Journal of Symbolic Logic

Parsimony hierarchies for inductive inference

Andris Ambainis, John Case, Sanjay Jain, and Mandayam Suraj

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Abstract

Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and “nearly” minimal size, i.e., within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A lim-computable function is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be “not-so-nearly” minimal size, e.g., to be within a lim-computable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are lim-computable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.

Article information

Source
J. Symbolic Logic, Volume 69, Issue 1 (2004), 287-327.

Dates
First available in Project Euclid: 2 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1080938842

Digital Object Identifier
doi:10.2178/jsl/1080938842

Mathematical Reviews number (MathSciNet)
MR2039362

Zentralblatt MATH identifier
1068.68071

Subjects
Primary: 68Q32: Computational learning theory [See also 68T05]

Keywords
Computational learning theory minimal size program constructive ordinal notations limiting computable function

Citation

Ambainis, Andris; Case, John; Jain, Sanjay; Suraj, Mandayam. Parsimony hierarchies for inductive inference. J. Symbolic Logic 69 (2004), no. 1, 287--327. doi:10.2178/jsl/1080938842. https://projecteuclid.org/euclid.jsl/1080938842


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