Published as: K. M. Podnieks. Probabilistic synthesis of enumerated classes of functions. Dokl. Akad. Nauk SSSR, 1975, Vol. 223, N5, pp. 1071–1074 (in Russian), English translation in: Soviet Math. Dokl., 1975, Vol.16, N4, pp. 1042–1045. Proofs were published as: K. M. Podnieks. Probabilistic synthesis of programs. In: Theory of Algorithms and Programs, Vol. 3, Latvia State University, 1977, pp.

... [Show full abstract] 57–88 (in Russian). [[[[[]]]]]
The following model of inductive inference is considered. Arbitrary
numbering tau = {tau-0, tau-1, tau-2, ...} of total functions N->N is fixed. A "black box"
outputs the values f(0), f(1), ..., f(m), ... of some function f from the numbering tau.
Processing these values by some algorithm (a strategy) F we try to identify a tau-index of
f (i.e. a number n such that f = tau-n). Strategy F outputs an infinite sequence of
hypotheses h-0, h-1, ..., h-m, .... If lim h-m = n and tau-n = f, we say that F identifies in the limit
tau-index of f. The complexity of identification is measured by the number of mind changes,
i.e. by F-tau(f) = card{m | h-m <> h-m+1 }. One can verify easily that for any
numbering tau there exists a deterministic strategy F such that F-tau(tau-n) <= n for all n.
This estimate is exact [Ba 74]. In the current paper the corresponding
exact estimate ln n + o(log n) is obtained for probabilistic strategies. [[[[[]]]]]
English translation with proofs: K. Podnieks. INDUCTIVE INFERENCE OF FUNCTIONS BY PROBABILISTIC STRATEGIES, 1992.