Frequency prediction of functions ⋆ Kaspars Balodis, Ilja Kucevalovs, R¯ usi¸ nˇ s Freivalds Institute of Mathematics and Computer Science, University of Latvia, Rai¸ na bulv¯ aris 29, Riga, LV-1459, Latvia Abstract. Prediction of functions is one of processes considered in in- ductive inference. There is a ”black box” with a given total function f in it. The result of the inductive inference machine F (<f (0),f (1), ··· ,f (n) > ) is expected to be f (n + 1). Deterministic and probabilistic prediction of functions has been widely studied. Frequency computation is a mech- anism used to combine features of deterministic and probabilistic algo- rithms. Frequency computation has been used for several types of induc- tive inference, especially, for learning via queries. We study frequency prediction of functions and show that that there exists an interesting hierarchy of predictable classes of functions. 1 Introduction Physicists are well aware that physical indeterminism is a complicated phe- nomenon and probabilistical models are merely reasonably good approximations of reality. The problem ”What is randomness?” has always been interesting not only for philosophers and physicists but also for computer scientists. The term ”nondeterministic algorithm” has been deliberately coined to differ from ”inde- terminism”. Probabilistic (randomized) algorithms is one of central notions in Theory of Computation. However, since long ago computer scientists have attempted to develop notions and technical implementations of these notions that would be similar to but not equal to randomization. The notion of frequency computation was introduced by G. Rose [28] as an attempt to have an absolutely deterministic mechanism with properties similar to probabilistic algorithms. The definition was as follows. A function f : w → w is (m, n)-computable, where 1 ≤ m ≤ n, iff there exists a recursive function R: w n → w n such that, for all n-tuples (x 1 , ··· ,x n ) of distinct natural numbers, card{i :(R(x 1 , ··· ,x n )) i = f (x i )}≥ m. R. McNaughton cites in his survey [25] a problem (posed by J. Myhill) whether f has to be recursive if m is close to n. This problem was answered by B.A. Trakhtenbrot [31] by showing that f is recursive whenever 2m>n. On ⋆ The research was supported by Grant No. 09.1570 from the Latvian Council of Science and by Project 2009/0216/1DP/1.1.1.2.0/09/IPIA/VIA/044 from the Eu- ropean Social Fund.