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Ordinal Mind Change Complexity of Language Identification
"... The approach of ordinal mind change complexity, introduced by Freivalds and Smith, uses (notations for) constructive ordinals to bound the number of mind changes made by a learning machine. This approach provides a measure of the extent to which a learning machine has to keep revising its estimate o ..."
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Cited by 18 (6 self)
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The approach of ordinal mind change complexity, introduced by Freivalds and Smith, uses (notations for) constructive ordinals to bound the number of mind changes made by a learning machine. This approach provides a measure of the extent to which a learning machine has to keep revising its estimate of the number of mind changes it will make before converging to a correct hypothesis for languages in the class being learned. Recently, this notion, which also yields a measure for the difficulty of learning a class of languages, has been used to analyze the learnability of rich concept classes. The present paper further investigates the utility of ordinal mind change complexity. It is shown that for identification from both positive and negative data and n ≥ 1, the ordinal mind change complexity of the class of languages formed by unions of up to n + 1 pattern languages is only ω ×O notn(n) (where notn(n) is a notation for n, ω is a notation for the least limit ordinal and ×O represents ordinal multiplication). This result nicely extends an observation of Lange and Zeugmann
Inductive Inference with Procrastination: Back to Definitions
 Fundamenta Informaticae
, 1999
"... In this paper, we reconsider the denition of procrastinating learning machines. In the original denition of Freivalds and Smith [FS93], constructive ordinals are used to bound mindchanges. We investigate possibility of using arbitrary linearly ordered sets to bound mindchanges in similar way. It ..."
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Cited by 8 (2 self)
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In this paper, we reconsider the denition of procrastinating learning machines. In the original denition of Freivalds and Smith [FS93], constructive ordinals are used to bound mindchanges. We investigate possibility of using arbitrary linearly ordered sets to bound mindchanges in similar way. It turns out that using certain ordered sets it is possible to dene inductive inference types dierent from the previously known ones. We investigate properties of the new inductive inference types and compare them to other types. This research was supported by Latvian Science Council Grant No.93.599 and NSF Grant 9421640. Some of the results from this paper were presented earlier [AFS96]. y The third author was supported in part by NSF Grant 9301339. 1 Introduction We study inductive inference using the model developed by Gold [Gol67]. There is a well known hierarchy of larger and larger classes of learnable sets of phenomena based on the number of time a learning machine is all...
Probabilistic and Team PFINtype Learning: General Properties
"... We consider the probability hierarchy for Popperian FINite learning and study the general properties of this hierarchy. We prove that the probability hierarchy is decidable, i.e. there exists an algorithm that receives p1 and p2 and answers whether PFINtype learning with the probability of success ..."
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Cited by 3 (3 self)
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We consider the probability hierarchy for Popperian FINite learning and study the general properties of this hierarchy. We prove that the probability hierarchy is decidable, i.e. there exists an algorithm that receives p1 and p2 and answers whether PFINtype learning with the probability of success p1 is equivalent to PFINtype learning with the probability of success p2. To prove our result, we analyze the topological structure of the probability hierarchy. We prove that it is wellordered in descending ordering and orderequivalent to ordinal ffl0. This shows that the structure of the hierarchy is very complicated. Using similar methods, we also prove that, for PFINtype learning, team learning and probabilistic learning are of the same power.
Parsimony Hierarchies for Inductive Inference
 JOURNAL OF SYMBOLIC LOGIC
, 2004
"... 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 parsi ..."
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Cited by 2 (1 self)
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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 the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the limcomputable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.
Abstract
"... 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
Mind Change Optimal Learning: . . .
, 2007
"... Learning theories play a significant role to machine learning as computability and complexity theories to software engineering. Gold’s language learning paradigm is one cornerstone of modern learning theories. The aim of this thesis is to establish an inductive principle in Gold’s language learning ..."
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Learning theories play a significant role to machine learning as computability and complexity theories to software engineering. Gold’s language learning paradigm is one cornerstone of modern learning theories. The aim of this thesis is to establish an inductive principle in Gold’s language learning paradigm to guide the design of machine learning algorithms. We follow the common practice of using the number of mind changes to measure complexity of Gold’s language learning problems, and study efficient learning with respect to mind changes. Our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem, but also locally in subproblems after receiving some evidence. Formalizing this idea leads to the notion of mind change optimality. We characterize mind change complexity of language collections with Cantor’s classic concept of accumulation order. We show that the characteristic property of mind change optimal learners is that they output conjectures (languages) with maximal accumulation order. Therefore, we obtain an inductive principle in Gold’s language learning paradigm based on the simple topological concept accumulation order. The new
Mind Change Complexity of Learning Logic Programs
"... The present paper motivates the study of mind change complexity for learning minimal models of lengthbounded logic programs. It establishes ordinal mind change complexity bounds for learnability of these classes both from positive facts and from positive and negative facts. Building on Angluin’s no ..."
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The present paper motivates the study of mind change complexity for learning minimal models of lengthbounded logic programs. It establishes ordinal mind change complexity bounds for learnability of these classes both from positive facts and from positive and negative facts. Building on Angluin’s notion of finite thickness and Wright’s work on finite elasticity, Shinohara defined the property of bounded finite thickness to give a sufficient condition for learnability of indexed families of computable languages from positive data. This paper shows that an effective version of Shinohara’s notion of bounded finite thickness gives sufficient conditions for learnability with ordinal mind change bound, both in the context of learnability from positive data and for learnability from complete (both positive and negative) data. Let ω be a notation for the first limit ordinal. Then, it is shown that if a language defining framework yields a uniformly decidable family of languages and has effective bounded finite thickness, then for each natural number m> 0, the class of languages defined by formal systems of length ≤ m: • is identifiable in the limit from positive data with a mind change bound of ω m; • is identifiable in the limit from both positive and negative data with an ordinal mind change bound of ω × m. The above sufficient conditions are employed to give an ordinal mind change bound for learnability of minimal models of various classes of lengthbounded Prolog programs, including Shapiro’s linear programs, Arimura and Shinohara’s depthbounded linearlycovering programs, and Krishna Rao’s depthbounded linearlymoded programs. It is also noted that the bound for learning from positive data is tight for the example classes considered.