<s>
In	O
theoretical	O
computer	O
science	O
and	O
formal	O
language	O
theory	O
,	O
a	O
weighted	B-General_Concept
automaton	I-General_Concept
or	O
weighted	B-General_Concept
finite-state	I-General_Concept
machine	I-General_Concept
is	O
a	O
generalization	B-Algorithm
of	O
a	O
finite-state	B-Architecture
machine	I-Architecture
in	O
which	O
the	O
edges	O
have	O
weights	O
,	O
for	O
example	O
real	O
numbers	O
or	O
integers	O
.	O
</s>
<s>
Finite-state	B-Architecture
machines	I-Architecture
are	O
only	O
capable	O
of	O
answering	O
decision	O
problems	O
;	O
they	O
take	O
as	O
input	O
a	O
string	O
and	O
produce	O
a	O
Boolean	O
output	O
,	O
i.e.	O
</s>
<s>
In	O
contrast	O
,	O
weighted	B-General_Concept
automata	I-General_Concept
produce	O
a	O
quantitative	O
output	O
,	O
for	O
example	O
a	O
count	O
of	O
how	O
many	O
answers	O
are	O
possible	O
on	O
a	O
given	O
input	O
string	O
,	O
or	O
a	O
probability	O
of	O
how	O
likely	O
the	O
input	O
string	O
is	O
according	O
to	O
a	O
probability	O
distribution	O
.	O
</s>
<s>
The	O
definition	O
of	O
a	O
weighted	B-General_Concept
automaton	I-General_Concept
is	O
generally	O
given	O
over	O
an	O
arbitrary	O
semiring	O
,	O
an	O
abstract	O
set	O
with	O
an	O
addition	O
operation	O
and	O
a	O
multiplication	O
operation	O
.	O
</s>
<s>
The	O
weighted	B-General_Concept
automaton	I-General_Concept
thus	O
defines	O
a	O
function	O
from	O
to	O
.	O
</s>
<s>
Weighted	B-General_Concept
automata	I-General_Concept
generalize	B-Algorithm
deterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
(	O
DFAs	B-General_Concept
)	O
and	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
(	O
NFAs	O
)	O
,	O
which	O
correspond	O
to	O
weighted	B-General_Concept
automata	I-General_Concept
over	O
the	O
Boolean	O
semiring	O
,	O
where	O
addition	O
is	O
logical	O
disjunction	O
and	O
multiplication	O
is	O
logical	O
conjunction	O
.	O
</s>
<s>
In	O
the	O
DFA	B-General_Concept
case	O
,	O
there	O
is	O
only	O
one	O
accepting	O
path	O
for	O
any	O
input	O
string	O
,	O
so	O
disjunction	O
is	O
not	O
applied	O
.	O
</s>
<s>
When	O
the	O
weights	O
are	O
real	O
numbers	O
and	O
the	O
outgoing	O
weights	O
for	O
each	O
state	O
add	O
to	O
one	O
,	O
weighted	B-General_Concept
automata	I-General_Concept
can	O
be	O
considered	O
a	O
probabilistic	O
model	O
and	O
are	O
also	O
known	O
as	O
probabilistic	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
Weighted	B-General_Concept
automata	I-General_Concept
have	O
applications	O
in	O
natural	B-Language
language	I-Language
processing	I-Language
where	O
they	O
are	O
used	O
to	O
assign	O
weights	O
to	O
words	O
and	O
sentences	O
,	O
as	O
well	O
as	O
in	O
image	B-General_Concept
compression	I-General_Concept
.	O
</s>
<s>
Since	O
their	O
introduction	O
,	O
many	O
extensions	O
have	O
been	O
proposed	O
,	O
for	O
example	O
nested	O
weighted	B-General_Concept
automata	I-General_Concept
,	O
cost	O
register	O
automata	O
,	O
and	O
weighted	O
finite-state	B-Architecture
transducers	I-Architecture
.	O
</s>
<s>
Researchers	O
have	O
studied	O
weighted	B-General_Concept
automata	I-General_Concept
from	O
the	O
perspective	O
of	O
learning	O
a	O
machine	O
from	O
its	O
input-output	O
behavior	O
(	O
see	O
computational	O
learning	O
theory	O
)	O
and	O
studying	O
decidability	O
questions	O
.	O
</s>
<s>
A	O
weighted	B-General_Concept
automaton	I-General_Concept
over	O
is	O
a	O
tuple	O
where	O
:	O
</s>
<s>
The	O
underlying	O
NFA	B-General_Concept
of	O
is	O
an	O
NFA	B-General_Concept
formed	O
by	O
removing	O
all	O
transitions	O
with	O
weight	O
and	O
then	O
erasing	O
all	O
of	O
the	O
weights	O
on	O
the	O
transitions	O
,	O
so	O
that	O
the	O
new	O
transition	O
set	O
lies	O
in	O
.	O
</s>
<s>
A	O
weighted	B-General_Concept
automaton	I-General_Concept
is	O
deterministic	B-General_Concept
if	O
the	O
underlying	O
NFA	B-General_Concept
is	O
deterministic	B-General_Concept
and	O
unambiguous	B-General_Concept
if	O
the	O
underlying	O
NFA	B-General_Concept
is	O
unambiguous	B-General_Concept
.	O
</s>
<s>
This	O
does	O
not	O
increase	O
the	O
expressiveness	O
of	O
weighted	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
The	O
requirement	O
that	O
there	O
is	O
a	O
zero	O
element	O
for	O
is	O
sometimes	O
omitted	O
;	O
in	O
this	O
case	O
the	O
machine	O
defines	O
a	O
partial	B-Algorithm
function	I-Algorithm
from	O
to	O
rather	O
than	O
a	O
total	B-Algorithm
function	I-Algorithm
.	O
</s>
