<s>
A	O
finite-state	B-Architecture
transducer	I-Architecture
(	O
FST	O
)	O
is	O
a	O
finite-state	B-Architecture
machine	I-Architecture
with	O
two	O
memory	O
tapes	O
,	O
following	O
the	O
terminology	O
for	O
Turing	B-Architecture
machines	I-Architecture
:	O
an	O
input	O
tape	O
and	O
an	O
output	O
tape	O
.	O
</s>
<s>
This	O
contrasts	O
with	O
an	O
ordinary	O
finite-state	B-Architecture
automaton	I-Architecture
,	O
which	O
has	O
a	O
single	O
tape	O
.	O
</s>
<s>
An	O
FST	O
is	O
a	O
type	O
of	O
finite-state	B-Architecture
automaton	I-Architecture
(	O
FSA	O
)	O
that	O
maps	O
between	O
two	O
sets	O
of	O
symbols	O
.	O
</s>
<s>
In	O
morphological	B-General_Concept
parsing	I-General_Concept
,	O
an	O
example	O
would	O
be	O
inputting	O
a	O
string	O
of	O
letters	O
into	O
the	O
FST	O
,	O
the	O
FST	O
would	O
then	O
output	O
a	O
string	O
of	O
morphemes	O
.	O
</s>
<s>
The	O
class	O
of	O
languages	O
generated	O
by	O
finite	B-Architecture
automata	I-Architecture
is	O
known	O
as	O
the	O
class	O
of	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
In	O
general	O
,	O
a	O
transducer	O
computes	O
a	O
relation	B-Algorithm
between	O
two	O
formal	O
languages	O
.	O
</s>
<s>
Each	O
string-to-string	O
finite-state	B-Architecture
transducer	I-Architecture
relates	O
the	O
input	O
alphabet	O
Σ	O
to	O
the	O
output	O
alphabet	O
Γ	O
.	O
</s>
<s>
Relations	O
R	O
on	O
Σ	O
*×Γ*	O
that	O
can	O
be	O
implemented	O
as	O
finite-state	B-Architecture
transducers	I-Architecture
are	O
called	O
rational	O
relations	O
.	O
</s>
<s>
Rational	O
relations	O
that	O
are	O
partial	B-Algorithm
functions	I-Algorithm
,	O
i.e.	O
</s>
<s>
Finite-state	B-Architecture
transducers	I-Architecture
are	O
often	O
used	O
for	O
phonological	O
and	O
morphological	O
analysis	O
in	O
natural	B-Language
language	I-Language
processing	I-Language
research	O
and	O
applications	O
.	O
</s>
<s>
Formally	O
,	O
a	O
finite	B-Architecture
transducer	I-Architecture
T	O
is	O
a	O
6-tuple	O
(	O
)	O
such	O
that	O
:	O
</s>
<s>
(	O
where	O
ε	O
is	O
the	O
empty	O
string	O
)	O
is	O
the	O
transition	O
relation	B-Algorithm
.	O
</s>
<s>
NOTE	O
:	O
This	O
definition	O
of	O
finite	B-Architecture
transducer	I-Architecture
is	O
also	O
called	O
letter	O
transducer	O
(	O
Roche	O
and	O
Schabes	O
1997	O
)	O
;	O
alternative	O
definitions	O
are	O
possible	O
,	O
but	O
can	O
all	O
be	O
converted	O
into	O
transducers	O
following	O
this	O
one	O
.	O
</s>
<s>
Define	O
the	O
extended	O
transition	O
relation	B-Algorithm
as	O
the	O
smallest	O
set	O
such	O
that	O
:	O
</s>
<s>
The	O
extended	O
transition	O
relation	B-Algorithm
is	O
essentially	O
the	O
reflexive	O
transitive	O
closure	O
of	O
the	O
transition	O
graph	O
that	O
has	O
been	O
augmented	O
to	O
take	O
edge	O
labels	O
into	O
account	O
.	O
</s>
<s>
The	O
behavior	O
of	O
the	O
transducer	O
T	O
is	O
the	O
rational	O
relation	B-Algorithm
 [ T ] 	O
defined	O
as	O
follows	O
:	O
if	O
and	O
only	O
if	O
there	O
exists	O
and	O
such	O
that	O
.	O
</s>
<s>
Finite	B-Architecture
