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Turing degrees and the Ershov hierarchy
 in Proceedings of the Tenth Asian Logic Conference, Kobe, Japan, 16 September 2008, World Scienti…c
"... Abstract. An nr.e. set can be defined as the symmetric difference of n recursively enumerable sets. The classes of these sets form a natural hierarchy which became a wellstudied topic in recursion theory. In a series of groundbreaking papers, Ershov generalized this hierarchy to transfinite level ..."
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Abstract. An nr.e. set can be defined as the symmetric difference of n recursively enumerable sets. The classes of these sets form a natural hierarchy which became a wellstudied topic in recursion theory. In a series of groundbreaking papers, Ershov generalized this hierarchy to transfinite levels based on Kleene’s notations of ordinals and this work lead to a fruitful study of these sets and their manyone and Turing degrees. The Ershov hierarchy is a natural measure of complexity of the sets below the halting problem. In this paper, we survey the early work by Ershov and others on this hierarchy and present the most fundamental results. We also provide some pointers to concurrent work in the field. 1.
Parsimony hierarchies for inductive inference
 Journal of Symbolic Logic
"... 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 requ ..."
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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 limcomputable 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 “notsonearly ” minimal size, e.g., to be within a limcomputable 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 limcomputable 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 limcomputability, the power of
On a generalized notion of mistake bounds
 Information and Computation
"... This paper proposes the use of constructive ordinals as mistake bounds in the online learning model. This approach elegantly generalizes the applicability of the online mistake bound model to learnability analysis of very expressive concept classes like pattern languages, unions of pattern languag ..."
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This paper proposes the use of constructive ordinals as mistake bounds in the online learning model. This approach elegantly generalizes the applicability of the online mistake bound model to learnability analysis of very expressive concept classes like pattern languages, unions of pattern languages, elementary formal systems, and minimal models of logic programs. The main result in the paper shows that the topological property of effective finite bounded thickness is a sufficient condition for online learnability with a certain ordinal mistake bound. An interesting characterization of the online learning model is shown in terms of the identification in the limit framework. It is established that the classes of languages learnable in the online model with a mistake bound of α are exactly the same as the classes of languages learnable in the limit from both positive and negative data by a Popperian, consistent learner with a mind change bound of α. This result nicely builds a bridge between the two models. 1
Counting Extensional Differences in BCLearning
 PROCEEDINGS OF THE 5TH INTERNATIONAL COLLOQUIUM ON GRAMMATICAL INFERENCE (ICGI 2000), SPRINGER LECTURE NOTES IN A. I. 1891
, 2000
"... Let BC be the model of behaviourally correct function learning as introduced by Barzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BCn to models from the ..."
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Let BC be the model of behaviourally correct function learning as introduced by Barzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BCn to models from the literature and discuss confidence, team learning, and finitely defective hypotheses. Among other things, we prove that there is a tradeoff between the number of semantic mind changes and the number of anomalies in the hypotheses. We also discuss consequences for language learning. In particular we show that, in contrast to the case of function learning, the family of classes that are confidently BClearnable from text is not closed under finite unions. Keywords. Models of grammar induction, inductive inference, behaviourally correct learning.
Learning How to Separate
"... The main question addressed in the present work is how to find effectively a recursive function separating two sets drawn arbitrarily from a given collection of disjoint sets. In particular, it is investigated in which cases it is possible to satisfy the following additional constraints: confidence ..."
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The main question addressed in the present work is how to find effectively a recursive function separating two sets drawn arbitrarily from a given collection of disjoint sets. In particular, it is investigated in which cases it is possible to satisfy the following additional constraints: confidence where the learner converges on all datasequences; conservativeness where the learner abandons only definitely wrong hypotheses; consistency where also every intermediate hypothesis is consistent with the data seen so far; setdriven learners whose hypotheses are independent of the order and the number of repetitions of the dataitems supplied; learners where either the last or even all hypotheses are programs of total recursive functions. The present work gives an overview of the relations between these notions and succeeds to answer many questions by finding ways to carry over the corresponding results from other scenarios within inductive inference. Nevertheless, the relations...
Counting Extensional Differences in BCLearning
"... Let BC be the model of behaviourally correct function learning as introduced by Barzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BC n to models from the lit ..."
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Let BC be the model of behaviourally correct function learning as introduced by Barzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BC n to models from the literature and discuss confidence, team learning, and finitely defective hypotheses. Among other things, we prove that there is a tradeoff between the number of semantic mind changes and the number of anomalies in the hypotheses. We also discuss consequences for language learning. In particular we show that, in contrast to the case of function learning, the family of classes that are confidently BClearnable from text is not closed under finite unions.
Counting Extensional Differences in BCLearning \Lambda
"... University of Heidelberg Sebastiaan A. Terwijn x Vrije Universiteit Amsterdam ..."
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University of Heidelberg Sebastiaan A. Terwijn x Vrije Universiteit Amsterdam
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"... Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on at most a bits? Let 0 ≤ a ≤ n − 1. We show that any deterministic protocol, as well as any errorfree quantum protocol (C ∗ version), for this problem requires at least n − 2 bits of communication. We ..."
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Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on at most a bits? Let 0 ≤ a ≤ n − 1. We show that any deterministic protocol, as well as any errorfree quantum protocol (C ∗ version), for this problem requires at least n − 2 bits of communication. We show the same bounds for the problem of determining if two strings differ in exactly a bits. We also prove a lower bound of n/2 − 1 for errorfree Q ∗ quantum protocols. Our results are obtained by lowerbounding the ranks of the appropriate matrices. 1
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"... Abstract The main question addressed in the present work is how to find effectively a recursive function separating two sets drawn arbitrarily from a given collection of disjoint sets. In particular, it is investigated in which cases it is possible to satisfy the following additional constraints: co ..."
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Abstract The main question addressed in the present work is how to find effectively a recursive function separating two sets drawn arbitrarily from a given collection of disjoint sets. In particular, it is investigated in which cases it is possible to satisfy the following additional constraints: confidence where the learner converges on all datasequences; conservativeness where the learner abandons only definitely wrong hypotheses; consistency where also every intermediate hypothesis is consistent with the data seen so far; setdriven learners whose hypotheses are independent of the order and the number of repetitions of the dataitems supplied; learners where either the last or even all hypotheses are programs of total recursive functions. The present work gives an overview of the relations between these notions and succeeds to answer many questions by finding ways to carry over the corresponding results from other scenarios within inductive inference. Nevertheless, the relations between conservativeness and setdriven inference needed a novel approach which enabled to show the following two major results: (1) There is a class for which recursive separators can be found in a confident and setdriven way, but no conservative learner finds a (not necessarily total) separator for this class. (2) There is a class for which recursive separators can be found in a confident and conservative way, but no setdriven learner finds a (not necessarily total) separator for this class.
Counting Extensional Differences in BCLearning
"... Let BC be the model of behaviourally correct function learning as introduced by Bārzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BCn to models from the lit ..."
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Let BC be the model of behaviourally correct function learning as introduced by Bārzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BCn to models from the literature and discuss confidence, team learning, and finitely defective hypotheses. Among other things, we prove that there is a tradeoff between the number of semantic mind changes and the number of anomalies in the hypotheses. We also discuss consequences for language learning. In particular we show that, in contrast to the case of function learning, the family of classes that are confidently BClearnable from text is not closed under finite unions.