<s>
In	O
automata	B-Application
theory	I-Application
,	O
a	O
nested	B-Application
stack	I-Application
automaton	I-Application
is	O
a	O
finite	B-Architecture
automaton	I-Architecture
that	O
can	O
make	O
use	O
of	O
a	O
stack	B-Application
containing	O
data	O
which	O
can	O
be	O
additional	O
stacks	B-Application
.	O
</s>
<s>
Like	O
a	O
stack	B-Application
automaton	O
,	O
a	O
nested	B-Application
stack	I-Application
automaton	I-Application
may	O
step	O
up	O
or	O
down	O
in	O
the	O
stack	B-Application
,	O
and	O
read	O
the	O
current	O
symbol	O
;	O
in	O
addition	O
,	O
it	O
may	O
at	O
any	O
place	O
create	O
a	O
new	O
stack	B-Application
,	O
operate	O
on	O
that	O
one	O
,	O
eventually	O
destroy	O
it	O
,	O
and	O
continue	O
operating	O
on	O
the	O
old	O
stack	B-Application
.	O
</s>
<s>
This	O
way	O
,	O
stacks	B-Application
can	O
be	O
nested	O
recursively	O
to	O
an	O
arbitrary	O
depth	O
;	O
however	O
,	O
the	O
automaton	O
always	O
operates	O
on	O
the	O
innermost	O
stack	B-Application
only	O
.	O
</s>
<s>
A	O
nested	B-Application
stack	I-Application
automaton	I-Application
is	O
capable	O
of	O
recognizing	O
an	O
indexed	O
language	O
,	O
and	O
in	O
fact	O
the	O
class	O
of	O
indexed	O
languages	O
is	O
exactly	O
the	O
class	O
of	O
languages	O
accepted	O
by	O
one-way	O
nondeterministic	O
nested	B-Application
stack	I-Application
automata	I-Application
.	O
</s>
<s>
Nested	B-Application
stack	I-Application
automata	I-Application
should	O
not	O
be	O
confused	O
with	O
embedded	B-Application
pushdown	I-Application
automata	I-Application
,	O
which	O
have	O
less	O
computational	O
power	O
.	O
</s>
<s>
Q	O
,	O
Σ	O
,	O
and	O
Γ	O
is	O
a	O
nonempty	O
finite	O
set	O
of	O
states	O
,	O
input	O
symbols	O
,	O
and	O
stack	B-Application
symbols	O
,	O
respectively	O
,	O
</s>
<s>
[	O
is	O
used	O
as	O
left	O
endmarker	O
for	O
both	O
the	O
input	O
string	O
and	O
a	O
(	O
sub	O
)	O
stack	B-Application
string	O
,	O
</s>
<s>
]	O
is	O
used	O
as	O
the	O
final	O
endmarker	O
of	O
the	O
string	O
denoting	O
the	O
whole	O
stack	B-Application
.	O
</s>
<s>
An	O
extended	O
input	O
alphabet	O
is	O
defined	O
by	O
Σ	O
 '	O
=	O
Σ	O
∪	O
{[,]},	O
an	O
extended	O
stack	B-Application
alphabet	O
by	O
Γ	O
 '	O
=	O
Γ	O
∪	O
 {   ]  } 	O
,	O
and	O
the	O
set	O
of	O
input	O
move	O
directions	O
by	O
D	O
=	O
 { -1 , 0 , +1 } 	O
.	O
</s>
<s>
Q	O
×	O
Σ	O
 '	O
×	O
[ Γ	O
into	O
subsets	O
of	O
Q	O
×	O
D	O
×	O
[ Γ*	O
(	O
pushdown	B-Application
mode	O
)	O
,	O
Q	O
×	O
Σ	O
 '	O
×	O
Γ	O
 '	O
into	O
subsets	O
of	O
Q	O
×	O
D	O
×	O
D	O
(	O
reading	O
mode	O
)	O
,	O
Q	O
×	O
Σ	O
 '	O
×	O
[ Γ	O
 '	O
into	O
subsets	O
of	O
Q	O
×	O
D	O
×	O
 { +1 } 	O
(	O
reading	O
mode	O
)	O
,	O
Q	O
×	O
Σ	O
 '	O
×	O
 {   ]  } 	O
into	O
subsets	O
of	O
Q	O
×	O
D	O
×	O
 { -1 } 	O
(	O
reading	O
mode	O
)	O
,	O
Q	O
×	O
Σ	O
 '	O
×	O
(Γ	O
 '	O
∪	O
[ Γ	O
 '	O
)	O
into	O
subsets	O
of	O
Q	O
×	O
D	O
×	O
 [ Γ* ] 	O
(	O
stack	B-Application
creation	O
mode	O
)	O
,	O
and	O
Q	O
×	O
Σ	O
 '	O
×	O
 {  [  ]  } 	O
into	O
subsets	O
of	O
Q	O
×	O
D	O
×	O
{ε},	O
(	O
stack	B-Application
destruction	O
mode	O
)	O
,	O
</s>
<s>
the	O
current	O
stack	B-Application
symbol	O
,	O
</s>
<s>
the	O
direction	O
in	O
which	O
to	O
move	O
on	O
the	O
stack	B-Application
,	O
or	O
the	O
string	O
of	O
symbols	O
to	O
replace	O
the	O
topmost	O
stack	B-Application
symbol	O
.	O
</s>
<s>
Z0	O
∈	O
Γ	O
is	O
the	O
initial	O
stack	B-Application
symbol	O
,	O
</s>
<s>
Z1X2	O
...	O
Xj	O
Xm	O
...	O
is	O
the	O
stack	B-Application
,	O
including	O
substacks	O
;	O
for	O
convenience	O
,	O
X1	O
=	O
[	O
Z1	O
and	O
Xm	O
=	O
]	O
is	O
defined	O
.	O
</s>
<s>
The	O
current	O
position	O
in	O
the	O
stack	B-Application
,	O
viz	O
.	O
</s>
<s>
When	O
automata	O
are	O
allowed	O
to	O
re-read	O
their	O
input	O
(	O
"	O
two-way	B-General_Concept
automata	I-General_Concept
"	O
)	O
,	O
nested	O
stacks	B-Application
do	O
not	O
result	O
in	O
additional	O
language	O
recognition	O
capabilities	O
,	O
compared	O
to	O
plain	O
stacks	B-Application
.	O
</s>
<s>
Gilman	O
and	O
Shapiro	O
used	O
nested	B-Application
stack	I-Application
automata	I-Application
to	O
solve	O
the	O
word	O
problem	O
in	O
certain	O
groups	O
.	O
</s>
