Algebraic simplification in computer algebra: an analysis of bottom-up algorithms

We consider a class of simplification algorithms for algebraic and logical expressions which are of systematic use in Computer Algebra systems. This class is basically characterized by the fact that algorithms operate in a bottom-up recursive way on the expressions, i.e. start from the atomic terms -constants and variables- and perform the simplifications on larger and larger terms until the whole expression is ultimately proceeded; no backtracking or iterated process should intervein in the simplification. We show that under these quite general assumptions, it is possible to analyze precisely, and almost automatically, the average size of the resulting expressions -the gain in space- and the average time complexity of the process, which happens to be linear, whereas the worst-case behaviour is not in general