<s>
In	O
automata	B-Application
theory	I-Application
,	O
a	O
field	O
of	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
signal	B-Application
automaton	I-Application
is	O
a	O
finite	B-Architecture
automaton	I-Architecture
extended	O
with	O
a	O
finite	O
set	O
of	O
real-valued	O
clocks	B-Application
.	O
</s>
<s>
During	O
a	O
run	O
of	O
a	O
signal	B-Application
automaton	I-Application
,	O
clock	B-Application
values	O
increase	O
all	O
with	O
the	O
same	O
speed	O
.	O
</s>
<s>
Along	O
the	O
transitions	O
of	O
the	O
automaton	O
,	O
clock	B-Application
values	O
can	O
be	O
compared	O
to	O
integers	O
.	O
</s>
<s>
Further	O
,	O
clocks	B-Application
can	O
be	O
reset	O
.	O
</s>
<s>
Before	O
formally	O
defining	O
what	O
a	O
signal	B-Application
automaton	I-Application
is	O
,	O
an	O
example	O
will	O
be	O
given	O
.	O
</s>
<s>
As	O
for	O
finite	B-Architecture
automaton	I-Architecture
,	O
incoming	O
arrows	O
represents	O
initial	O
locations	O
and	O
double	O
circle	O
represents	O
accepting	O
locations	O
.	O
</s>
<s>
However	O
,	O
contrary	O
to	O
finite	B-Architecture
automata	I-Architecture
,	O
letters	O
occurs	O
in	O
locations	O
and	O
not	O
in	O
transition	O
.	O
</s>
<s>
The	O
symbol	O
represents	O
a	O
clock	B-Application
.	O
</s>
<s>
This	O
clock	B-Application
allow	O
to	O
measure	O
the	O
time	O
since	O
the	O
last	O
time	O
where	O
was	O
emitted	O
.	O
</s>
<s>
Formally	O
,	O
a	O
signal	B-Application
automaton	I-Application
is	O
a	O
tuple	O
that	O
consists	O
of	O
the	O
following	O
components	O
:	O
</s>
<s>
is	O
a	O
finite	O
set	O
called	O
the	O
clocks	B-Application
of	O
.	O
</s>
<s>
is	O
the	O
powerset	B-General_Concept
of	O
.	O
</s>
<s>
An	O
edge	O
from	O
is	O
a	O
transition	O
from	O
locations	O
to	O
which	O
reset	O
the	O
clocks	B-Application
of	O
.	O
</s>
<s>
A	O
pair	O
with	O
a	O
location	O
and	O
a	O
clock	B-Application
valuation	O
is	O
called	O
either	O
an	O
extended	O
state	O
or	O
a	O
state	O
.	O
</s>
<s>
Here	O
lies	O
one	O
of	O
the	O
biggest	O
difference	O
between	O
signal-automata	O
and	O
finite	B-Architecture
automata	I-Architecture
.	O
</s>
<s>
In	O
a	O
finite	B-Architecture
automaton	I-Architecture
,	O
at	O
some	O
point	O
of	O
the	O
execution	O
,	O
the	O
state	O
is	O
entirely	O
described	O
by	O
the	O
number	O
of	O
letter	O
read	O
and	O
by	O
a	O
finite	O
number	O
of	O
possible	O
values	O
,	O
which	O
are	O
actually	O
called	O
"	O
states	O
"	O
.	O
</s>
<s>
Thus	O
,	O
the	O
word	O
"	O
finite	O
"	O
in	O
the	O
name	O
"	O
finite	B-Architecture
automata	I-Architecture
"	O
.	O
</s>
<s>
However	O
,	O
as	O
it	O
is	O
explained	O
in	O
the	O
section	O
"	O
run	O
"	O
below	O
,	O
in	O
order	O
to	O
resume	O
clocks	B-Application
are	O
used	O
to	O
determine	O
which	O
transitions	O
can	O
