Reading the Generalizer's Mind

In his new commentary, Damper re-emphasises his claim that parity is not a generalisation problem. But when proper account is taken of the arguments he puts forward, we nd that the proposed conclusion is not the only one that can be drawn. In adding the word`still' to the title of his ongoing commentary, Damper re-emphasises his claim that`parity is not a generalisation problem.' His view is that in our response we failed to accept or even properly address his argument. However, as we hinted in the rst paragraph of R6, the interpretation of his claim is not a straightforward matter. The namèparity' refers to the truth function whose rule is that the output is true if an odd number of the inputs are true. Parity functions may be of any arity but we are particularly familiar with the 2-place variant, also known asèxclusive-or' (XOR). Now, of course, a function is a function is a function. It is not, in itself, a `problem'. And incontrovertibly, therefore, it cannot be a `generalisation prob-lem'. However the parity function (like any other function) is easily used as the basis for a generalisation problem. The procedure is straightforward: we take the complete mapping for a given parity function (e.g., all 16 input/output associations for the 4-bit parity function) and we present a subset of these cases to the generalisation mechanism, e.g., supervised learner. We then test the generalisation performance of the mechanism by examining its response on the unseen cases, or thèvalidation' set as Damper calls it. This is the standard method for presenting a generalisation problem to a supervised learning mechanism. And we note that in both his commentaries, Damper describes the way he used the method to test the generalisation abilities of a backpropagation network. His account in his rst commentary is particularly clear: he describes how he provided his backpropagation network with just`the rst three lines of the xor] truth table' so as to see whether it could`generalise on the 2-variable parity (XOR) problem.' Clearly, then, Damper is familiar with the procedure via which a parity mapping is used as the basis for the presentation of a gener-alisation problem. His intention thus cannot be to argue that parity cannot be used as the basis of a generalisation problem. What, then, are we to make of his declaration that`parity is not a generalisation problem.' A clue to his intentions comes in his second …