<s>
In	O
model	B-Application
checking	I-Application
,	O
a	O
field	O
of	O
computer	B-General_Concept
science	I-General_Concept
,	O
Timed	B-Application
Propositional	I-Application
Temporal	I-Application
Logic	I-Application
(	O
TPTL	B-Application
)	O
is	O
an	O
extension	O
of	O
Linear	O
Temporal	O
Logic	O
(	O
LTL	O
)	O
in	O
which	O
variables	O
are	O
introduced	O
to	O
measure	O
times	O
between	O
two	O
events	O
.	O
</s>
<s>
For	O
example	O
,	O
while	O
LTL	O
allows	O
to	O
state	O
that	O
each	O
event	O
p	O
is	O
eventually	O
followed	O
by	O
an	O
event	O
q	O
,	O
TPTL	B-Application
furthermore	O
allows	O
to	O
give	O
a	O
time	O
limit	O
for	O
q	O
to	O
occur	O
.	O
</s>
<s>
The	O
future	O
fragment	O
of	O
TPTL	B-Application
is	O
defined	O
similarly	O
to	O
linear	O
temporal	O
logic	O
,	O
in	O
which	O
furthermore	O
,	O
clock	B-Application
variables	O
can	O
be	O
introduced	O
and	O
compared	O
to	O
constants	O
.	O
</s>
<s>
Formally	O
,	O
given	O
a	O
set	O
of	O
clocks	B-Application
,	O
MTL	O
is	O
built	O
up	O
from	O
:	O
</s>
<s>
a	O
clock	B-Application
comparison	O
,	O
with	O
,	O
a	O
number	O
and	O
being	O
a	O
comparison	O
operator	O
such	O
as	O
,	O
,	O
=	O
,	O
or	O
.	O
</s>
<s>
a	O
freeze	O
quantification	O
operator	O
,	O
for	O
a	O
TPTL	B-Application
formula	O
with	O
set	O
of	O
clocks	B-Application
.	O
</s>
<s>
A	O
closed	O
formula	O
is	O
a	O
formula	O
over	O
an	O
empty	O
set	O
of	O
clocks	B-Application
.	O
</s>
<s>
A	O
model	O
of	O
a	O
TPTL	B-Application
formula	O
is	O
such	O
a	O
function	O
.	O
</s>
<s>
Usually	O
,	O
is	O
either	O
a	O
timed	B-Application
word	I-Application
or	O
a	O
signal	B-Application
.	O
</s>
<s>
Let	O
a	O
set	O
of	O
clocks	B-Application
.	O
</s>
<s>
Let	O
a	O
clock	B-Application
valuation	O
over	O
.	O
</s>
<s>
We	O
are	O
now	O
going	O
to	O
explain	O
what	O
it	O
means	O
that	O
a	O
TPTL	B-Application
formula	O
holds	O
at	O
time	O
for	O
a	O
valuation	O
.	O
</s>
<s>
Let	O
and	O
be	O
two	O
formulas	O
over	O
the	O
set	O
of	O
clocks	B-Application
,	O
a	O
formula	O
over	O
the	O
set	O
of	O
clocks	B-Application
,	O
,	O
,	O
a	O
number	O
and	O
being	O
a	O
comparison	O
operator	O
such	O
as	O
,	O
,	O
=	O
,	O
or	O
:	O
</s>
<s>
Metric	B-Application
temporal	I-Application
logic	I-Application
is	O
another	O
extension	O
of	O
LTL	O
which	O
allow	O
to	O
measure	O
time	O
.	O
</s>
<s>
TPTL	B-Application
is	O
as	O
least	O
as	O
expressive	O
as	O
MTL	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
MTL	O
formula	O
is	O
equivalent	O
to	O
the	O
TPTL	B-Application
formula	O
where	O
is	O
a	O
new	O
variable	O
.	O
</s>
<s>
It	O
follows	O
that	O
any	O
other	O
operator	O
introduced	O
in	O
the	O
page	O
MTL	O
,	O
such	O
as	O
and	O
can	O
also	O
be	O
defined	O
as	O
TPTL	B-Application
formulas	O
.	O
</s>
<s>
TPTL	B-Application
is	O
strictly	O
more	O
expressive	O
than	O
MTL	O
both	O
over	O
timed	B-Application
words	I-Application
and	O
over	O
signals	O
.	O
</s>
<s>
Over	O
timed	B-Application
words	I-Application
,	O
no	O
MTL	O
formula	O
is	O
equivalent	O
to	O
.	O
</s>
<s>
Over	O
signal	B-Application
,	O
there	O
are	O
no	O
MTL	O
formula	O
equivalent	O
to	O
,	O
which	O
states	O
that	O
the	O
last	O
atomic	O
proposition	O
before	O
time	O
point	O
1	O
is	O
an	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
for	O
an	O
arbitrary	O
LTL	O
formula	O
,	O
if	O
and	O
only	O
if	O
,	O
where	O
is	O
considered	O
as	O
a	O
TPTL	B-Application
formula	O
with	O
non-strict	O
operator	O
,	O
and	O
is	O
the	O
only	O
function	O
defined	O
on	O
the	O
empty	O
set	O
.	O
</s>
