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The present paper deals with the best-case, worst-case and average-case behavior of Lange and Wiehagen's (1991) pattern language learning algorithm with respect to its total learning time. Pattern lan...guages have been introduced by Angluin (1980) and are defined as follows: Let $A = { 0,1,..}$ be any non-empty finite alphabet containing at least two elements. Furthermore, let $X = { x_i mid i in IN}$ be an infinite set of variables such that $A cap X = phi$. Patterns are non-empty strings from $A cup X$. $L(pi)$, the language generated by pattern $pi$ is the set of strings which can be obtained by substituting non-null strings from $A^*$ for the variables of the pattern pi. Lange and Wiehagen's (1991) algorithm learns the class of all pattern languages from text in the limit. We analyze this algorithm with respect to its total learning time behavior, i.e., the overall time taken by the algorithm until convergence. For every pattern $pi$ containing $k$ different variables it is shown that the total learning time is $O(log _mid A mid (mid A mid + k) mid pi mid ^2)$ in the best-case and unbounded in the worst-case. Furthermore, we estimate the expectation of the total learning time. In particular, it is shown that Lange and Wiehagen's algorithm possesses an expected total learning time of $O(2^kk^2 mid A mid mid pi mid ^2 log _mid A mid (k mid A mid))$ with respect to the uniform distribution.続きを見る
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