<s>
In	O
mathematics	O
and	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
probabilistic	B-General_Concept
automaton	I-General_Concept
(	O
PA	O
)	O
is	O
a	O
generalization	O
of	O
the	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
;	O
it	O
includes	O
the	O
probability	O
of	O
a	O
given	O
transition	O
into	O
the	O
transition	B-Architecture
function	I-Architecture
,	O
turning	O
it	O
into	O
a	O
transition	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
the	O
probabilistic	B-General_Concept
automaton	I-General_Concept
also	O
generalizes	O
the	O
concepts	O
of	O
a	O
Markov	O
chain	O
and	O
of	O
a	O
subshift	O
of	O
finite	O
type	O
.	O
</s>
<s>
The	O
languages	O
recognized	O
by	O
probabilistic	B-General_Concept
automata	I-General_Concept
are	O
called	O
stochastic	B-General_Concept
languages	I-General_Concept
;	O
these	O
include	O
the	O
regular	B-General_Concept
languages	I-General_Concept
as	O
a	O
subset	O
.	O
</s>
<s>
The	O
number	O
of	O
stochastic	B-General_Concept
languages	I-General_Concept
is	O
uncountable	O
.	O
</s>
<s>
In	O
recent	O
years	O
,	O
a	O
variant	O
has	O
been	O
formulated	O
in	O
terms	O
of	O
quantum	O
probabilities	O
,	O
the	O
quantum	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
For	O
a	O
given	O
initial	O
state	O
and	O
input	O
character	O
,	O
a	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
DFA	B-General_Concept
)	O
has	O
exactly	O
one	O
next	O
state	O
,	O
and	O
a	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
NFA	O
)	O
has	O
a	O
set	O
of	O
next	O
states	O
.	O
</s>
<s>
A	O
probabilistic	B-General_Concept
automaton	I-General_Concept
(	O
PA	O
)	O
instead	O
has	O
a	O
weighted	O
set	O
(	O
or	O
vector	O
)	O
of	O
next	O
states	O
,	O
where	O
the	O
weights	O
must	O
sum	O
to	O
1	O
and	O
therefore	O
can	O
be	O
interpreted	O
as	O
probabilities	O
(	O
making	O
it	O
a	O
stochastic	O
vector	O
)	O
.	O
</s>
<s>
However	O
,	O
this	O
is	O
somewhat	O
misleading	O
,	O
as	O
the	O
PA	O
utilizes	O
the	O
notion	O
of	O
the	O
real	O
numbers	O
to	O
define	O
the	O
weights	O
,	O
which	O
is	O
absent	O
in	O
the	O
definition	O
of	O
both	O
DFAs	B-General_Concept
and	O
NFAs	O
.	O
</s>
<s>
This	O
additional	O
freedom	O
enables	O
them	O
to	O
decide	O
languages	O
that	O
are	O
not	O
regular	O
,	O
such	O
as	O
the	O
p-adic	B-General_Concept
languages	I-General_Concept
with	O
irrational	O
parameters	O
.	O
</s>
<s>
As	O
such	O
,	O
PAs	O
are	O
more	O
powerful	O
than	O
both	O
DFAs	B-General_Concept
and	O
NFAs	O
(	O
which	O
are	O
famously	O
equally	O
powerful	O
)	O
.	O
</s>
<s>
The	O
probabilistic	B-General_Concept
automaton	I-General_Concept
may	O
be	O
defined	O
as	O
an	O
extension	O
of	O
a	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
,	O
together	O
with	O
two	O
probabilities	O
:	O
the	O
probability	O
of	O
a	O
particular	O
state	O
transition	O
taking	O
place	O
,	O
and	O
with	O
the	O
initial	O
state	O
replaced	O
by	O
a	O
stochastic	O
vector	O
giving	O
the	O
probability	O
of	O
the	O
automaton	O
being	O
in	O
a	O
given	O
initial	O
state	O
.	O
</s>
<s>
Such	O
a	O
transition	B-Algorithm
matrix	I-Algorithm
is	O
always	O
defined	O
for	O
a	O
non-deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
The	O
initial	O
state	O
of	O
a	O
probabilistic	B-General_Concept
automaton	I-General_Concept
is	O
given	O
by	O
a	O
row	O
vector	O
,	O
whose	O
components	O
are	O
the	O
probabilities	O
of	O
the	O
individual	O
initial	O
states	O
,	O
that	O
add	O
to	O
1	O
:	O
</s>
<s>
In	O
particular	O
,	O
the	O
state	O
of	O
a	O
probabilistic	B-General_Concept
automaton	I-General_Concept
is	O
always	O
a	O
stochastic	O
vector	O
,	O
since	O
the	O
product	O
of	O
any	O
two	O
stochastic	B-Algorithm
matrices	I-Algorithm
is	O
a	O