be	O
taken	O
.	O
</s>
<s>
Thus	O
,	O
in	O
order	O
to	O
know	O
the	O
state	O
of	O
the	O
automaton	O
,	O
you	O
must	O
both	O
now	O
in	O
which	O
location	O
you	O
are	O
,	O
and	O
the	O
clock	B-Application
valuation	O
.	O
</s>
<s>
As	O
for	O
finite	B-Architecture
automata	I-Architecture
,	O
a	O
run	O
is	O
essentially	O
a	O
sequence	O
of	O
locations	O
,	O
such	O
that	O
there	O
exists	O
a	O
transition	O
between	O
two	O
locations	O
.	O
</s>
<s>
Some	O
time	O
elapses	O
while	O
in	O
a	O
location	O
;	O
the	O
clock	B-Application
constraints	O
labelling	O
a	O
location	O
or	O
its	O
successor	O
may	O
constraint	O
the	O
time	O
spent	O
in	O
a	O
single	O
location	O
.	O
</s>
<s>
The	O
signal	B-Application
defined	O
by	O
this	O
run	O
is	O
the	O
function	O
defined	O
above	O
.	O
</s>
<s>
It	O
is	O
said	O
that	O
the	O
run	O
defined	O
above	O
is	O
a	O
run	O
for	O
the	O
signal	B-Application
.	O
</s>
<s>
The	O
notion	O
of	O
accepting	O
run	O
is	O
defined	O
as	O
in	O
finite	B-Architecture
automata	I-Architecture
for	O
finite	O
words	O
and	O
as	O
in	O
Büchi	B-General_Concept
automata	I-General_Concept
for	O
infinite	O
words	O
.	O
</s>
<s>
A	O
signal	B-Application
is	O
said	O
to	O
be	O
accepted	O
by	O
a	O
signal	B-Application
automaton	I-Application
if	O
there	O
exists	O
a	O
run	O
of	O
on	O
accepting	O
it	O
.	O
</s>
<s>
As	O
in	O
the	O
case	O
of	O
finite	O
and	O
Büchi	B-General_Concept
automaton	I-General_Concept
,	O
a	O
signal-automaton	O
may	O
be	O
deterministic	O
or	O
non-deterministic	O
.	O
</s>
<s>
the	O
zone	O
defined	O
by	O
the	O
clock	B-Application
constraint	O
,	O
</s>
<s>
the	O
zone	O
defined	O
by	O
the	O
clock	B-Application
constraint	O
where	O
the	O
constraints	O
on	O
clocks	B-Application
of	O
are	O
removed	O
,	O
</s>
<s>
the	O
zone	O
defined	O
by	O
the	O
clock	B-Application
constraint	O
where	O
the	O
constraints	O
on	O
clocks	B-Application
of	O
are	O
removed	O
,	O
</s>
<s>
the	O
zone	O
defined	O
by	O
the	O
clock	B-Application
constraint	O
where	O
the	O
constraints	O
on	O
clocks	B-Application
of	O
are	O
removed	O
,	O
</s>
<s>
Depending	O
on	O
the	O
authors	O
,	O
the	O
exact	O
definition	O
of	O
signal	B-Application
automata	O
may	O
be	O
slightly	O
different	O
.	O
</s>
<s>
A	O
bipartite	O
signal	B-Application
automaton	I-Application
is	O
a	O
signal	B-Application
automaton	I-Application
in	O
which	O
the	O
run	O
alternates	O
between	O
open	O
intervals	O
and	O
singular	O
intervals	O
(	O
i.e.	O
</s>
<s>
In	O
order	O
to	O
ensure	O
that	O
each	O
singular	O
location	O
is	O
indeed	O
singular	O
,	O
for	O
each	O
location	O
,	O
there	O
must	O
be	O
a	O
clock	B-Application
which	O
is	O
reset	O
when	O
entering	O
and	O
such	O
that	O
the	O
clock	B-Application
constraint	O
of	O
contains	O
.	O
</s>
