Binary periodic signals and flows

TL;DR

This paper explores the properties of binary periodic signals and flows in asynchronous systems, analyzing their periodicity, eventual behavior, and how modifications affect their periodicity, with applications in systems theory and computer science.

Contribution

It provides a comprehensive analysis of periodicity in boolean signals and introduces methods to modify their periodic properties within asynchronous systems.

Findings

Characterization of eventually constant signals

Analysis of periodic and eventually periodic signals

Methods to alter periodicity properties

Abstract

The concept of boolean autonomous deterministic regular asynchronous system has its origin in switching theory, the theory of modeling the switching circuits from the digital electrical engineering. The attribute boolean vaguely refers to the Boole algebra with two elements; autonomous means that there is no input; determinism means the existence of a unique state function; and regular indicates the existence of a function $\Phi:\{0,1\}^{n}\rightarrow\{0,1\}^{n},\Phi=(\Phi_{1},...,\Phi_{n})$ whose coordinates iterate independently on each other. Time is discrete or continuous. The flows are these that result by analogy with the dynamical systems. The 'nice' discrete time and real time functions that the (boolean) asynchronous systems work with are called signals and periodicity is a very important feature in Nature. In the first two Chapters we give the most important concepts concerning the signals and periodicity. The periodicity properties are used to characterize the eventually constant signals in Chapter 3 and the constant signals in Chapter 4. Chapters 5,...,8 are dedicated to the eventually periodic points, eventually periodic signals, periodic points and periodic signals. Chapter 9 shows constructions that, given an (eventually) periodic point, by changing some values of the signal, change the periodicity properties of the point. The book addresses to researchers in systems theory and computer science, but it is also interesting to those that study periodicity itself. From this last perspective, the binary signals may be thought of as functions with finitely many values.