Calculating the Mind Change Complexity of Learning Algebraic Structures

This paper studies algorithmic learning theory applied to algebraic structures. In previous papers, we have defined our framework, where a learner, given a family of structures, receives larger and larger pieces of an arbitrary copy of a structure in the family and, at each stage, is required to out...

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Bibliographic Details
Published inRevolutions and Revelations in Computability Vol. 13359; pp. 1 - 12
Main Authors Bazhenov, Nikolay, Cipriani, Vittorio, San Mauro, Luca
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 2022
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Algorithmic learning theory
Computable structures
Inductive inference
Mind change complexity
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ISBN3031087399
9783031087394
ISSN0302-9743
1611-3349
DOI10.1007/978-3-031-08740-0_1

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Summary:This paper studies algorithmic learning theory applied to algebraic structures. In previous papers, we have defined our framework, where a learner, given a family of structures, receives larger and larger pieces of an arbitrary copy of a structure in the family and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. The learning is successful if there is a learner that eventually stabilizes to a correct conjecture. Here, we analyze the number of mind changes that are needed to learn a given family K $$\mathfrak {K}$$ . We give a descriptive set-theoretic interpretation of such mind change complexity. We also study how bounding the Turing degree of learners affects the mind change complexity of a given family of algebraic structures.
Bibliography:Bazhenov was supported by the Ministry of Education and Science of the Republic of Kazakhstan, grant AP08856493 “Positive graphs and computable reducibility on them as mathematical model of databases”. Cipriani’s research was partially supported by the Italian PRIN 2017 Grant “Mathematical Logic: models, sets, computability”. We also thank the anonymous referees for their careful reading of the paper and the valuable suggestions.
Original Abstract: This paper studies algorithmic learning theory applied to algebraic structures. In previous papers, we have defined our framework, where a learner, given a family of structures, receives larger and larger pieces of an arbitrary copy of a structure in the family and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. The learning is successful if there is a learner that eventually stabilizes to a correct conjecture. Here, we analyze the number of mind changes that are needed to learn a given family K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {K}$$\end{document}. We give a descriptive set-theoretic interpretation of such mind change complexity. We also study how bounding the Turing degree of learners affects the mind change complexity of a given family of algebraic structures.
ISBN:3031087399
9783031087394
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-031-08740-0_1