Colt Proceedings 1990 by Mark A. Fulk (Eds.)

By Mark A. Fulk (Eds.)

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One can show the following lemma, stating that if all the dependency matrix assignments agree on every variable not appearing in φ, then under that projection φ is equivalent to φ*. 29 30 Hancock L e m m a 7 Suppose φ is a well structured hypothesis for μ-FDT φ* over Vn, with dependency matrix D. If V CVn are the variables appearing in φ, and there exists a unique partial assignment ρ where D[i,j]v<~* = ρ for all Xi,Xj G V (not necessarily distinct), then φ = φ+\ρ. 3 The Main Routines The first routine we present is I N S E R T .

Repeat until a does not change, (a) For each k φ i,j where a(Xk) Φ b(Xk), i. If Xi is in S(axk<—,xk) or 5 ( α χ ; . - , χ . , χ ^ ^ ) update a to that assignment. 2. χ 1 ιχ Γ Λ, - x m) > then update a to that assignment. 3. xfc ) update a to that assignment. 4. Return a. Figure 2: Make assignment a as close to b as possible while sensitive to Xi. 2 more of the variables besides X , and X j , which gives the η bounds. < ,xi, and thus can be immediately changed to agree with b. (Once a agrees with 6 on some variable, that value will never change).

W e present an identification algorithm using these two types of queries that runs in time polynomial in the number of variables, and show that no such polynomial time algorithm exists that uses either membership or equivalence queries alone (in the latter case under the stipulation that the hypotheses are drawn from the same representation class). 1 Introduction A n interesting model for learning is that of exact identification using membership and equivalence queries, pioneered by Dana Angluin.

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