<s>
Any	O
signal	B-Application
automaton	I-Application
can	O
be	O
transformed	O
into	O
an	O
equivalent	O
bipartite	O
signal	B-Application
automaton	I-Application
.	O
</s>
<s>
It	O
suffices	O
to	O
replace	O
each	O
location	O
by	O
a	O
pair	O
of	O
locations	O
and	O
introduce	O
a	O
new	O
clock	B-Application
,	O
such	O
that	O
for	O
each	O
,	O
.	O
</s>
<s>
Nearby	O
is	O
pictured	O
a	O
bipartite	O
automaton	O
equivalent	O
to	O
the	O
signal	B-Application
automaton	I-Application
from	O
the	O
example	O
section	O
.	O
</s>
<s>
The	O
notion	O
of	O
product	O
of	O
finite	B-Architecture
automaton	I-Architecture
is	O
extended	O
to	O
signal	B-Application
automaton	I-Application
.	O
</s>
<s>
The	O
main	O
difference	O
between	O
synchronization	O
and	O
product	O
is	O
that	O
,	O
when	O
two	O
finite	B-Architecture
automata	I-Architecture
read	O
the	O
same	O
word	O
,	O
they	O
take	O
transition	O
simultaneously	O
.	O
</s>
<s>
This	O
is	O
not	O
the	O
case	O
anymore	O
for	O
signal	B-Application
automata	O
,	O
since	O
they	O
can	O
take	O
transition	O
at	O
arbitrary	O
time	O
.	O
</s>
<s>
Thus	O
,	O
the	O
transition	O
relation	O
of	O
a	O
signal	B-Application
automaton	I-Application
may	O
allow	O
transition	O
to	O
be	O
taken	O
in	O
one	O
or	O
two	O
automata	O
.	O
</s>
<s>
Let	O
and	O
two	O
signal	B-Application
automata	O
,	O
their	O
synchronization	O
is	O
the	O
signal	B-Application
automaton	I-Application
,	O
where	O
contains	O
the	O
following	O
transitions	O
:	O
</s>
<s>
Timed	B-Application
automata	I-Application
is	O
another	O
extension	O
of	O
finite	B-Architecture
automata	I-Architecture
,	O
which	O
adds	O
a	O
notion	O
of	O
time	O
to	O
words	O
.	O
</s>
<s>
We	O
now	O
state	O
some	O
of	O
the	O
main	O
differences	O
between	O
timed	B-Application
automata	I-Application
and	O
signal	B-Application
automata	O
.	O
</s>
<s>
In	O
timed	B-Application
automata	I-Application
,	O
letters	O
are	O
emitted	O
on	O
the	O
transitions	O
and	O
not	O
in	O
the	O
locations	O
.	O
</s>
<s>
As	O
explained	O
above	O
when	O
comparing	O
signal	B-Application
automata	O
to	O
finite	B-Architecture
automata	I-Architecture
,	O
letters	O
are	O
emitted	O
on	O
transitions	O
when	O
the	O
words	O
are	O
emitted	O
discretely	O
,	O
as	O
for	O
words	O
and	O
timed-words	B-Application
while	O
they	O
are	O
emitted	O
on	O
locations	O
when	O
letters	O
are	O
emitted	O
continuously	O
,	O
as	O
for	O
signals	O
.	O
</s>
<s>
In	O
timed	B-Application
automata	I-Application
,	O
guards	O
are	O
only	O
checked	O
on	O
transitions	O
.	O
</s>
<s>
This	O
simplifies	O
the	O
definition	O
of	O
deterministic	O
automaton	O
,	O
since	O
it	O
means	O
that	O
the	O
constraint	O
must	O
be	O
satisfied	O
before	O
the	O
clocks	B-Application
are	O
restarted	O
.	O
</s>
