<s>
In	O
automata	B-Application
theory	I-Application
,	O
an	O
unambiguous	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
UFA	O
)	O
is	O
a	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
NFA	B-General_Concept
)	O
such	O
that	O
each	O
word	O
has	O
at	O
most	O
one	O
accepting	O
path	O
.	O
</s>
<s>
Each	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
DFA	B-General_Concept
)	O
is	O
an	O
UFA	O
,	O
but	O
not	O
vice	O
versa	O
.	O
</s>
<s>
DFA	B-General_Concept
,	O
UFA	O
,	O
and	O
NFA	B-General_Concept
recognize	O
exactly	O
the	O
same	O
class	O
of	O
formal	O
languages	O
.	O
</s>
<s>
On	O
the	O
one	O
hand	O
,	O
an	O
NFA	B-General_Concept
can	O
be	O
exponentially	O
smaller	O
than	O
an	O
equivalent	O
DFA	B-General_Concept
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
some	O
problems	O
are	O
easily	O
solved	O
on	O
DFAs	B-General_Concept
and	O
not	O
on	O
UFAs	O
.	O
</s>
<s>
For	O
example	O
,	O
given	O
an	O
automaton	O
A	O
,	O
an	O
automaton	O
A	O
which	O
accepts	O
the	O
complement	O
of	O
A	O
can	O
be	O
computed	O
in	O
linear	O
time	O
when	O
A	O
is	O
a	O
DFA	B-General_Concept
,	O
it	O
is	O
not	O
known	O
whether	O
it	O
can	O
be	O
done	O
in	O
polynomial	O
time	O
for	O
UFA	O
.	O
</s>
<s>
Hence	O
UFAs	O
are	O
a	O
mix	O
of	O
the	O
worlds	O
of	O
DFA	B-General_Concept
and	O
of	O
NFA	B-General_Concept
;	O
in	O
some	O
cases	O
,	O
they	O
lead	O
to	O
smaller	O
automata	O
than	O
DFA	B-General_Concept
and	O
quicker	O
algorithms	O
than	O
NFA	B-General_Concept
.	O
</s>
<s>
An	O
NFA	B-General_Concept
is	O
represented	O
formally	O
by	O
a	O
5-tuple	O
,	O
.	O
</s>
<s>
An	O
UFA	O
is	O
an	O
NFA	B-General_Concept
such	O
that	O
,	O
for	O
each	O
word	O
,	O
there	O
exists	O
at	O
most	O
one	O
sequence	O
of	O
states	O
,	O
in	O
with	O
the	O
following	O
conditions	O
:	O
</s>
<s>
The	O
figures	O
show	O
a	O
DFA	B-General_Concept
and	O
a	O
UFA	O
accepting	O
this	O
language	O
for	O
n	O
=	O
2	O
.	O
</s>
<s>
The	O
minimal	O
DFA	B-General_Concept
accepting	O
has	O
2n	O
states	O
,	O
one	O
for	O
each	O
subset	O
of	O
1	O
...	O
n	O
.	O
</s>
<s>
Three	O
PSPACE-hard	O
problems	O
for	O
general	O
NFA	B-General_Concept
belong	O
to	O
PTIME	O
for	O
DFA	B-General_Concept
and	O
are	O
now	O
considered	O
.	O
</s>
<s>
The	O
number	O
of	O
words	O
of	O
length	O
n	O
accepted	O
by	O
an	O
automaton	O
can	O
be	O
computed	O
in	O
polynomial	O
time	O
using	O
dynamic	B-Algorithm
programming	I-Algorithm
,	O
which	O
ends	O
the	O
proof	O
.	O
</s>
<s>
For	O
a	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
with	O
states	O
and	O
an	O
letter	O
alphabet	O
,	O
it	O
is	O
decidable	O
in	O
time	O
whether	O
is	O
unambiguous	O
.	O
</s>
<s>
The	O
notion	O
of	O
unambiguity	O
extends	O
to	O
finite	B-Architecture
state	I-Architecture
transducers	I-Architecture
and	O
weighted	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
If	O
a	O
finite	B-Architecture
state	I-Architecture
transducer	I-Architecture
T	O
is	O
unambiguous	O
,	O
then	O
each	O
input	O
word	O
is	O
associated	O
by	O
T	O
to	O
at	O
most	O
one	O
output	O
word	O
.	O
</s>
<s>
If	O
a	O
weighted	B-General_Concept
automaton	I-General_Concept
A	O
is	O
unambiguous	O
,	O
then	O
the	O
set	O
of	O
weight	O
does	O
not	O
need	O
to	O
be	O
a	O
semiring	O
,	O
instead	O
it	O
suffices	O
to	O
consider	O
a	O
monoid	O
.	O
</s>
<s>
Leung	O
proved	O
that	O
a	O
DFA	B-General_Concept
equivalent	O
to	O
an	O
-state	O
UFA	O
requires	O
states	O
in	O
the	O
worst	O
case	O
,	O
and	O
that	O
a	O
UFA	O
equivalent	O
to	O
a	O
finitely	O
ambiguous	O
-state	O
NFA	B-General_Concept
requires	O
states	O
in	O
the	O
worst	O
case	O
.	O
</s>
<s>
For	O
a	O
one-letter	O
alphabet	O
Okhotin	O
proved	O
that	O
a	O
DFA	B-General_Concept
equivalent	O
to	O
an	O
-state	O
UFA	O
requires	O
states	O
in	O
the	O
worst	O
case	O
.	O
</s>
