Towards an algebraic theory of Boolean circuits

Boolean circuits are used to represent programs on finite data. Reversible Boolean circuits and quantum Boolean circuits have been introduced to modelize some physical aspects of computation. Those notions are essential in complexity theory, but we claim that a deep mathematical theory is needed to make progress in this area. For that purpose, the recent developments of knot theory is a major source of inspiration. Following the ideas of Burroni, we consider logical gates as generators for some algebraic structure with two compositions, and we are interested in the relations satisfied by those generators. For that purpose, we introduce canonical forms and rewriting systems. Up to now, we have mainly studied the basic case and the linear case, but we hope that our methods can be used to get presentations by generators and relations for the (reversible) classical case and for the (unitary) quantum case. (C) 2003 Elsevier B.V. All rights reserved.