State	I-Architecture
Transducers	I-Architecture
can	O
be	O
weighted	O
,	O
where	O
each	O
transition	O
is	O
labelled	O
with	O
a	O
weight	O
in	O
addition	O
to	O
the	O
input	O
and	O
output	O
labels	O
.	O
</s>
<s>
A	O
Weighted	O
Finite	B-Architecture
State	I-Architecture
Transducer	I-Architecture
(	O
WFST	O
)	O
over	O
a	O
set	O
K	O
of	O
weights	O
can	O
be	O
defined	O
similarly	O
to	O
an	O
unweighted	O
one	O
as	O
an	O
8-tuple	O
,	O
where	O
:	O
</s>
<s>
Two	O
typical	O
semirings	O
used	O
in	O
practice	O
are	O
the	O
log	O
semiring	O
and	O
tropical	O
semiring	O
:	O
nondeterministic	B-General_Concept
automata	I-General_Concept
may	O
be	O
regarded	O
as	O
having	O
weights	O
in	O
the	O
Boolean	O
semiring	O
.	O
</s>
<s>
The	O
following	O
operations	O
defined	O
on	O
finite	B-Architecture
automata	I-Architecture
also	O
apply	O
to	O
finite	B-Architecture
transducers	I-Architecture
:	O
</s>
<s>
This	O
definition	O
uses	O
the	O
same	O
notation	O
used	O
in	O
mathematics	O
for	O
relation	B-Algorithm
composition	O
.	O
</s>
<s>
Determinization	B-General_Concept
.	O
</s>
<s>
The	O
powerset	B-General_Concept
construction	I-General_Concept
can	O
be	O
extended	O
to	O
transducers	O
,	O
or	O
even	O
weighted	O
transducers	O
,	O
but	O
sometimes	O
fails	O
to	O
halt	O
;	O
indeed	O
,	O
some	O
non-deterministic	O
transducers	O
do	O
not	O
admit	O
equivalent	O
deterministic	O
transducers	O
.	O
</s>
<s>
Removal	O
of	O
epsilon-transitions	B-General_Concept
.	O
</s>
<s>
It	O
is	O
decidable	O
whether	O
the	O
relation	B-Algorithm
 [ T ] 	O
of	O
a	O
transducer	O
T	O
is	O
empty	O
.	O
</s>
<s>
Equivalence	O
is	O
however	O
decidable	O
in	O
the	O
special	O
case	O
where	O
the	O
relation	B-Algorithm
 [ T ] 	O
of	O
a	O
transducer	O
T	O
is	O
a	O
(	O
partial	O
)	O
function	O
.	O
</s>
<s>
If	O
one	O
defines	O
the	O
alphabet	O
of	O
labels	O
,	O
finite-state	B-Architecture
transducers	I-Architecture
are	O
isomorphic	O
to	O
NDFA	B-General_Concept
over	O
the	O
alphabet	O
,	O
and	O
may	O
therefore	O
be	O
determinized	O
(	O
turned	O
into	O
deterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
over	O
the	O
alphabet	O
)	O
and	O
subsequently	O
minimized	O
so	O
that	O
they	O
have	O
the	O
minimum	O
number	O
of	O
states	O
.	O
</s>
<s>
FSTs	O
are	O
used	O
in	O
the	O
lexical	B-Application
analysis	I-Application
phase	O
of	O
compilers	B-Language
to	O
associate	O
semantic	O
value	O
with	O
the	O
discovered	O
tokens	O
.	O
</s>
<s>
Context-sensitive	O
rewriting	O
rules	O
of	O
the	O
form	O
a	O
→	O
b	O
/	O
c	O
_	O
d	O
,	O
used	O
in	O
linguistics	O
to	O
model	O
phonological	O
rules	O
and	O
sound	O
change	O
,	O
are	O
computationally	O
equivalent	O
to	O
finite-state	B-Architecture
transducers	I-Architecture
,	O
provided	O
that	O
application	O
is	O
nonrecursive	O
,	O
i.e.	O
</s>
<s>
Weighted	O
FSTs	O
found	O
applications	O
in	O
natural	B-Language
language	I-Language
processing	I-Language
,	O
including	O
machine	B-Application
translation	I-Application
,	O
and	O
in	O
machine	O
learning	O
.	O
</s>
