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Latin American Symposium on Theoretical Informatics

LATIN 2002: LATIN 2002: Theoretical Informatics pp 386–399Cite as

Category, Measure, Inductive Inference: A Triality Theorem and Its Applications

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2286))

Abstract

The famous Sierpinski-Erdös Duality Theorem [Sie34b], [Erd43] states, informally, that any theorem about effiective measure 0 and/or first category sets is also true when all occurrences of “effective measure 0” are replaced by “first category” and vice versa. This powerful and nice result shows that “measure” and “category” are equally useful notions neither of which can be preferred to the other one when making formal the intuitive notion “almost all sets.” Effective versions of measure and category are used in recursive function theory and related areas, and resource-bounded versions of the same notions are used in Theory of Computation. Again they are dual in the same sense.

We show that in the world of recursive functions there is a third equipotent notion dual to both measure and category. This new notion is related to learnability (also known as inductive inference or identifiability). We use the term “triality” to describe this three-party duality.

This project was supported by Latvian Science Council Grant No. 01.0354 and Contract IST-1999-11234 from the European Commission

This project was supported by NSF Grant CCR-9732692

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Author information

Authors and Affiliations

  1. Institute of Math and Computer Science, University of Latvia, Raiņa bulvāris 29, 1459, Riga, LV, Latvia

    Rūsiņš Freivalds

  2. Department of Computer Science, University of Maryland, A.V.Williams Bldg., 20742, College Park, MD, USA

    Carl H. Smith

Authors
  1. Rūsiņš Freivalds
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  2. Carl H. Smith
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Editor information

Editors and Affiliations

  1. Compaq Cambridge Research Laboratory, One Cambridge Center, 02142-1612, Cambridge, MA, USA

    Sergio Rajsbaum

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© 2002 Springer-Verlag Berlin Heidelberg

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Freivalds, R., Smith, C.H. (2002). Category, Measure, Inductive Inference: A Triality Theorem and Its Applications. In: Rajsbaum, S. (eds) LATIN 2002: Theoretical Informatics. LATIN 2002. Lecture Notes in Computer Science, vol 2286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45995-2_35

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  • DOI: https://doi.org/10.1007/3-540-45995-2_35

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  • Print ISBN: 978-3-540-43400-9

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