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The structure of intrinsic complexity of learning

Published online by Cambridge University Press:  12 March 2014

Sanjay Jain  and
Arun Sharma
Sanjay Jain
Affiliation:
Department of Information Systems and Computer Science, National University of Singapore, Singapore 119260, Republic of Singapore E-mail: sanjay@iscs.nus.sg
Arun Sharma
Affiliation:
School of Computer Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia, E-mail: arun@cse.unsw.edu.au
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Abstract

Limiting identification of r.e. indexes for r.e. languages (from a presentation of elements of the language) and limiting identification of programs for computable functions (from a graph of the function) have served as models for investigating the boundaries of learnability. Recently, a new approach to the study of “intrinsic” complexity of identification in the limit has been proposed. This approach, instead of dealing with the resource requirements of the learning algorithm, uses the notion of reducibility from recursion theory to compare and to capture the intuitive difficulty of learning various classes of concepts. Freivalds, Kinber, and Smith have studied this approach for function identification and Jain and Sharma have studied it for language identification.

The present paper explores the structure of these reducibilities in the context of language identification. It is shown that there is an infinite hierarchy of language classes that represent learning problems of increasing difficulty. It is also shown that the language classes in this hierarchy are incomparable, under the reductions introduced, to the collection of pattern languages.

Richness of the structure of intrinsic complexity is demonstrated by proving that any finite, acyclic, directed graph can be embedded in the reducibility structure. However, it is also established that this structure is not dense. The question of embedding any infinite, acyclic, directed graph is open.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 4 , December 1997 , pp. 1187 - 1201
Copyright
Copyright © Association for Symbolic Logic 1997

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References

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