# The power of procrastination in inductive inference: How it depends on used ordinal notations

## Abstract

We consider inductive inference with procrastination. Usually it is defined using constructive ordinals. For constructive ordinals there exist many different systems of notations. In this paper we study how the power of inductive inference depends on used system of notations.

We prove that for constructive ordinals *α* smaller than *ω*^{2} each set of total recursive functions which is *EX*_{ α }-identifiable in one system of notations is *EX*_{ α }-identifiable in arbitrary system of notations. For \(EX_{\omega ^2 } \)-identification such property does not hold.

Also, we consider the question whether, among all systems of notations there exists the strongest and the weakest system. We prove that there exist such system of notations *S* that arbitrary set of functions which is *EX*_{ α }-identifiable in some system of notations is *EX*_{ α }-identifiable in *S*, too. If *ω*^{2}*≤α<2ω*^{2}, there exist such system *A* that each set of functions which is *EX*_{ α }-identifiable in *A* is *EX*_{ α }-identifiable in arbitrary system of notations.

Further, we consider possible tradeoffs between using larger ordinals and using more complicated systems of notations. We prove that, in general, there are no such tradeoffs.

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## References

- 1.K. Apsitis,
*Derived sets and inductive inference.*, Proceedings of AII'94, Lecture Notes in Artificial Inteligence, vol. 872.Google Scholar - 2.A. Church,
*The constructive second number class.*, Bull. Amer. Math. Soc., vol. 44(1938), pp.224–232Google Scholar - 3.A. Church, S. Kleene,
*Formal definitions in the theory of ordinal numbers.*, Fund. Math., vol. 28(1937), pp. 11–21Google Scholar - 4.R. V. Freivalds and E. B. Kinber,
*Identification in the limit of minimal Goedel numbers*. In “Theory of Algorithms and Programs”, vol. 3, Latvia State University, Riga, 1977.Google Scholar - 5.R. Freivalds, C.H. Smith,
*The Role of Procrastination in Machine Learning*. Information and Computation, vol. 107(1993), pp. 237–271Google Scholar - 6.S. Kleene,
*On notation for ordinal numbers.*, J. Symbolic Logic, vol.3(1938), pp. 150–155Google Scholar - 7.H. Rogers,
*Theory of Recursive Functions and Effective Computability*. Mcgraw-Hill, 1967.Google Scholar