<s>
A	O
tree	B-Application
stack	I-Application
automaton	I-Application
(	O
plural	O
:	O
tree	O
stack	B-Application
automata	O
)	O
is	O
a	O
formalism	O
considered	O
in	O
automata	B-Application
theory	I-Application
.	O
</s>
<s>
It	O
is	O
a	O
finite	B-Architecture
state	I-Architecture
automaton	I-Architecture
with	O
the	O
additional	O
ability	O
to	O
manipulate	O
a	O
tree-shaped	O
stack	B-Application
.	O
</s>
<s>
It	O
is	O
an	O
automaton	O
with	O
storage	O
whose	O
storage	O
roughly	O
resembles	O
the	O
configurations	O
of	O
a	O
thread	B-Application
automaton	I-Application
.	O
</s>
<s>
A	O
restricted	O
class	O
of	O
tree	O
stack	B-Application
automata	O
recognises	O
exactly	O
the	O
languages	O
generated	O
by	O
multiple	O
context-free	O
grammars	O
(	O
or	O
linear	O
context-free	O
rewriting	O
systems	O
)	O
.	O
</s>
<s>
is	O
a	O
partial	B-Algorithm
function	I-Algorithm
from	O
strings	O
of	O
positive	O
integers	O
to	O
the	O
set	O
}	O
with	O
prefix-closed	O
domain	B-Algorithm
(	O
called	O
tree	O
)	O
,	O
</s>
<s>
is	O
an	O
element	O
of	O
the	O
domain	B-Algorithm
of	O
(	O
called	O
stack	B-Application
pointer	O
)	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
tree	O
stacks	B-Application
over	O
is	O
denoted	O
by	O
.	O
</s>
<s>
The	O
set	O
of	O
predicates	B-Algorithm
on	O
,	O
denoted	O
by	O
,	O
contains	O
the	O
following	O
unary	O
predicates	B-Algorithm
:	O
</s>
<s>
which	O
is	O
true	O
for	O
any	O
tree	O
stack	B-Application
over	O
,	O
</s>
<s>
which	O
is	O
true	O
for	O
some	O
tree	O
stack	B-Application
if	O
,	O
</s>
<s>
The	O
set	O
of	O
instructions	O
on	O
,	O
denoted	O
by	O
,	O
contains	O
the	O
following	O
partial	B-Algorithm
functions	I-Algorithm
:	O
</s>
<s>
which	O
adds	O
for	O
a	O
given	O
tree	O
stack	B-Application
a	O
pair	O
to	O
the	O
tree	O
and	O
sets	O
the	O
stack	B-Application
pointer	O
to	O
(	O
i.e.	O
</s>
<s>
it	O
pushes	O
to	O
the	O
-th	O
child	O
position	O
)	O
if	O
is	O
not	O
yet	O
in	O
the	O
domain	B-Algorithm
of	O
,	O
</s>
<s>
which	O
replaces	O
the	O
current	O
stack	B-Application
pointer	O
by	O
(	O
i.e.	O
</s>
<s>
it	O
moves	O
the	O
stack	B-Application
pointer	O
to	O
the	O
-th	O
child	O
position	O
)	O
if	O
is	O
in	O
the	O
domain	B-Algorithm
of	O
,	O
</s>
<s>
which	O
removes	O
the	O
last	O
symbol	O
from	O
the	O
stack	B-Application
pointer	O
(	O
i.e.	O
</s>
<s>
which	O
replaces	O
the	O
symbol	O
currently	O
under	O
the	O
stack	B-Application
pointer	O
by	O
,	O
</s>
<s>
,	O
,	O
and	O
are	O
finite	O
sets	O
(	O
whose	O
elements	O
are	O
called	O
states	O
,	O
stack	B-Application
symbols	O
,	O
and	O
input	O
symbols	O
,	O
respectively	O
)	O
,	O
</s>
<s>
Tree	O
stack	B-Application
automata	O
are	O
equivalent	O
to	O
Turing	B-Architecture
machines	I-Architecture
.	O
</s>
<s>
A	O
tree	B-Application
stack	I-Application
automaton	I-Application
is	O
called	O
-restricted	O
for	O
some	O
positive	O
natural	O
number	O
if	O
,	O
during	O
any	O
run	O
of	O
the	O
automaton	O
,	O
any	O
position	O
of	O
the	O
tree	O
stack	B-Application
is	O
accessed	O
at	O
most	O
times	O
from	O
below	O
.	O
</s>
<s>
1-restricted	O
tree	O
stack	B-Application
automata	O
are	O
equivalent	O
to	O
pushdown	B-Application
automata	I-Application
and	O
therefore	O
also	O
to	O
context-free	O
grammars	O
.	O
</s>
<s>
-restricted	O
tree	O
stack	B-Application
automata	O
are	O
equivalent	O
to	O
linear	O
context-free	O
rewriting	O
systems	O
and	O
multiple	O
context-free	O
grammars	O
of	O
fan-out	O
at	O
most	O
(	O
for	O
every	O
positive	O
integer	O
)	O
.	O
</s>
