<s>
In	O
automata	B-Application
theory	I-Application
,	O
an	O
alternating	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
AFA	O
)	O
is	O
a	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
whose	O
transitions	O
are	O
divided	O
into	O
existential	B-Algorithm
and	O
universal	O
transitions	O
.	O
</s>
<s>
For	O
example	O
,	O
let	O
A	O
be	O
an	O
alternating	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
For	O
an	O
existential	B-Algorithm
transition	O
,	O
A	O
nondeterministically	O
chooses	O
to	O
switch	O
the	O
state	O
to	O
either	O
or	O
,	O
reading	O
a	O
.	O
</s>
<s>
Thus	O
,	O
behaving	O
like	O
a	O
regular	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
A	O
accepts	O
a	O
word	O
w	O
,	O
if	O
there	B-Algorithm
exists	I-Algorithm
a	O
run	O
tree	O
on	O
w	O
such	O
that	O
every	O
path	O
ends	O
in	O
an	O
accepting	O
state	O
.	O
</s>
<s>
A	O
basic	O
theorem	O
states	O
that	O
any	O
AFA	O
is	O
equivalent	O
to	O
a	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
DFA	B-General_Concept
)	O
,	O
hence	O
AFAs	O
accept	O
exactly	O
the	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
An	O
alternative	O
model	O
which	O
is	O
frequently	O
used	O
is	O
the	O
one	O
where	O
Boolean	O
combinations	O
are	O
in	O
disjunctive	B-Application
normal	I-Application
form	I-Application
so	O
that	O
,	O
e.g.	O
,	O
would	O
represent	O
.	O
</s>
<s>
Alternating	O
finite	O
automata	O
can	O
be	O
extended	O
to	O
accept	O
trees	O
in	O
the	O
same	O
way	O
as	O
tree	B-Application
automata	I-Application
,	O
yielding	O
alternating	B-Application
tree	I-Application
automata	I-Application
.	O
</s>
<s>
An	O
alternating	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
AFA	O
)	O
is	O
a	O
6-tuple	O
,	O
</s>
<s>
is	O
a	O
finite	O
set	O
of	O
existential	B-Algorithm
states	O
.	O
</s>
<s>
Even	O
though	O
AFA	O
can	O
accept	O
exactly	O
the	O
regular	B-General_Concept
languages	I-General_Concept
,	O
they	O
are	O
different	O
from	O
other	O
types	O
of	O
finite	O
automata	O
in	O
the	O
succinctness	O
of	O
description	O
,	O
measured	O
by	O
the	O
number	O
of	O
their	O
states	O
.	O
</s>
<s>
converts	O
an	O
AFA	O
with	O
states	O
to	O
a	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
NFA	O
)	O
with	O
up	O
to	O
states	O
by	O
performing	O
a	O
similar	O
kind	O
of	O
powerset	O
construction	O
as	O
used	O
for	O
the	O
transformation	O
of	O
an	O
NFA	O
to	O
a	O
DFA	B-General_Concept
.	O
</s>
