A Normal Form for Function Rings of Piecewise Functions

Computer algebra systems often have to deal with piecewise continuous functions. These are, for example, the absolute value function, signum, piecewise defined functions but also functions that are the supremum or infimum of two functions. We present a new algebraic approach to these types of problems. This paper presents a normal form for a function ring containing piecewise polynomial functions of a real variable. We give a complete rule system to compute the normal form of an expression. The main result is that this normal form can be used to decide extensional equality of two piecewise functions. Also we define supremum and infimum for piecewise functions; in fact, we show that the function ring forms a lattice. Additionally, a method to solve equalities and inequalities in this function ring is presented. Finally, we give a “user interface” to the algebraic representation of the piecewise functions.