Representing and Learning a Large System of Number Concepts with Latent Predicate Networks

Conventional models of exemplar or rule-based concept learning tend to focus on the acquisition of one concept at a time. They often underemphasize the fact that we learn many concepts as part of large systems rather than as isolated individuals. In such cases, the challenge of learning is not so much in providing stand-alone definitions, but in describing the richly structured relations between concepts. The natural numbers are one of the first such abstract conceptual systems children learn, serving as a serious case study in concept representation and acquisition (Carey, 2009; Fuson, 1988; Gallistel & Gelman, 2005). Even so, models of natural number learning focused on single-concept acquisition have largely ignored two challenges related to natural number’s status as a system of concepts: 1) there is an unbounded set of exact number concepts, each with distinct semantic content; and 2) people can reason flexibly about any of these concepts (even fictitious ones like eighteen-gazillion). To succeed, models must instead learn the structure of the entire infinite set of number concepts, focusing on how relationships between numbers support reference and generalization. Here, we suggest that the latent predicate network (LPN) – a probabilistic context-sensitive grammar formalism – facilitates tractable learning and reasoning for natural number concepts (Dechter, Rule, & Tenenbaum, 2015). We show how to express several key numerical relationships in our framework, and how a Bayesian learning algorithm for LPNs can model key phenomena observed in children learning to count. These results suggest that LPNs might serve as a computational mechanism by which children learn abstract numerical knowledge from utterances about number.

[1]  Y. Hirai,et al.  Modeling the acquisition of counting with an associative network , 2004, Biological Cybernetics.

[2]  James W. Stigler,et al.  Counting in Chinese: Cultural variation in a basic cognitive skill , 1987 .

[3]  S. Dehaene,et al.  The Number Sense: How the Mind Creates Mathematics. , 1998 .

[4]  K. Holyoak,et al.  The Cambridge handbook of thinking and reasoning , 2005 .

[5]  Noah D. Goodman,et al.  Bootstrapping in a language of thought: A formal model of numerical concept learning , 2012, Cognition.

[6]  Pierre Boullier,et al.  Range Concatenation Grammars , 2000, IWPT.

[7]  E. Spelke,et al.  Language and Conceptual Development series Core systems of number , 2004 .

[8]  S. Carey The Origin of Concepts , 2000 .

[9]  Kenichi Kurihara,et al.  Variational Bayes via propositionalized probability computation in PRISM , 2008, Annals of Mathematics and Artificial Intelligence.

[10]  Robert S. Siegler,et al.  The Development of Numerical Understandings , 1982 .

[11]  Noah D. Goodman,et al.  Theory learning as stochastic search in the language of thought , 2012 .

[12]  K. Fuson Children's Counting and Concepts of Number , 1987 .

[13]  Joshua B. Tenenbaum,et al.  Latent Predicate Networks: Concept Learning with Probabilistic Context-Sensitive Grammars , 2015, AAAI Spring Symposia.

[14]  Thomas L. Griffiths,et al.  A Rational Analysis of Rule-Based Concept Learning , 2008, Cogn. Sci..

[15]  G. Miller,et al.  Cognitive science. , 1981, Science.

[16]  James R. Hurford,et al.  The linguistic theory of numerals , 1975 .

[17]  John Richards,et al.  The Acquisition and Elaboration of the Number Word Sequence , 1982 .

[18]  A. Gopnik The Scientist as Child , 1996, Philosophy of Science.

[19]  C. Gallistel,et al.  Mathematical Cognition , 2005 .