<s>
Metric	B-Application
temporal	I-Application
logic	I-Application
(	O
MTL	O
)	O
is	O
a	O
special	O
case	O
of	O
temporal	O
logic	O
.	O
</s>
<s>
Full	O
MTL	O
over	O
infinite	O
timed	B-Application
words	I-Application
is	O
undecidable	O
.	O
</s>
<s>
The	O
full	O
metric	B-Application
temporal	I-Application
logic	I-Application
is	O
defined	O
similarly	O
to	O
linear	O
temporal	O
logic	O
,	O
where	O
a	O
set	O
of	O
non-negative	O
real	O
number	O
is	O
added	O
to	O
temporal	O
modal	O
operators	O
U	O
and	O
S	O
.	O
Formally	O
,	O
MTL	O
is	O
built	O
up	O
from	O
:	O
</s>
<s>
The	O
past	O
fragment	O
of	O
metric	B-Application
temporal	I-Application
logic	I-Application
,	O
denoted	O
as	O
past-MTL	O
is	O
defined	O
as	O
the	O
restriction	O
of	O
the	O
full	O
metric	B-Application
temporal	I-Application
logic	I-Application
without	O
the	O
until	O
operator	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
future	O
fragment	O
of	O
metric	B-Application
temporal	I-Application
logic	I-Application
,	O
denoted	O
as	O
future-MTL	O
is	O
defined	O
as	O
the	O
restriction	O
of	O
the	O
full	O
metric	B-Application
temporal	I-Application
logic	I-Application
without	O
the	O
since	O
operator	O
.	O
</s>
<s>
either	O
a	O
timed	B-Application
word	I-Application
or	O
a	O
signal	B-Application
.	O
</s>
<s>
Over	O
timed	B-Application
words	I-Application
,	O
this	O
formula	O
always	O
hold	O
.	O
</s>
<s>
We	O
now	O
consider	O
this	O
formula	O
over	O
timed	B-Application
words	I-Application
and	O
over	O
signals	O
.	O
</s>
<s>
We	O
consider	O
timed	B-Application
words	I-Application
first	O
.	O
</s>
<s>
Let	O
a	O
timed	B-Application
word	I-Application
and	O
in	O
its	O
domain	O
of	O
definition	O
.	O
</s>
<s>
Over	O
timed	B-Application
words	I-Application
,	O
the	O
formula	O
holds	O
if	O
and	O
only	O
if	O
also	O
holds	O
.	O
</s>
<s>
Let	O
us	O
now	O
consider	O
signal	B-Application
.	O
</s>
<s>
The	O
equivalence	O
mentioned	O
above	O
does	O
not	O
hold	O
anymore	O
over	O
signal	B-Application
.	O
</s>
<s>
This	O
is	O
due	O
to	O
the	O
fact	O
that	O
,	O
using	O
the	O
variables	O
introduced	O
above	O
,	O
there	O
may	O
exists	O
an	O
infinite	O
number	O
of	O
correct	O
values	O
for	O
,	O
due	O
to	O
the	O
fact	O
that	O
the	O
domain	O
of	O
definition	O
of	O
a	O
signal	B-Application
is	O
continuous	O
.	O
</s>
<s>
Some	O
more	O
example	O
can	O
be	O
found	O
on	O
article	O
of	O
fragments	O
of	O
MITL	B-Application
,	O
such	O
as	O
metric	B-Application
interval	I-Application
temporal	I-Application
logic	I-Application
.	O
</s>
<s>
Logic	O
(	O
MITL	B-Application
)	O
.	O
</s>
<s>
Some	O
other	O
subsets	O
of	O
MITL	B-Application
are	O
defined	O
in	O
the	O
article	O
MITL	B-Application
.	O
</s>
<s>
Both	O
over	O
timed-words	B-Application
and	O
over	O
signals	O
,	O
it	O
is	O
less	O
expressive	O
than	O
Full-MTL	O
.	O
</s>
<s>
Over	O
signals	O
,	O
ECL	O
is	O
as	O
expressive	O
as	O
MITL	B-Application
and	O
as	O
MITL0	O
.	O
</s>
<s>
If	O
is	O
not	O
a	O
singleton	O
and	O
is	O
a	O
MITL	B-Application
formula	O
,	O
is	O
defined	O
as	O
a	O
MITL	B-Application
formula	O
.	O
</s>
<s>
If	O
is	O
a	O
singleton	O
,	O
then	O
is	O
equivalent	O
to	O
which	O
is	O
a	O
MITL-formula	O
.	O
</s>
