Introduction to Knowledge Spaces

Prerequisite for demand structures: A set of problems can be analyzed in order to reveal so-called problem components inherent all of the problems.
Components may be considered dimensions (or sets) that have certain features (elements) which can vary on these dimensions. These features are called attributes.
Each problem is conceptualized as a specific combination of such attributes, i. e., each problem has a certain value on each of the components.
If a problem analysis has been carried out and attributes have been identified, then, an ordering of problems can be established.

Definition:

Let c be a problem component and D_c a set of demands. A mapping is called a demand assignment for c. Each attribute is assigned a non-empty subset of D_c.
If a demand assignment has been made, an order is established on the attributes of each of the components involved by comparing the respective demand sets for attributes with respect to set inclusion.

Definition:

An attribute order on a problem component c is imposed by the condition .
An attribute c_i is `subordinate' to an attribute c_j if and only if the set of demands assigned to c_i is a subset of the set of demands assigned to c_j.
The set inclusion relation fulfills the three axioms (reflexivity, antisymmetry, transitivity) of a partially ordered set.
According to the second definition on that slide, this ordering is induced on the set of attributes.

Example:

Consider that two problem components c and c' have been identified with the following attributes:

and

. Let

and

the corresponding sets of demands. The demand assignments

are given in following Table.

	c₁	c₂	c₃
	{d₁}	{d₂}	{d₁,d₂,d₃}

	c'₁	c'₂	c_'3
	{d₅}	{d₆}	{d₆,d₆,d₇}

With regard to the definition before, the resulting attribute orders and are visualized in this figures as upward drawings.

Definition:

Let be ordered sets. The Cartesian or direct product can be made into an ordered set by imposing the coordinatewise order defined by

This procedure can be used to establish a partial order on the product of the ordered attribute sets. Each of the resulting elements of the product corresponds to (at least) one problem.

Example

Upward Drawings for the attribute orders and imposed by an ordering of the demand sets which are given aside.

The figure below visualizes the resulting problem order for the last example shown in the figure above.

Upward Drawing for the problem order constructed componentwise relation from the attribute orders.
The procedure described here leads to a partial order on a set of problems or a surmise relation, respectively. A corresponding quasi-ordinal knowledge space can be constructed for such a relation.
Held (1993) also described a procedure with which, under certain conditions, a surmise system can be established on a problem set by a suitable demand assignment.