Algebraic constructions for algorithms (Extended Abstract)

In the last decade or so there has been an out-growth of research in algebraic complexity of computations which showed how to derive algorithms systematically. One of the features of these derivation is their reliance on algebraic and symbolic constructions. I would like, in this paper, to survey some of the symbolic and algebraic constructions which are used, and then draw some conclusions on the implications of this development for symbol manipulation systems. Let me start by sketching some of these derivations.