Abstract
Based on the observation that the category of concept spaces with the positive information topology is equivalent to the category of countably based T 0 topological spaces, we investigate further connections between the learning in the limit model of inductive inference and topology. In particular, we show that the “texts” or “positive presentations” of concepts in inductive inference can be viewed as special cases of the “admissible representations” of computable analysis. We also show that several structural properties of concept spaces have well known topological equivalents. In addition to topological methods, we use algebraic closure operators to analyze the structure of concept spaces, and we show the connection between these two approaches. The goal of this paper is not only to introduce new perspectives to learning theorists, but also to present the field of inductive inference in a way more accessible to domain theorists and topologists.
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References
Angluin, D.: Inductive inference of formal languages from positive data. Information and Control 45, 117–135 (1980)
Angluin, D.: Inference of reversible languages. Journal of the ACM 29, 741–765 (1982)
Aull, C.E., Thron, W.J.: Separation axioms between T 0 and T 1. Indag. Math. 24, 26–37 (1963)
Brown, D.J., Suszko, R.: Abstract logics. Dissertationes Mathematicae 102, 9–41 (1973)
Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, Heidelberg (1981)
de Brecht, M., Kobayashi, M., Tokunaga, H., Yamamoto, A.: Inferability of closed set sytems from positive data. In: Washio, T., Satoh, K., Takeda, H., Inokuchi, A. (eds.) JSAI 2006. LNCS (LNAI), vol. 4384, pp. 265–275. Springer, Heidelberg (2007)
de Brecht, M., Yamamoto, A.: Mind change complexity of inferring unbounded unions of pattern languages from positive data. In: Balcázar, J.L., Long, P.M., Stephan, F. (eds.) ALT 2006. LNCS (LNAI), vol. 4264, pp. 124–138. Springer, Heidelberg (2006)
Freivalds, R., Smith, C.H.: On the role of procrastination for machine learning. Information and Computation 107, 237–271 (1993)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.W.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)
Gold, E.M.: Language identification in the limit. Information and Control 10, 447–474 (1967)
Jain, S., Kinber, E., Wiehagen, R.: Language learning from texts: Degrees of intrinsic complexity and their characterizations. Journal of Computer and System Sciences 63, 305–354 (2001)
Jain, S., Sharma, A.: The intrinsic complexity of language identification. Journal of Computer and System Sciences 52, 393–402 (1996)
Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)
Kechris, A.: Classical Descriptive Set Theory. Springer, Heidelberg (1995)
Kelley, J.: General Topology. Springer, Heidelberg (1975)
Kelly, K.: The Logic of Reliable Inquiry. Oxford University Press, Oxford (1996)
Kobayashi, S.: Approximate identification, finite elasticity and lattice structure of hypothesis space. Technical Report, CSIM 96-04, Dept. of Compt. Sci. and Inform. Math., Univ. of Electro- Communications (1996)
Luo, W., Schulte, O.: Mind change efficient learning. Information and Computation 204, 989–1011 (2006)
Mac Lane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)
Martin, E., Osherson, D.: Elements of Scientific Inquiry. MIT Press, Cambridge (1998)
Martin, E., Sharma, A., Stephan, F.: Unifying logic, topology and learning in parametric logic. Theoretical Computer Science 350, 103–124 (2006)
Motoki, T., Shinohara, T., Wright, K.: The correct definition of finite elasticity: Corrigendum to identification of unions. In: Proc. COLT 1991, p. 375 (1991)
Sato, M., Mukouchi, Y., Zheng, D.: Characteristic sets for unions of regular pattern languages and compactness. In: Richter, M.M., Smith, C.H., Wiehagen, R., Zeugmann, T. (eds.) ALT 1998. LNCS (LNAI), vol. 1501, pp. 220–233. Springer, Heidelberg (1998)
Sato, M., Umayahara, K.: Inductive inferability for formal languages from positive data. IEICE Trans. Inf. and Syst. E75-D(4), 415–419 (1992)
Schröder, M.: Extended admissibility. Theoretical Computer Science 284, 519–538 (2002)
Stephan, F., Ventsov, Y.: Learning algebraic structures from text. Theoretical Computer Science 268, 221–273 (2001)
Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)
Wright, K.: Identification of unions of languages drawn from an identifiable class. In: Proc. COLT 1989, pp. 328–333 (1989)
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de Brecht, M., Yamamoto, A. (2008). Topological Properties of Concept Spaces. In: Freund, Y., Györfi, L., Turán, G., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 2008. Lecture Notes in Computer Science(), vol 5254. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87987-9_31
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DOI: https://doi.org/10.1007/978-3-540-87987-9_31
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