Symbolic Dynamics and Coding Applications

This article is a summary of the plenary lecture that I gave at the 1995 IEEE International Symposium on Information Theory. The lecture was intended to give a concrete and elementary introduction to the subject of symbolic dynamics. From my point of view, symbolic dynamics is the subject of \coding" from one \constrained set" of sequences to another. Of course, there are restrictions on what I mean by \coding" and \constrained set," and these are spelled out in section 4 below. I will begin with a very concrete unsolved graph-theoretic problem, called the Road Problem, which rst arose in symbolic dynamics. Then I will motivate symbolic dynamics from two v ery dierent points of view: (1) modeling of dynamical systems, which goes back to Hadamard almost 100 years ago, and (2) coding for data recording channels, which is of course much more recent. Finally, I will introduce some of the basic concepts and fundamental results in the subject and brie BLOCKINy indicate how they pertain to data recording. For introductory reading on symbolic dynamics and its applications, requiring only a minimum of mathematical background, I unapologetically recommend the recent textbook, An Introduction to Symbolic Dynamics and Coding [9], by Doug Lind and myself. In addition there are some other very good expositions such as the monograph [4] and the article [14] (section IV). For expositions which focus more exclusively on data recording applications, the reader may consult [10] or [11]. All of these references contain extensive bibliographies. 1. The Road Problem Suppose you are driving to Cleveland, but have lost your way. Y ou call the auto club and ask for directions. The auto club operator gives you directions, and you now happily nd yourself in Cleveland. But you never told the operator where you were calling from (Memphis, Los Angeles, Peoria?). How can this be? Well, this would be possible if the interstate highway system were a network of roads labeled in such a w a y that one particular sequence of instructions would direct drivers starting at any city to all arrive a t a 1