## Computing and Reasoning with Piecewise Algebraic Functions

*Michael Bulmer*

`mrb@maths.uq.edu.au`

**University of Queensland**

Australia

*Desmond Fearnley-Sander*

`dfs@hilbert.maths.utas.edu.au`

**University of Tasmania**

Australia

*Tim Stokes*

`stokes@prodigal.murdoch.edu.au`

**Murdoch University**

Australia

### Abstract

A main concern of elementary algebra is the interplay between algorithms
and their associated functions. For example, the identity

x^2-1 = (x-1)(x+1)

captures the fact that two different algorithms have the same input-output
behaviour. We advocate adoption of this perspective in the teaching of
algebra. In the present paper, we address a question that then suggests
itself: can the calculus of polynomials be extended in a simple way to take
in other basic algebraic functions such as the absolute value function.
The theory of Boolean affine combinations, which we introduced in [1],
supports such an extension.
While the theory is applicable to arbitrary (many-sorted) universal
algebras, our presentation here focusses on real piecewise-defined
polynomials, and is illustrated by simple algebraic proofs of theorems such
as ||x|| = |x| and the triangle inequality. A prototype system has been
implemented (in Mathematica), which supports both the evaluation of
piecewise-defined algebraic functions and automatic reasoning about the
properties of such functions. The trace of a session with this package is
included.

References:
[1] M. Bulmer, D. Fearnley-Sander and T. Stokes,
`Towards a Calculus of Algorithms', Bull. Austral. Math. Soc.,
50 (1994), pp. 81--89.

[2] H. Ehrig and B. Mahr, Fundamentals of algebraic
specification, EATCS monographs on theoretical computer science, v. 6
(Springer-Verlag, Berlin, 1985).

[3] H. Lausch and W. Nobauer, Algebra of polynomials,
North-Holland Mathematical Library 5 (North-Holland, Amsterdam, 1973).

[4] A.G. Pinus, `Boolean Constructions in Universal Algebra',
Russ. Math. Surv. 47, 4 (1992), 157--198.

Key Words: algebraic computation, Boolean affine combination, Boolean
power, piecewise polynomial, Mathematica.

© ATCM, Inc. 2001.