Abstract
A one-sided classifier for a given class of languages converges to 1 on every language from the class and outputs 0 infinitely often on languages outside the class. A two-sided classifier, on the other hand, converges to 1 on languages from the class and converges to 0 on languages outside the class. The present paper investigates one-sided and two-sided classification for classes of computable languages. Theorems are presented that help assess the classifiability of natural classes. The relationships of classification to inductive learning theory and to structural complexity theory in terms of Turing degrees are studied. Furthermore, the special case of classification from only positive data is also investigated.
Supported by a grant from the Australian Research Council.
Supported by the Deutsche Forschungsgemeinschaft (DFG) grant Am 60/9-1.
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References
Leonard Adleman and Manuel Blum: Inductive Inference and Unsolvability. Journal of Symbolic Logic 56 (1991) 891–900.
Dana Angluin: Inductive Inference of Formal Languages from Positive Data, Information and Control 45 (1980) 117–135.
Shai Ben-David: Can Finite Samples Detect Singularities of Real-Valued Functions? Proceedings of the 24th Annual ACM Symposium on the Theory of Computer Science, Victoria, B.C., (1992) 390–399.
Lenore Blum and Manuel Blum: Toward a mathematical Theory of Inductive Inference. Information and Control, 28 (1975) 125–155.
J. Richard Büchi: On a decision method in restricted second order arithmetic. In Proceedings of the International Congress on Logic, Methodology and Philosophy of Science, Standford University Press, Standford, California, 1960.
J. Richard Büchi and Lawrence H. Landweber: Definability in the Monadic Second Order Theory of Successor. Journal of Symbolic Logic 34 (1969) 166–170.
John Case, Sanjay Jain and Arun Sharma: On Learning Limiting Programs. International Journal of Foundations of Computer Science, 3 (1992) 93–115.
John Case, Efim Kinber, Arun Sharma and Frank Stephan. On the Classification of Computable Languages. Technical Report No. 9603, School of Computer Science and Engineering, The University of New South Wales, Sydney NSW 2052, Australia.
William Gasarch, Mark Pleszkoch, Frank Stephan and Mahendran Velauthapillai: Classification Using Information. To appear in: Annals of Mathematics and Artificial Intelligence.
E. Mark Gold: Language Identification in the Limit. Information and Control, 10 (1967) 447–474.
Peter G. Hinman: Recursion-Theoretic Hierarchies. Springer-Verlag, Heidelberg, 1978.
Klaus Peter Jantke: Monotonic and Non-Monotonic Inductive Inference. New Generation Computing 8 (1991) 349–360.
Kevin Kelly: The Logic of Reliable Inquiry. Oxford University Press, Oxford, to appear.
Lawrence H. Landweber: Decision Problems for ω-Automata. Mathematical Systems Theory, 3 (1969) 376–384.
Robert McNaughton: Testing and Generating Infinite Sequences by a Finite Automaton. Information and Control 9 (1966) 434–448.
Eliana Minicozzi: Some Natural Properties of Strong Identification in Inductive Inference. Theoretical Computer Science 2 (1976) 345–360.
Maurice Nivat and Dominique Perrin (editors): Automata on Infinite Words. Lecture Notes to Computer Science 192, Springer-Verlag, Heidelberg, 1984.
Piergiorgio Odifreddi: Classical recursion theory. North-Holland, Amsterdam, 1989.
Daniel N. Osherson, Michael Stob and Scott Weinstein: Systems that learn. Bradford / MIT Press, London, 1986.
Hartley Rogers, Jr.: Theory of Recursive Functions and Effective Computability. McGraw-Hill Book Company, New York, 1967.
Gerald E. Sacks: Higher Recursion Theory, Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg, 1990.
Robert I. Soare: Recursively enumerable sets and degrees. Springer-Verlag, Heidelberg, 1987.
Frank Stephan: On One-Sided Versus Two-Sided Classification. Forschungsbericht 25 / 1996 des Mathematischen Instituts der Universität Heidelberg, Heidelberg, 1996.
Rolf Wiehagen and Carl H. Smith: Generalization versus Classification. Journal of Experimental and Theoretical Artificial Intelligence, 7, 1995. Shorter version in Proceedings 5th Annual Workshop on Computational Learning Theory, (1992) 224–230. ACM Press, New York.
Carl H. Smith, Rolf Wiehagen and Thomas Zeugmann: Classifying Predicates and Languages. To appear in: International Journal of Foundations of Computer Science.
Boris A. Trakhtenbrot: Finite Automata and the Logic of One Place Predicates. Siberian Mathematical Journal 3 (1962) 103–131 [in Russian].
Thomas Zeugmann and Steffen Lange: A Guided Tour Across the Boundaries of Learning Recursive Languages. Algorithmic Learning for Knowledge-Based Systems (K. P. Jantke and S. Lange, Eds.), Lecture Notes in Computer Science 961 (1995) 193–262.
Thomas Zeugmann, Steffen Lange and Shyam Kapur: Characterizations of Monotonic and Dual Monotonic Language Learning, Information and Computation 120 (1995) 155–173.
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Case, J., Kinber, E., Sharma, A., Stephan, F. (1997). On the classification of computable languages. In: Reischuk, R., Morvan, M. (eds) STACS 97. STACS 1997. Lecture Notes in Computer Science, vol 1200. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023462
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DOI: https://doi.org/10.1007/BFb0023462
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