Surmise Systems

Abstract

When a knowledge structure is a quasi ordinal space, it can be faithfully represented by its surmise relation (cf. Theorem 1.49). In fact, as illustrated by Example 1.46, an ordinal space is completely recoverable from the Hasse diagram of the surmise relation. However, for knowledge structures in general, and even for knowledge spaces, the information provided by the surmise relation may be incomplete. In this chapter, we introduce the ‘surmise system’, a concept generalizing that of a surmise relation, and allowing more than one possible learning history for an item¹. We then derive, in the style of Theorem 1.49, a one-to-one correspondence between knowledge spaces and surmise systems. The surmise systems are closely related to the AND/OR graphs encountered in artificial intelligence. A section of this chapter is devoted to clarifying the relationship between the two concepts. This chapter also contains a discussion of the particular surmise systems which arise from well-graded knowledge spaces. Other highlights are: a generalization of the concept of a Hasse diagram, and a study of intractable ‘cyclic’ histories which leads us to formulate conditions precluding such situations.