Knowledge space

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

An important subclass of knowledge spaces, the well-graded knowledge spaces or learning spaces, can be defined as satisfying two mathematical axioms: A set family satisfying these two axioms forms a mathematical structure known as an antimatroid.
 * 1) If $$S$$ and $$T$$ are both feasible subsets of concepts, then $$S\cup T$$ 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 $$S$$ is a nonempty feasible subset of concepts, then there is some concept x in S such that $$S\setminus\{x\}$$ is also feasible. In educational terms: any feasible state of knowledge can be reached by learning one concept at a time.

Knowledge space is also a term used with a different meaning in philosophy by Pierre Lévy in his 1997 book Collective Intelligence.