Abstract
The surprising fact that Hilbert’s basis theorem in algebra shows identifiabilty of ideals of polynomials in the limit from positive data is derived by the correspondence between ideals and languages in the context of machine learning. This correspondence also reveals the difference between the two and raises new problems to be solved in both of algebra and machine learning. In this article we solve the problem of providing a concrete form of the characteristic set of a union of two polynomial ideals. Our previous work showed that the finite basis of every polynomial ideal is its characteristic set, which ensures that the class of ideals of polynomials is identifiable from positive data. Union or set-theoretic sum is a basic set operation, and it could be conjectured that there is some effective method which produces a characteristic set of a union of two polynomial ideals if both of the basis of ideals are given. Unfortunately, we cannot find a previous work which gives a general method for how to find characteristic sets of unions of languages even though the languages are in a class identifiable from positive data. We give methods for computing a characteristic set of the union of two polynomial ideals.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Angluin, D.: Inductive Inference of Formal Languages from Positive Data. Information and Control 45, 117–135 (1980)
Buchberger, B.: Groebner Bases: An Algorithmic Method in Polynomial Ideal Theory. In: Bose, N.K. (ed.) Multidimensional System Theory, pp. 184–232. D. Reidel Publishing Company, Dordrecht (1985)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, Heidelberg (1992) (revised version, 1997)
de Brecht, M., Kobayashi, M., Tokunaga, H., Yamamoto, A.: Inferability of Closed Set Systems 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)
Gold, D.: Language Identfication in the Limit. Information and Control 10, 447–474 (1967)
Hayashi, S.: Mathematics based on Incremental Learning –Excluded middle and Inductive inference. Theoretical Computer Science 350(1) (2006)
Jain, S., Osherson, D., Royer, J.S., Sharma, A.: Systems That Learn – An introduction to Learning Theory, 2nd edn. MIT Press, Cambridge (1999)
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)
Sakakibara, Y., Kobayashi, S., Yokomori, T.: Computational Learning Theory (in Japanese), Baifukan (2001)
Kobayashi, M., Tokunaga, H., Yamamoto, A.: Ideals of Polynomial Rings and Learning from Positive Data (in Japanese). In: Proc. 8th IBIS, pp. 129–134 (2005)
Motoki, T., Shinohara, T., Wright, K.: The Correct Definition of Finite Elasticity: Corrigendum to Identification of Unions. In: Proceedings of COLT 1991, pp. 587–626. Morgan-Kaufman, San Francisco (1991)
Stephan, F., Ventsov, Y.: Learning Algebraic Structures from Text. Theoretical Computer Science 268, 221–273 (2001)
van der Waerden, B.L.: Moderne Algebra, vol. II. Spiringer, Heidelberg (1940)
Wright, K.: Identification of Unions of Languages Drawn from an Identifiable Class. In: Proceedings of COLT 1989, pp. 328–388. Morgan Kaufmann, San Francisco (1989)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Takamatsu, I., Kobayashi, M., Tokunaga, H., Yamamoto, A. (2008). Computing Characteristic Sets of Bounded Unions of Polynomial Ideals. In: Satoh, K., Inokuchi, A., Nagao, K., Kawamura, T. (eds) New Frontiers in Artificial Intelligence. JSAI 2007. Lecture Notes in Computer Science(), vol 4914. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78197-4_30
Download citation
DOI: https://doi.org/10.1007/978-3-540-78197-4_30
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78196-7
Online ISBN: 978-3-540-78197-4
eBook Packages: Computer ScienceComputer Science (R0)