1985), "Spaces for the assessment of knowledge", International Journal of Man-Machine Studies, 23 (2): 175–196, doi:10.1016/S0020-7373(85)80031-6.
3. ^ Falmagne, J.-Cl.; Albert, D.; Doble, C.; Eppstein, D.; Hu, X. (2013), Knowledge Spaces. Applications in Education, Springer.
4. ^ A bibliography on knowledge spaces maintained by Cord Hockemeyer contains over 400 publications on the subject.
5. ^ Introduction to Knowledge Spaces: Theory and Applications, Christof Körner, Gudrun Wesiak, and Cord Hockemeyer, 1999 and 2001.
6. ^ Schrepp, M. (2005), "About the connection between knowledge structures and latent class models", Methodology, 1 (3): 92–102, doi:10.1027/1614-2241.1.3.92.
7. ^ Falmagne, Jean-Claude; Doignon, Jean-Paul (2010-09-10). Learning Spaces: Interdisciplinary Applied Mathematics. Springer Science & Business Media. ISBN 9783642010392.
8. ^ Koppen, M. (1993), "Extracting human expertise for constructing knowledge spaces: An algorithm", Journal of Mathematical Psychology, 37: 1–20, doi:10.1006/jmps.1993.1001.
9. ^ Koppen, M.; Doignon, J.-P. (1990), "How to build a knowledge space by querying an expert", Journal of Mathematical Psychology, 34 (3): 311–331, doi:10.1016/0022-2496(90)90035-8.
10. ^ Schrepp, M.; Held, T. (1995), "A simulation study concerning the effect of errors on the establishment of knowledge spaces by querying experts", Journal of Mathematical Psychology, 39 (4): 376–382, doi:10.1006/jmps.1995.1035
11. ^ Schrepp, M. (1999), "Extracting knowledge structures from observed data", British journal of mathematical and statistical psychology, 52 (2): 213–224, doi:10.1348/000711099159071
12. ^ Schrepp, M. (2003), "A method for the analysis of hierarchical dependencies between items of a questionnaire" (PDF), Methods of Psychological Research Online, 19: 43–79
13. ^ Albert, D.; Lukas, J. (1999), Knowledge Spaces: Theories, Empirical Research, Applications, Lawrence Erlbaum Associates, Mahwah, NJ