Information Technology Reference
In-Depth Information
The following definition of
flow
generalizes the concept of autonomous system, by
using a next-state-function equipped with a real number as further parameter.
Definition 5.7.
A
flow
D
is a triple
D
=(
S
,
T
,
δ
)
given by i) a set
S
of states, called
t
∈
phase space
, ii) a set
T
⊆
R
of
time intervals
such that 0
∈
T
and when
t
,
t
∈
T
,thenalso
t
+
T
, and iii) a
dynamics
δ
, which is a function
δ
:
S
×
T
→
S
providing the state
δ
(
s
,
t
)
which the system assumes after time
t
, starting from
s
.It
t
∈
t
)=
is required that
δ
(
s
,
0
)=
s
,andforany
s
∈
S
and
t
,
T
we have
δ
(
δ
(
s
,
t
)
,
t
)
.
δ
(
s
,
t
+
When
T
is a subset of
Z
,then
D
is a
discrete
flow.
A timed dynamical system
is
totally discrete
when both sets
S
and
T
are
finite or enumerable sets. Even though it sounds a little strange, a real number dec-
imal representation can be seen as a totally discrete dynamical system, whenever
we identify it with a sequence providing its decimal digits. In fact in this case
S
(
S
,
T
,
δ
)
. Infinite words, trees, or graphs are totally discrete
dynamical systems on suitable discrete phase spaces, where the dynamics is the
function giving the structure at some step of its generation. Examples of totally dis-
crete dynamical systems are: deductive and rewriting systems, Chomsky grammars,
automata, L systems, cellular automata (see Chap. 6).
Any computation is a kind of dynamics. However, the two concepts are based
on different perspectives. In a computation an input is provided to the system (in
an initial state) and an output is expected at the end of the process if the system
reaches a state which is considered as final, according to some termination crite-
rion. This means that the dynamics underlying the move of a computation from an
initial to a final state is functional to the desired relation between input and output,
and the sooner a final state is obtained, the better the computation is realized. On
the contrary, in a dynamical system the focus of the process is on the dynamics
itself. Input and output can be even disposed of, but when they are present, they
are used in order to guarantee that dynamics could proceed by satisfying some be-
havioral requirements (e.g., acquiring or expelling substances). A major example
of this dynamical perspective are living organisms. They are open systems, that is,
they need inputs and outputs, to ensure that a dynamics, with specific features, can
be maintained in time, as long as possible. In this perspective, it is very important
to individuate dynamical properties which are relevant in life phenomena, and to
discover principles according to which some artificial dynamics exhibits behaviors
with these properties.
The definition of autonomous systems implicitly assumes that the application of
=
{
0
,
1
,
2
,...
9
}
and
T
= N
δ
can be considered uniform, that is, the same temporal interval is assumed for
any application of
. In the late 16th century, Galileo discovered the pendulum, as
a physical phenomenon for which this uniformity feature may be assumed. This
discovery is the beginning of the experimental science, which is intrinsically based
on the measure of time.
Does time come before any dynamics, or vice versa is time an abstraction orig-
inated by dynamical phenomena? In fact, time apart from dynamics seems to be
a metaphysical abstraction. From an epistemological point of view, we only have
clocks, and any physical notion of time is always related to a
clock
, that is, a system
δ
