<s>
In	O
theoretical	O
computer	O
science	O
,	O
a	O
transition	B-Application
system	I-Application
is	O
a	O
concept	O
used	O
in	O
the	O
study	O
of	O
computation	O
.	O
</s>
<s>
It	O
is	O
used	O
to	O
describe	O
the	O
potential	O
behavior	O
of	O
discrete	B-Application
systems	I-Application
.	O
</s>
<s>
It	O
consists	O
of	O
states	B-Application
and	O
transitions	O
between	O
states	B-Application
,	O
which	O
may	O
be	O
labeled	O
with	O
labels	O
chosen	O
from	O
a	O
set	O
;	O
the	O
same	O
label	O
may	O
appear	O
on	O
more	O
than	O
one	O
transition	O
.	O
</s>
<s>
Transition	B-Application
systems	I-Application
coincide	O
mathematically	O
with	O
abstract	O
rewriting	O
systems	O
(	O
as	O
explained	O
further	O
in	O
this	O
article	O
)	O
and	O
directed	O
graphs	O
.	O
</s>
<s>
They	O
differ	O
from	O
finite-state	B-Architecture
automata	I-Architecture
in	O
several	O
ways	O
:	O
</s>
<s>
The	O
set	O
of	O
states	B-Application
is	O
not	O
necessarily	O
finite	O
,	O
or	O
even	O
countable	O
.	O
</s>
<s>
No	O
"	O
start	O
"	O
state	O
or	O
"	O
final	O
"	O
states	B-Application
are	O
given	O
.	O
</s>
<s>
Transition	B-Application
systems	I-Application
can	O
be	O
represented	O
as	O
directed	O
graphs	O
.	O
</s>
<s>
Formally	O
,	O
a	O
transition	B-Application
system	I-Application
is	O
a	O
pair	O
where	O
is	O
a	O
set	O
of	O
states	B-Application
and	O
,	O
the	O
transition	O
relation	O
,	O
is	O
a	O
subset	O
of	O
.	O
</s>
<s>
A	O
labelled	B-Application
transition	I-Application
system	I-Application
is	O
a	O
tuple	O
where	O
is	O
a	O
set	O
of	O
states	B-Application
,	O
is	O
a	O
set	O
of	O
labels	O
,	O
and	O
,	O
the	O
labelled	O
transition	O
relation	O
,	O
is	O
a	O
subset	O
of	O
.	O
</s>
<s>
Labelled	O
transitions	O
systems	O
were	O
originally	O
introduced	O
as	O
named	O
transition	B-Application
systems	I-Application
.	O
</s>
<s>
Labelled	O
state	B-Application
transition	I-Application
systems	I-Application
on	O
with	O
labels	O
from	O
correspond	O
one-to-one	B-Algorithm
with	O
functions	O
,	O
where	O
is	O
the	O
(	O
covariant	O
)	O
powerset	O
functor	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
a	O
labelled	O
state	B-Application
transition	I-Application
system	I-Application
is	O
a	O
coalgebra	O
for	O
the	O
functor	O
.	O
</s>
<s>
Some	O
are	O
simple	O
,	O
such	O
as	O
observing	O
that	O
a	O
labelled	B-Application
transition	I-Application
system	I-Application
where	O
the	O
set	O
of	O
labels	O
consists	O
of	O
only	O
one	O
element	O
is	O
equivalent	O
to	O
an	O
unlabelled	O
transition	B-Application
system	I-Application
.	O
</s>
<s>
As	O
a	O
mathematical	O
object	O
,	O
an	O
unlabeled	O
transition	B-Application
system	I-Application
is	O
identical	O
with	O
an	O
(	O
unindexed	O
)	O
abstract	O
rewriting	O
system	O
.	O
</s>
<s>
If	O
we	O
consider	O
the	O
rewriting	O
relation	O
as	O
an	O
indexed	O
set	O
of	O
relations	O
,	O
as	O
some	O
authors	O
do	O
,	O
then	O
a	O
labeled	B-Application
transition	I-Application
system	I-Application
is	O
equivalent	O
to	O
an	O
abstract	O
rewriting	O
system	O
with	O
the	O
indices	O
being	O
the	O
labels	O
.	O
</s>
<s>
In	O
a	O
transition	B-Application
system	I-Application
one	O
is	O
interested	O
in	O
interpreting	O
the	O
labels	O
as	O
actions	O
,	O
whereas	O
in	O
an	O
abstract	O
rewriting	O
system	O
the	O
focus	O
is	O
on	O
how	O
objects	O
may	O
be	O
transformed	O
(	O
rewritten	O
)	O
into	O
others	O
.	O
</s>
<s>
In	O
model	B-Application
checking	I-Application
,	O
a	O
transition	B-Application
system	I-Application
is	O
sometimes	O
defined	O
to	O
include	O
an	O
additional	O
labeling	O
function	O
for	O
the	O
states	B-Application
as	O
well	O
,	O
resulting	O
in	O
a	O
notion	O
that	O
encompasses	O
that	O
of	O
Kripke	B-Application
structure	I-Application
.	O
</s>
<s>
Action	B-Application
languages	I-Application
are	O
extensions	O
of	O
transition	B-Application
systems	I-Application
,	O
adding	O
a	O
set	O
of	O
fluents	O
F	O
,	O
a	O
set	O
of	O
values	O
V	O
,	O
and	O
a	O
function	O
that	O
maps	O
F	O
S	O
to	O
V	O
.	O
</s>
