The role of the principal part in factorizing block maps

Abstract

An n-block is a sequence b1 … bn where bi ε {0,1} for 1 ≤ i ≤ n, and an n-block map is a function from the set of n-blocks to the set {0,1}. Block maps can be used to study endomorphisms of the shift dynamical system [7] and shift register sequences [4]. Composition of endomorphisms and cascading of shift registers [6] can be studied via composition of block maps. If fog is a given block map which is linear in the first variable and if g is also given, then f is unique and can be determined. If fog and f are given then g is unique and can be determined, at least up to a constant [10]. However, little is known about the level of computational complexity of the general task of searching for factors of a given block map. In this paper I identify a substantial class of maps which are linear in the first variable but not linear, for which there are effective methods of searching for factors. The methods are based on the presentation of block maps via successive principal parts, introduced in [9]. There I define the principal vector of a block map. One of the key results gives conditions under which there is a particularly close relationship between the principal vectors of two block maps f and g and their composition fog. In the second section of this paper I show that the same relationship holds under weaker conditions.