# Strong separation of learning classes

## Abstract

Suppose LC_{1} and LC_{2} are two machine learning classes each based on a criterion of success. Suppose, for every machine which learns a class of functions according to the LC_{1} criterion of success, there is a machine which learns this class according to the LC_{2} criterion. In the case where the converse does *not* hold LC_{1} is said to be *separated* from LC_{2}. It is shown that for many such separated learning classes from the literature a much *stronger* separation holds: (∀∁ ∈ LC_{1})(∃∁′ ∈ (LC_{2}−LC_{1}))[*∁*′ ⊃ *∁*]. It is also shown that there is a pair of separated learning classes from the literature for which the stronger separation just above does not hold. A philosophical heuristic toward the design of artificially intelligent learning programs is presented with each strong separation result.

## Keywords

Recursive Function Computable Function Inductive Inference Total Function Learning Class## Preview

Unable to display preview. Download preview PDF.

## References

- [AS83]D. Angluin and C. Smith. A survey of inductive inference: Theory and methods.
*Computing Surveys*, 15:237–289, 1983.Google Scholar - [Bar74]J. M. Barzdin. Two theorems on the limiting synthesis of functions.
*In Theory of Algorithms and Programs, Latvian State University, Riga*, 210:82–88, 1974. In Russian.Google Scholar - [BB75]L. Blum and M. Blum. Toward a mathematical theory of inductive inference.
*Information and Control*, 28:125–155, 1975.Google Scholar - [Blu67a]M. Blum. A machine, independent theory of the complexity of recursive functions.
*Journal of the ACM*, 14:322–336, 1967.Google Scholar - [Blu67b]M. Blum. On the size of machines.
*Information and Control*, 11:257–265, 1967.Google Scholar - [BP73]J. M. Barzdin and K. Podnieks. The theory of inductive inference. In
*Mathematical Foundations of Computer Science*, 1973.Google Scholar - [Cas74]J. Case. Periodicity in generations of automata.
*Mathematical Systems Theory*, 8:15–32, 1974.Google Scholar - [Cas86]J. Case. Learning machines. In W. Demopoulos and A. Marras, editors,
*Language Learning and Concept Acquisition*. Ablex Publishing Company, 1986.Google Scholar - [Cas88]J. Case. The power of vacillation. In D. Haussler and L. Pitt, editors,
*Proceedings of the Workshop on Computational Learning Theory*, pages 133–142. Morgan Kaufmann Publishers, Inc., 1988.Google Scholar - [Che81]K. Chen.
*Tradeoffs in Machine Inductive Inference*. PhD thesis, SUNY at Buffalo, 1981.Google Scholar - [CJS91]J. Case, S. Jain, and A. Sharma. Complexity issues for vacillatory function identification. In
*Proceedings, Foundations of Software Technology and Theoretical Computer Science, Eleventh Conference, New Delhi, India. Lecture Notes in Computer Science 560*, pages 121–140. Springer-Verlag, December 1991.Google Scholar - [CNM79]J. Case and S. Ngo Manguelle. Refinements of inductive inference by Popperian machines. Technical Report 152, SUNY/Buffalo, 1979.Google Scholar
- [CS83]J. Case and C. Smith. Comparison of identification criteria for machine inductive inference.
*Theoretical Computer Science*, 25:193–220, 1983.Google Scholar - [Fel72]J. Feldman. Some decidability results on grammatical inference and complexity.
*Information and Control*, 20:244–262, 1972.Google Scholar - [Ful85]M. Fulk.
*A Study of Inductive Inference machines*. PhD thesis, SUNY at Buffalo, 1985.Google Scholar - [Gol67]E. M. Gold. Language identification in the limit.
*Information and Control*, 10:447–474, 1967.Google Scholar - [HU79]J. Hopcroft and J. Ullman.
*Introduction to Automata Theory Languages and Computation*. Addison-Wesley Publishing Company, 1979.Google Scholar - [Kop6l]Zdenek Kopal.
*Numerical Analysis*. Chapman and Hall Ltd., London, 1961.Google Scholar - [KW80]R. Klette and R. Wiehagen. Research in the theory of inductive inference by GDR mathematicians — A survey.
*Information Sciences*, 22:149–169, 1980.Google Scholar - [Mar89]Y. Marcoux. Composition is almost as good as s-1-1. In
*Proceedings, Structure in Complexity Theory-Fourth Annual Conference*. IEEE Computer Society Press, 1989.Google Scholar - [MY78]M. Machtey and P. Young.
*An Introduction to the General Theory of Algorithms*. North Holland, New York, 1978.Google Scholar - [OSW86]D. Osherson, M. Stob, and S. Weinstein.
*Systems that Learn, An Introduction to Learning Theory for Cognitive and Computer Scientists*. MIT Press, Cambridge, Mass., 1986.Google Scholar - [Pop68]K. Popper.
*The Logic of Scientific Discovery*. Harper Torch Books, New York, second edition, 1968.Google Scholar - [Ric80]G. Riccardi.
*The Independence of Control Structures in Abstract Programming Systems*. PhD thesis, SUNY/ Buffalo, 1980.Google Scholar - [Ric81]G. Riccardi. The independence of control structures in abstract programming systems.
*Journal of Computer and System Sciences*, 22:107–143, 1981.Google Scholar - [Rog58]H. Rogers. Gödel numberings of partial recursive functions.
*Journal of Symbolic Logic*, 23:331–341, 1958.Google Scholar - [Rog67]H. Rogers.
*Theory of Recursive Functions and Effective Computability*. McGraw Hill, New York, 1967. Reprinted. MIT Press. 1987.Google Scholar - [Roy87]J. Royer.
*A Connotational Theory of Program Structure*. Lecture Notes in Computer Science 273. Springer Verlag, 1987.Google Scholar - [Wie78]R. Wiehagen.
*Zur Therie der Algrithmischen Erkennung*. PhD thesis, Humboldt-Universitat, Berlin, 1978.Google Scholar