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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.〔A (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, (RATH ) and ALEKS.〔(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.〔.〕 ==Definitions== Some basic definitions used in the knowledge space approach - *A tuple consisting of a non-empty set and a set of subsets from is called a ''knowledge structure'' if contains the empty set and . *A knowledge structure is called a ''knowledge space'' if it is closed under union, i.e. if implies . *A knowledge space is called a ''quasi-ordinal knowledge space'' if it is in addition closed under intersection, i.e. if implies . 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: # If and are both feasible subsets of concepts, then 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 is a nonempty feasible subset of concepts, then there is some concept ''x'' in ''S'' such that 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「knowledge space」の詳細全文を読む スポンサード リンク
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