stochastic	B-Algorithm
matrix	I-Algorithm
,	O
and	O
the	O
product	O
of	O
a	O
stochastic	O
vector	O
and	O
a	O
stochastic	B-Algorithm
matrix	I-Algorithm
is	O
again	O
a	O
stochastic	O
vector	O
.	O
</s>
<s>
Formally	O
,	O
the	O
definition	O
of	O
a	O
probabilistic	B-General_Concept
automaton	I-General_Concept
does	O
not	O
require	O
the	O
mechanics	O
of	O
the	O
non-deterministic	O
automaton	O
,	O
which	O
may	O
be	O
dispensed	O
with	O
.	O
</s>
<s>
Formally	O
,	O
a	O
probabilistic	B-General_Concept
automaton	I-General_Concept
PA	O
is	O
defined	O
as	O
the	O
tuple	O
.	O
</s>
<s>
The	O
set	O
of	O
languages	O
recognized	O
by	O
probabilistic	B-General_Concept
automata	I-General_Concept
are	O
called	O
stochastic	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
They	O
include	O
the	O
regular	B-General_Concept
languages	I-General_Concept
as	O
a	O
subset	O
.	O
</s>
<s>
By	O
abuse	O
of	O
notation	O
,	O
can	O
also	O
be	O
understood	O
to	O
be	O
the	O
column	O
vector	O
that	O
is	O
the	O
membership	B-General_Concept
function	I-General_Concept
for	O
;	O
that	O
is	O
,	O
it	O
has	O
a	O
1	O
at	O
the	O
places	O
corresponding	O
to	O
elements	O
in	O
,	O
and	O
a	O
zero	O
otherwise	O
.	O
</s>
<s>
Every	O
regular	B-General_Concept
language	I-General_Concept
is	O
stochastic	O
,	O
and	O
more	O
strongly	O
,	O
every	O
regular	B-General_Concept
language	I-General_Concept
is	O
η-stochastic	O
.	O
</s>
<s>
A	O
weak	O
converse	O
is	O
that	O
every	O
0-stochastic	O
language	O
is	O
regular	O
;	O
however	O
,	O
the	O
general	O
converse	O
does	O
not	O
hold	O
:	O
there	O
are	O
stochastic	B-General_Concept
languages	I-General_Concept
that	O
are	O
not	O
regular	O
.	O
</s>
<s>
Every	O
stochastic	B-General_Concept
language	I-General_Concept
is	O
representable	O
by	O
a	O
Rabin	O
automaton	O
.	O
</s>
<s>
If	O
is	O
an	O
isolated	O
cut-point	O
,	O
then	O
is	O
a	O
regular	B-General_Concept
language	I-General_Concept
.	O
</s>
<s>
The	O
p-adic	B-General_Concept
languages	I-General_Concept
provide	O
an	O
example	O
of	O
a	O
stochastic	B-General_Concept
language	I-General_Concept
that	O
is	O
not	O
regular	O
,	O
and	O
also	O
show	O
that	O
the	O
number	O
of	O
stochastic	B-General_Concept
languages	I-General_Concept
is	O
uncountable	O
.	O
</s>
<s>
That	O
is	O
,	O
a	O
p-adic	B-General_Concept
language	I-General_Concept
is	O
merely	O
the	O
set	O
of	O
real	O
numbers	O
in	O
[	O
0	O
,	O
1 ]	O
,	O
written	O
in	O
base-p	O
,	O
such	O
that	O
they	O
are	O
greater	O
than	O
.	O
</s>
<s>
It	O
is	O
straightforward	O
to	O
show	O
that	O
all	O
p-adic	B-General_Concept
languages	I-General_Concept
are	O
stochastic	O
.	O
</s>
<s>
In	O
particular	O
,	O
this	O
implies	O
that	O
the	O
number	O
of	O
stochastic	B-General_Concept
languages	I-General_Concept
is	O
uncountable	O
.	O
</s>
<s>
A	O
p-adic	B-General_Concept
language	I-General_Concept
is	O
regular	O
if	O
and	O
only	O
if	O
is	O
rational	O
.	O
</s>
<s>
The	O
probabilistic	B-General_Concept
automaton	I-General_Concept
has	O
a	O
geometric	O
interpretation	O
:	O
the	O
state	O
vector	O
can	O
be	O
understood	O
to	O
be	O
a	O
point	O
that	O
lives	O
on	O
the	O
face	O
of	O
the	O
standard	O
simplex	O
,	O
opposite	O
to	O
the	O
orthogonal	O
corner	O
.	O
</s>
<s>
When	O
the	O
cut-point	O
is	O
suitably	O
generalized	O
,	O
one	O
has	O
a	O
topological	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
An	O
example	O
of	O
such	O
a	O
generalization	O
is	O
the	O
quantum	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
;	O
here	O
,	O
the	O
automaton	O
state	O
is	O
represented	O
by	O
a	O
point	O
in	O
complex	O
projective	O
space	O
,	O
while	O
the	O
transition	O
matrices	O
are	O
a	O
fixed	O
set	O
chosen	O
from	O
the	O
unitary	O
group	O
.	O
</s>
