Learning of monotone functions with single error correction

Author:

Selezneva Svetlana N.1,Liu Yongqing1

Affiliation:

1. Lomonosov Moscow State University , Moscow , Russia

Abstract

Abstract Learning of monotone functions is a well-known problem. Results obtained by V. K. Korobkov and G. Hansel imply that the complexity φM (n) of learning of monotone Boolean functions equals C n n / 2 $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor} \end{array}$ + C n n / 2 + 1 $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor+1} \end{array}$ (φM (n) denotes the least number of queries on the value of an unknown monotone function on a given input sufficient to identify an arbitrary n-ary monotone function). In our paper we consider learning of monotone functions in the case when the teacher is allowed to return an incorrect response to at most one query on the value of an unknown function so that it is still possible to correctly identify the function. We show that learning complexity in case of the possibility of a single error is equal to the complexity in the situation when all responses are correct.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Discrete Mathematics and Combinatorics

Reference10 articles.

1. Korobkov V. K., “On monotone functions of the algebra logic”, Problemy kibernetiki, 1965, N 13, 5–28 (in Russian).

2. Korobkov V. K., “An estimate of the number of monotone functions of an algebra of logic and the complexity of the algorithm of finding the resolvent set for an aribitrary function of the algebra logic”, Dokl. Akad. Nauk SSSR, 150:4 (1963), 744–747 (in Russian).

3. Hansel G., “Sur le nombre des fonctions Booleennes monotones de n variables”, C. R. Acad. Sci. Paris, 262 (1966), 1088–1090.

4. Alekseev V. B., “On the deciphering some classes of monotone multivalued functions”, 16:1 (1976), 189–198 (in Russian).

5. Sapozhenko A. A., Goryainov M. V., “On decoding monotone functions on partially ordered sets”, Diskretnyy analiz i issle-dovanie operatsiy, 2:3 (1995), 79–80 (in Russian).

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