<s>
In	O
automata	B-Application
theory	I-Application
,	O
a	O
timed	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
timed	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>
Timed	B-Application
automata	I-Application
are	O
a	O
sub-class	O
of	O
a	O
type	O
hybrid	B-Application
automata	I-Application
.	O
</s>
<s>
Timed	B-Application
automata	I-Application
can	O
be	O
used	O
to	O
model	O
and	O
analyse	O
the	O
timing	O
behavior	O
of	O
computer	O
systems	O
,	O
e.g.	O
,	O
real-time	O
systems	O
or	O
networks	O
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
the	O
state	O
reachability	O
problem	O
for	O
timed	B-Application
automata	I-Application
is	O
decidable	O
,	O
which	O
makes	O
this	O
an	O
interesting	O
sub-class	O
of	O
hybrid	B-Application
automata	I-Application
.	O
</s>
<s>
There	O
exists	O
a	O
variety	O
of	O
tools	O
to	O
input	O
and	O
analyse	O
timed	B-Application
automata	I-Application
and	O
extensions	O
,	O
including	O
the	O
model	O
checkers	O
UPPAAL	B-Application
,	O
Kronos	O
,	O
and	O
the	O
schedulability	O
analyser	O
TIMES	O
.	O
</s>
<s>
Before	O
formally	O
defining	O
what	O
a	O
timed	B-Application
automaton	I-Application
is	O
,	O
some	O
examples	O
are	O
given	O
.	O
</s>
<s>
Consider	O
the	O
language	O
of	O
timed-words	B-Application
over	O
the	O
unary	O
alphabet	O
such	O
that	O
there	O
is	O
an	O
during	O
the	O
first	O
time	O
unit	O
,	O
and	O
there	O
is	O
less	O
than	O
one	O
time	O
unit	O
between	O
two	O
successive	O
.	O
</s>
<s>
The	O
timed	B-Application
automaton	I-Application
recognizing	O
this	O
language	O
,	O
pictured	O
nearby	O
,	O
use	O
a	O
single	O
clock	B-Application
,	O
which	O
should	O
never	O
be	O
equal	O
to	O
one	O
.	O
</s>
<s>
This	O
clock	B-Application
counts	O
the	O
time	O
since	O
the	O
start	O
of	O
the	O
run	O
if	O
no	O
were	O
emitted	O
,	O
or	O
from	O
the	O
last	O
emitted	O
otherwise	O
.	O
</s>
<s>
This	O
means	O
that	O
each	O
time	O
an	O
is	O
emitted	O
,	O
this	O
clock	B-Application
is	O
reset	O
to	O
zero	O
.	O
</s>
<s>
Consider	O
the	O
language	O
of	O
timed-words	B-Application
over	O
the	O
binary	O
alphabet	O
such	O
that	O
each	O
is	O
followed	O
by	O
a	O
in	O
the	O
next	O
time	O
unit	O
.	O
</s>
<s>
The	O
timed	B-Application
automaton	I-Application
recognizing	O
this	O
language	O
,	O
pictured	O
nearby	O
,	O
recalls	O
whether	O
there	O
was	O
a	O
which	O
was	O
not	O
followed	O
by	O
a	O
or	O
not	O
.	O
</s>
<s>
Furthermore	O
,	O
when	O
there	O
is	O
such	O
a	O
,	O
it	O
has	O
a	O
clock	B-Application
which	O
recall	O
the	O
time	O
elapsed	O
since	O
the	O
first	O
such	O
was	O
emitted	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
a	O
can	O
not	O
be	O
emitted	O
if	O
the	O
clock	B-Application
is	O
at	O
least	O
equal	O
to	O
one	O
,	O
and	O
thus	O
the	O
run	O
fails	O
.	O
</s>
<s>
Formally	O
,	O
a	O
timed	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
with	O
action	O
,	O
guard	O
and	O
clock	B-Application
resets	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
timed-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
know	O
in	O
which	O
location	O
you	O
are	O
,	O
and	O
the	O
clock	B-Application
valuation	O
.	O
</s>
<s>
Given	O
a	O
timed	B-Application
word	I-Application
with	O
,	O
an	O
increasing	O
sequence	O
of	O
non-negative	O
number	O
,	O
and	O
a	O
timed-automaton	O
as	O
above	O
,	O
a	O
run	O
is	O
a	O
sequence	O
of	O
the	O
form	O
satisfying	O
the	O
following	O
constraint	O
:	O
</s>
<s>
the	O
new	O
clock	B-Application
valuation	O
corresponds	O
to	O
,	O
in	O
which	O
time	O
units	O
passed	O
and	O
in	O
which	O
the	O
clocks	B-Application
of	O
where	O
reset	O
.	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>
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
timed-automaton	O
may	O
be	O
deterministic	O
or	O
non-deterministic	O
.	O
</s>
<s>
for	O
each	O
pair	O
of	O
transitions	O
and	O
,	O
the	O
set	O
of	O
clocks	B-Application
valuations	O
satisfying	O
is	O
disjoint	O
from	O
the	O
set	O
of	O
clocks	B-Application
valuations	O
satisfying	O
.	O
</s>
<s>
The	O
class	O
of	O
languages	O
recognized	O
by	O
non-deterministic	O
timed	B-Application
automata	I-Application
is	O
:	O
</s>
<s>
closed	O
under	O
union	O
,	O
indeed	O
,	O
the	O
disjoint	O
union	O
of	O
two	O
timed	B-Application
automata	I-Application
recognize	O
the	O
union	O
of	O
the	O
language	O
recognized	O
by	O
those	O
automata	O
.	O
</s>
<s>
The	O
computational	O
complexity	O
of	O
some	O
problems	O
related	O
to	O
timed	B-Application
automata	I-Application
are	O
now	O
given	O
.	O
</s>
<s>
The	O
emptiness	O
problem	O
for	O
timed	B-Application
automaton	I-Application
can	O
be	O
solved	O
by	O
constructing	O
a	O
region	O
automaton	O
and	O
checking	O
whether	O
it	O
accepts	O
the	O
empty	O
language	O
.	O
</s>
<s>
The	O
universality	O
problem	O
of	O
non-deterministic	O
timed	B-Application
automaton	I-Application
is	O
undecidable	O
,	O
and	O
more	O
precisely	O
Π	O
.	O
</s>
<s>
However	O
,	O
when	O
the	O
automaton	O
contains	O
a	O
single	O
clock	B-Application
,	O
the	O
property	O
is	O
decidable	O
,	O
however	O
it	O
is	O
not	O
primitive	B-Architecture
recursive	I-Architecture
.	O
</s>
