Geoffrey K. Pullum · 2008
LEARNABILITY. The mathematical theory of language learnability (also known as learnability theory, grammar induction, or grammatical inference) deals with idealized “learning procedures ” for acquiring grammars on the basis of exposure to evidence about languages. In one classic paradigm, presented in a seminal article by Mark Gold (1967), a learning procedure is taken to be an algorithm running an infinite loop on a never-ending stream of inputs. The inputs are grammatical strings chosen from a target language in a known class of languages. That language has to be identified by choosing a grammar for it from a known set of grammars. At each point in the process, any string in the language might be the next string that turns up (strings can turn up repeatedly). After each input the algorithm produces a guess at the grammar. Success in identifying a language consists in eventually guessing a grammar which is correct for the target language, and which is never subsequently abandoned in the face of additional input strings. A class of languages is called identifiable in the limit
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2021
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Frandsen, Gudmund Skovbjerg · 1985
Les Valiant has recently conceived a remarkable mathematical model of learnability. The originality appears through several facets of the model. Objects belonging to a specific concept are given a measure of naturalness in the form of a probability distribution. The learning of a concept takes place by means of a protocol that among other tools allows the use of a source of natural examples. A concept is learnable if a recognition algorithm can be synthesized within a polynomial number of steps. The recognition algorithm is allowed to be incorrect for an adjustable fraction of inputs measured with respect to naturalness.<br/><br/>Technically the model is based on the propositional logic over a finite number of Boolean variables. However, the underlying ideas are quite universal and can be realised by means of an almost arbitrary formal language, which we will demonstrate in this note. A single concept may include infinitely many objects within a formal language frame. Fortunately we can learn such concepts from finite sets of examples only. We shall prove a specific class of concepts to be learnable within the nontrivial formal language of predicate logic.
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SpringerReference · 2012
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Philip Duchastel · IGI Global eBooks · 2003
This chapter discusses usability in terms of the concept of learnability, that is, what makes the content of an online learning site learnable? It considers three core issues underlying learnability: learning theory, instructional design, and curriculum choices. For learning, it proposes a model integrating comprehension, interest, and memorizing. For design, it distinguishes between representational models, illustrative cases, and simple facts. For curriculum content, it suggests focusing on structural issues and contrasts content design with process design. In looking to the future, it emphasizes information design as the true basis for learnability and places online instruction within the larger context of all instructional design.
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Steve Hanneke, Shay Moran, Hilla Schefler et al. · ArXiv.org · 2025
This work explores the connection between differential privacy (DP) and online learning in the context of PAC list learning. In this setting, a $k$-list learner outputs a list of $k$ potential predictions for an instance $x$ and incurs a loss if the true label of $x$ is not included in the list. A basic result in the multiclass PAC framework with a finite number of labels states that private learnability is equivalent to online learnability [Alon, Livni, Malliaris, and Moran (2019); Bun, Livni, and Moran (2020); Jung, Kim, and Tewari (2020)]. Perhaps surprisingly, we show that this equivalence does not hold in the context of list learning. Specifically, we prove that, unlike in the multiclass setting, a finite $k$-Littlestone dimensio--a variant of the classical Littlestone dimension that characterizes online $k$-list learnability--is not a sufficient condition for DP $k$-list learnability. However, similar to the multiclass case, we prove that it remains a necessary condition. To demonstrate where the equivalence breaks down, we provide an example showing that the class of monotone functions with $k+1$ labels over $\mathbb{N}$ is online $k$-list learnable, but not DP $k$-list learnable. This leads us to introduce a new combinatorial dimension, the \emph{$k$-monotone dimension}, which serves as a generalization of the threshold dimension. Unlike the multiclass setting, where the Littlestone and threshold dimensions are finite together, for $k>1$, the $k$-Littlestone and $k$-monotone dimensions do not exhibit this relationship. We prove that a finite $k$-monotone dimension is another necessary condition for DP $k$-list learnability, alongside finite $k$-Littlestone dimension. Whether the finiteness of both dimensions implies private $k$-list learnability remains an open question.
