Information Technology Reference
In-Depth Information
From the previous equations the following equation holds:
Y
=
F
(
Y
)
and consequently, for every
n
∈
N
:
F
n
Y
=
(
Y
)
.
This means that
Y
, also called Curry-Feys
paradoxical operator
, provides an infi-
nite process of self-generation. The paradoxical nature of
Y
consists in its ability to
exhibit the property of producing a computation with the only result of producing
computations making the same task, along a non-terminating process which does
not return any definite result.
This argument shows, in abstract, algebraic terms, that application and replication
are the essential ingredients of an endless phenomenon of self-generation. However,
an important issue follows directly from this analysis. In fact, a difference between
life computations and artificial problem solving computations clearly arises. On the
one hand, the computations performed for solving problems are terminating pro-
cesses providing results when they stop. On the other hand, almost all computations
performed in life processes are aimed at keeping their livening nature over time, by
fulfilling some specific characteristics, but without terminating. Life in itself can be
seen as an endless phenomenon propagating in space and in time and evolving into
forms able to make this propagation ability more efficient and diversified.
In the development of modern computer science, four giants had a dominant role:
Alan Turing, Nobert Wiener, Claude Shannon, and John von Neumann. Their major
contributions were related to natural computing issues, along a network of ideas
and formalisms developed by other scholars at the borders of many disciplines:
mathematics, physics, logic, linguistics, and biology.
Alan Turing was the first scientist who elaborated in 1936 a general mathemat-
ical model of a computational device performing computations [214]. His model
was inspired by a deep behavioral analysis of mind activity of a human agent per-
forming calculations. A less known work of Turing was devoted to a mathematical
analysis of morphogenesis, as a consequence of numerical phenomena ruling the
threshold passage between different dynamical regimes described in terms of differ-
ential equations [158].
A joint work of Wiener with Rosenbluth, a Mexican physiologist, and Bigelow,
an American electrical engineer [150] (Wiener studied mathematics under Russell's
guidance) was entitled “Behavior, purpose and teleology” and was a philosophical
starting point of his Cybernetics [160], a discipline he founded for studying, in a
unified perspective, mechanisms of information processing in animal and artificial
systems (see also [117]).
Claude Shannon founded the modern information theory in his famous book-
let published in 1948 [212], “A mathematical theory of communication”, where
the mathematical analysis of quantitative principles of digital information was de-
fined. It is not well-known that Shannon studied in his PhD thesis some methods for
