Technical Report Announcement

[ANDERSON%BROWNCOG.BITNET@mitvma.mit.edu]

                     Technical Report 91-3 available from:

                Department of Cognitive and Linguistic Sciences

                Box 1978, Brown University, Providence, RI 02912





                        A Study in Numerical Perversity:

                    Teaching Arithmetic to a Neural Network



           James A. Anderson, Kathryn T. Spoehr, and David J. Bennett

                Department of Cognitive and Linguistic Sciences

                                    Box 1978

                                Brown University

                              Providence, RI 02912



                                    Abstract



             There are only a few hundred well-defined facts in

        elementary arithmetic, but humans find them hard to learn and

        hard to use.  One reason for this difficulty is that the

        structure of elementary arithmetic lends itself to severe

        associative interference.  If a neural network corresponds in

        any sense to brain-style computation, then we should expect

        similar difficulties teaching elementary arithmetic to a neural

        network.  We find this observation is correct for a simple

        network that was taught the multiplication tables.  We can

        enhance learning of arithmetic by forming a hybrid coding for

        the representation of number that contains a powerful analog or

        "sensory" component as well as a more abstract component.  When

        the simple network uses a hybrid representation, many of the

        effects seen in human arithmetic learning are reproduced,

        including overall error patterns and response time patterns for

        false products.  An extension of the arithmetic network is

        capable of being flexibly programmed to correctly answer

        questions involving terms such as "bigger" or "smaller."

        Problems can be answered correctly, even if the particular

        comparisons involved had not been learned previously.  Such a

        system is genuinely creative and flexible, though only in a

        limited domain.  It remains to be seen if the computational

        limitations of this approach are coincident with the limitations

        of human cognition.



             A version of this report will appear as a chapter in:

          "Neural Networks for Knowledge Representation and Inference"

               Edited by Daniel S. Levine and Manuel Aparicio, IV

                               To be published by

               Lawrence Erlbaum Associates, Hillsdale, New Jersey



             Copies can be obtained by sending an email message to:

                            LI700008 at brownvm.BITNET

                                     or to:

                            anderson at browncog.BITNET