Computational Complementarity and Sofic Shifts

Finite automata (with outputs but no initial states) have been extensively used as models of computational complementarity, a property which mimics the physical complementarity. All this work was focussed on \frames", i.e., on xed, static, local descriptions of the system behaviour. In this paper we are mainly interested in the asymptotical description of complementarity. To this aim we will study the asymptot-ical behaviour of two complementarity principles by associating to every incomplete deterministic automaton (with outputs, but no initial state) certain sooc shifts: au-tomata having the same behaviour correspond to a unique sooc shift. In this way, a class of sooc shifts reeecting complementarity will be introduced and studied. We will prove that there is a strong relation between \local complementarity", as it is perceived at the level of \frames", and \asymptotical complementarity" as it is described by the sooc shift. 1 Motivation Physical systems are normally described by measurements. For example, a gas is described by the position and momentum of its molecules and a swinging pendulum is characterized by its angle from the vertical and its angular velocity. In the simplest case the set of possible values describing a system can be arranged in a sequence, a lm which is innnite in both directions: each frame of the lm|describing the system in a xed interval of time|depends upon the previous one, usually in a continuous way. The notion of measurement is strongly connected to physical complementarity: the observer either experiences one certain type of observation, (exclusive) or a diierent, complementary one. The \folklore" understanding of complementarity, in general, and of Heisenberg's uncertainty relation, in particular, is the existence of certain (complementary) features of quantum systems which cannot be measured and predicted simultaneously with arbitrary accuracy. In other words, any description of properties of microscopic objects in classical terms generates pairs of complementary variables; the accuracy in one member