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Irfan Rafique, Jingnong Weng, Yunhong Wang et al. · 2012
Learnability has been regarded as a fundamental usability attribute and plays a significant role for the success of software applications. Good learnability not only leads to a better productivity quickly but also plays a vital role in initial adoption or rejection of a technology. To evaluate and hence improve learnability, understanding the factors affecting learnability is necessary. In this paper we develop a comprehensive model of learnability by identifying and then abstracting learnability attributes logically to lower level. We carry out a case study to prove adequacy of the proposed model. This learnability attribute model can set a baseline for future learnability understanding and evaluation.
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Katarzyna Chmielewska · Acta Polytechnica Hungarica · 2022
Mathability in its definition, refers to cognitive infocommunication and combines machine and the human cognitive capabilities that are essential for mathematics.Through the years, the notion evolved and its methods turned to be applicable for self-education and constructive learning.In this paper, we would like to investigate how some of the methods can be practiced to meet requirements of the contemporary labor market, especially if we consider as an aid of ICT.
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Sean A. Fulop, Nick Chater · Wiley Interdisciplinary Reviews Cognitive Science · 2013
Learnability theory is a body of mathematical and computational results concerning questions such as: when is learning possible? What prior information is required to support learning? What computational or other resources are required for learning to be possible? It is therefore complementary both to the computational project of building machine learning systems and to the scientific project of understanding learning in people and animals through observation and experiment. Learnability theory includes work within a variety of theoretical frameworks, including, for example, identification in the limit, and Bayesian learning, which idealize learning in different ways. Learnability theory addresses one of the foundational questions in cognitive science: to what extent can knowledge be derived from experience? WIREs Cogn Sci 2013, 4:299-306. doi: 10.1002/wcs.1228 For further resources related to this article, please visit the WIREs website.
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Robert Tetlow, P. Von zur Muehlen · Finance and Economics Discussion Series · 2005
In recent years, the learnability of rational expectations equilibria (REE) and determinacy of economic structures have rightfully joined the usual performance criteria among the sought-after goals of policy design. Some contributions to the literature, including Bullard and Mitra (2001) and Evans and Honkapohja (2002), have made significant headway in establishing certain features of monetary policy rules that facilitate learning. However a treatment of policy design for learnability in worlds where agents have potentially misspecified their learning models has yet to surface. This paper provides such a treatment. We begin with the notion that because the profession has yet to settle on a consensus model of the economy, it is unreasonable to expect private agents to have collective rational expectations. We assume that agents have only an approximate understanding of the workings of the economy and that their learning the reduced forms of the economy is subject to potentially destabilizing perturbations. The issue is then whether a central bank can design policy to account for perturbations and still assure the learnability of the model. Our test case is the standard New Keynesian business cycle model. For different parameterizations of a given policy rule, we use structured singular value analysis (from robust control theory) to find the largest ranges of misspecifications that can be tolerated in a learning model without compromising convergence to an REE.
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Clara Cohen, Syed Waqar Nabi, Catherine Higham · OSF Preprints (OSF Preprints)
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Shuxiao Gong · OSF Preprints (OSF Preprints) · 2024
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Syed Waqar Nabi, Catherine Higham, Clara Cohen · OSF Preprints (OSF Preprints) · 2021
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Gyora M. Benedek, Alon Itai · Journal of Computer and System Sciences · 1994
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Katarzyna Chmielewska · Acta Polytechnica Hungarica · 2022
Mathability in its definition, refers to cognitive infocommunication and combines machine and the human cognitive capabilities that are essential for mathematics. Through the years, the notion evolved and its methods turned to be applicable for self-education and constructive learning. In this paper, we would like to investigate how some of the methods can be practiced to meet requirements of the contemporary labor market, especially if we consider as an aid of ICT.
Cited by 6Open access
Robert Tetlow, Peter von zur Muehlen · Journal of Economic Dynamics and Control · 2008
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Robert Tetlow, Peter von zur Muehlen · SSRN Electronic Journal · 2006
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Jasper Lindenberg, Mark Neerincx, Tim F. Paymans et al. · 2004
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Peter C. Gordon · Developmental Psychology · 1990
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M. Rita Manzini · Mind & Language · 1991
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