Chapter 3


This chapter describes a connectionist model of memory for multiplication facts built using McClelland's "cascade" equations (McClelland 1979; McClelland & Rumerlahrt 1988). It is referred to as the "cascade model" or "cascade network" (no relation to cascade correlation, Fahlman & Lebiere 1990).

The model was originally designed with two objectives in mind. First, in contrast to the BSB model, the network should be able to capture occasional errors. That is, although the network should correctly learn all the multiplication facts, it should also produce errors when under time pressure. The second objective was to minimize the number of assumptions about connection and unit types. This objective was formulated in the context of the Campbell & Graham (1985) model, where many different knowledge sources were presented. The aim was to see how many of the assumptions could be omitted.

The above aims were only selected after it was observed that the architecture could potentially account for the phenomena. The network was first used as a "slave" network, providing arithmetic facts for a multicolumn arithmetic network---the one described in chapter 5, although the fact network was not used in the final multicolumn network. The aim was to investigate what kinds of factual knowledge would be useful to the multicolumn network, hence the same technology was used to build the fact network (a multilayer perceptron trained with backpropagation). The results from the work of Campbell & Graham (1985) suggested that the fact network could be tested for RT and errors. When this was done, a primitive RT measure showed a dip for 5s problems, prompting further investigation of the network.

Since then the architecture and representations have changed in many ways. For example, the first experiments used a one-of-N input encoding and represented answers in separate tens and units fields. It was found that in order to capture human performance, various changes needed to be made. This chapter first outlines the "finished product", after all the changes have been made. The motivation for the changes, and the results they gave, are presented in section 3.4.

Much of the literature has focussed on the problems 2x2 to 9x9. The first set of experiments did not simulate ones or zeros problems as human empirical data had only been found for 2x2 to 9x9 at that time. However, in light of the work of Harley (1991) and Miller et al (1984), simulations of zero and ones problems were performed. These experiments are presented in section 3.3. Finally, the system is compared to the other connectionist models, and issues arising from the model are discussed.

  • Architecture of the model
  • Recall
  • Training
  • Training set conditional probabilities
  • Simulations for 2x2 to 9x9
  • Training
  • Recall
  • Results
  • Comments
  • Simulations for 0x0 to 9x9
  • Training
  • Results
  • Comments
  • Futher experiments
  • One-of-N input encoding
  • Training without a frequency skew
  • McCloskey & Lindermann's input encoding
  • Predictions for the 10, 11 and 12 tables
  • Damaging the network
  • Discussion
  • Choice of output representation
  • Rule based processing
  • Verification and primed production tasks
  • Summary

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