Associative Arithmetic with Boltzmann Machines: The Role of Number Representations

This paper presents a study on associative mental arithmetic with mean-field Boltzmann Machines. We examined the role of number representations, showing theoretically and experimentally that cardinal number representations (e.g., numerosity) are superior to symbolic and ordinal representations w.r.t. learnability and cognitive plausibility. Only the network trained on numerosities exhibited the problem-size effect, the core phenomenon in human behavioral studies. These results urge a reevaluation of current cognitive models of mental arithmetic.

[1]  ROBERT S. MOYER,et al.  Time required for Judgements of Numerical Inequality , 1967, Nature.

[2]  G. Groen,et al.  A chronometric analysis of simple addition. , 1972 .

[3]  Geoffrey E. Hinton,et al.  A Learning Algorithm for Boltzmann Machines , 1985, Cogn. Sci..

[4]  Carsten Peterson,et al.  A Mean Field Theory Learning Algorithm for Neural Networks , 1987, Complex Syst..

[5]  M. McCloskey,et al.  Chapter 10 Mathnet: Preliminary Results From A Distributed Model Of Arithmetic Fact Retrieval , 1992 .

[6]  S. Dehaene,et al.  Cross-linguistic regularities in the frequency of number words , 1992, Cognition.

[7]  M. Ashcraft.  Cognitive arithmetic: A review of data and theory , 1992, Cognition.

[8]  S. Dehaene,et al.  Abstract representations of numbers in the animal and human brain , 1998, Trends in Neurosciences.

[9]  J. Whalen.  The Influence of Semantic Magnitude Representations on Arithmetic : Theory , Data , and Simulation , 1998 .

[10]  C. Gallistel,et al.  Nonverbal Counting in Humans: The Psychophysics of Number Representation , 1999 .

[11]  C. Gallistel,et al.  Non-verbal numerical cognition: from reals to integers , 2000, Trends in Cognitive Sciences.

[12]  B. Mazoyer,et al.  Neural Correlates of Simple and Complex Mental Calculation , 2001, NeuroImage.

[13]  M. Zorzi,et al.  Storage and retrieval of addition facts: The role of number comparison , 2001, The Quarterly journal of experimental psychology. A, Human experimental psychology.

[14]  Sven Blankenberger.  The arithmetic tie effect is mainly encoding-based , 2001, Cognition.

[15]  M. Brysbaert,et al.  Semantic priming in number naming , 2002, The Quarterly journal of experimental psychology. A, Human experimental psychology.

[16]  Geoffrey E. Hinton.  Training Products of Experts by Minimizing Contrastive Divergence , 2002, Neural Computation.

[17]  Geoffrey E. Hinton,et al.  A New Learning Algorithm for Mean Field Boltzmann Machines , 2002, ICANN.

[18]  Marco Zorzi,et al.  A Computational Model of Number Comparison , 2020, Proceedings of the Twenty First Annual Conference of the Cognitive Science Society.