# Enumerable classes of total recursive functions: Complexity of inductive inference

## Abstract

This paper includes some results on complexity of inductive inference for enumerable classes of total recursive functions, where enumeration is considered in more general meaning than usual recursive enumeration. The complexity is measured as the worst-case mindchange (error) number for the first n functions of the given class. Three generalizations are considered.

First: the numbering is computed in limit (with a fixed number of mind-changes). Then the complexity can be arbitrary fast growing recursive function. Second: a fixed number of functions are given by the enumbering function wrongly. In this case only universal strategies have large complexity function.

Third: every function given by the enumbering function can differ in a fixed number of points from the corresponding genuine function of the class. Two cases are considered: functions given by the enumbering function can be only partially defined or they must be total. In the first case there are unidentifiable classes. In the second case there are logarithmic algorithms for prediction and EX-identifying and linear algorithms for identifying of *τ*-indices.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.J. Barzdins. Limiting synthesis of
*τ*-indices. Theory of Algorithms and Programs, vol. 1, Latvia State University, 1974, pp. 112–116 (in Russian).Google Scholar - 2.J. Barzdins, R. Freivalds. On the prediction of general recursive functions. Soviet Math. Dokl. 1, 1972, pp. 1224–1228.Google Scholar
- 3.J. Barzdins, R. Freivalds. Prediction and limiting synthesis of effectively enumerable classes of functions. Theory of Algorithms and Programs, vol. 1, Latvia State University, 1974, pp. 101–111 (in Russian).Google Scholar
- 4.R. Freivalds, J. Barzdins, K. Podnieks. Inductive inference of recursive functions: complexity bounds. Baltic Computer Science, Lecture Notes in Computer Science, Vol. 502, Springer Verlag, 1991, pp. 111–155.Google Scholar
- 5.E.M. Gold. Language identification in the limit. Information and Control, 10:5, 1967, pp. 447–474.Google Scholar
- 6.A.N. Kolmogorov. Three approaches to the definition of the notion “quantity of information”. Problems Information Transmission 1 (1965), pp. 1–7.Google Scholar
- 7.P. Martin-Lof. On the notion of random sequence. Information and Control, 9 (1966), pp. 602–619.Google Scholar
- 8.H. Rogers, Jr. Theory of recursive functions and effective computability. McGraw-Hill, New York, 1967.Google Scholar