Binary Periodic Signals and Flows, Nova publishers, New York, 2016
Hardcover: https://www.novapublishers.com
E-book: https://www.novapublishers.com
The Boolean autonomous deterministic regular asynchronous systems have been defined
for the first time in our work Boolean dynamical systems, ROMAI Journal,
vol. 3, Nr. 2, 277-324, (2007), and a deeper study of such systems can be found in
"Asynchronous systems theory", second edition, LAP LAMBERT Academic Publishing, Saarbrucken,
(2012). The concept 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
$\Phi:\{0,1\}^{n}\rightarrow
\{0,1\}^{n},\Phi=(\Phi_{1},...,\Phi_{n})$
that 'generates' the system. The time set is discrete: {-1,0,1,...} or continuous: R. The system,
which is analogue to the (real, usual) dynamical systems, iterates (asynchronously) on each coordinate i, one of
- $\Phi_{i}$: we say that $\Phi$ is computed, at that time instant, on that coordinate;
- the projection of ${0,1}^{n}$ on the i-th coordinate: we use to say that $\Phi$ is not computed,
at that time instant, on that coordinate.
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 monograph continues with flows. Chapter 10 is dedicated to the computation functions, i.e. to the
functions that show when and how the function $\Phi$ is iterated (asynchronously). Chapter 11 introduces
the flows and Chapter 12 gives a wider point of view on the flows, which are interpreted as deterministic
asynchronous systems. Chapters 13,...,15 restate the topics from Chapters 3,...,8 in the special case when
the signals are flows and the main interest is periodicity.
The bibliography consists in general in works of (real, usual) dynamical systems and we use analogies.
The book ends with a list of notations, an index of notions and an appendix with lemmas. These lemmas are
frequently used in the exposure and some of them are interesting by themselves.
The book is structured in chapters, each chapter consists in several sections and each section is structured
in paragraphs. The chapters begin with an abstract. The paragraphs are of the following kinds: definitions,
notations, remarks, theorems, corollaries, lemmas, examples and propositions. Each kind of paragraph is numbered
separately on the others. Inside the paragraphs, the equations and, more generally, the most important statements
are numbered also. When we refer to the statement (x,y) this means the y-th statement of the x-th section of the
current chapter. Sometimes we write (x,y)_{page z} in order to indicate the page where the statement occurs.
We refer to a definition, theorem, example,... by indicating its number and, when necessary, its page.
An interesting open problem that we have found during these investigations is the following one. If a function
is periodic, then its values are periodic. Is the inverse implication true?
If all the values of a function are periodic, the possibility exists that the intersection of the sets of
their periods is empty and then the signal is not periodic.
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.
