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Example text

Thus, Q is necessarily finite. Axiom [MA] results immediately from the fact that, for any state K, there is an L1-chain from ∅ to K. Turning to the ∪-closure, we take any two states K and L in K and suppose that neither of them is empty or a subset of the other (otherwise ∪-closure holds trivially). Since ∅ ⊂ L, Axiom [L1] implies the existence of an L1–chain ∅ ⊂ {q1 } ⊂ · · · ⊂ {q1 , . . , qn } = L. Let j ∈ {1, . . , n} be the first index with qj ∈ / K. If j > 1; we have {q1 , . . , qj−1 } ⊂ K, and {q1 , .

A knowledge structure (Q, K) is called a quasi learning space if it satisfies the two following conditions. [L1∗ ] Quasi learning smoothness. 5) with p = e(K, L) and Ki = Ki−1 + {qi∗ } for some qi ∈ Q, 1 ≤ i ≤ p. 2) as a quasi L1–chain from K to L. [L2∗ ] Quasi learning consistency. If K, L are two states satisfying K ⊂ L and q is an item such that K + {q ∗ } ∈ K, then L ∪ {q ∗ } ∈ K. Our next definition introduces a nondiscriminative variant of the wellgradedness condition. 3 Definition. A family of sets F is quasi well-graded or a qwg-family if, for any two distinct states K, L ∈ F, there exists a finite sequence of states K = K0 , K1 , .

In some cases, the base can be considerably smaller than the knowledge space, which results in a substantial economy of storage in a computer memory. The extreme case is the power set of a set of n elements, where the 2n knowledge states can be subsumed by the family of the n singleton sets. 6. Other features of knowledge spaces are also important, however, and are dealt with in this chapter. In the next section, we mention in passing the problem of building a knowledge space in practice. The key idea is to code the information concerning the structure in the form of a relation R on the power set of the domain, with the following interpretation: ARB holds if failing all the items in A implies failing all those in B.

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