Function Discovery using Data Transformation

Function discovery is the problem of nding a symbolic formula for a function f : D1!D2 from a set of examples f(x; y) j y = f(x)g D1 D2. Acceptable solutions are restricted to formulas expressible in some given description language L. In most previous discovery systems|for example, Bacon [4], Abacus [1], Coper [2], Fahrenheit [3] and IDS [5]|the description language is restricted to contain only rational functions so that symbolic descriptions can easily be enumerated. This paper shows how the idea of data transformation, a technique pioneered in FFD [6], can be used as the basis of a far more comprehensive description language that includes all functions that can be transformed to rational functions by di erentiation and logarithm operations. The data transformation approach induces functions by searching through a space of transformations that can be expressed both symbolically and as operations on the example set. It expresses the solution implicitly, as a small system of simultaneous equations, some of which may be di erential equations. Solving these symbolically would (where possible) give a formula for the unknown function. In order to operationalize transformations while maintaining the ability of the example set to faithfully represent transformed functions, it is necessary to allow for new examples to be requested interactively. The main contribution of this paper is to de ne a transformation-based description language and characterize its representational power. We also brie y sketch a practical implementation of a function induction system that uses this approach.