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ε
/
z
−→
(
,
z
∈
Δ
∗
,
r
γ
∈
Γ
∗
.Sucharuleas
for any
a
∈
Σ
∈
Q
,
(
p
,
A
)
q
,
θ
)
is called an
ε
-
rule
.
The
deterministic pushdown automaton
(
dpda
for short)
M
=(
Q
,
Γ
,
Σ
,
δ
,
q
0
,
Z
0
,
F
)
with
,
θ
)
(
z
−→
(
a
/
a
−→
(
∈
Δ
∗
}
δ
=
{
(
p
,
A
)
q
p
,
A
)
q
,
θ
)
∈
μ
,
z
is called the
associated
dpda for the above dpdt
T
, and is just as in Ref. [5], Defini-
tion 2.1, pp.190-191. A dpdt or a dpda is said to be
real-time
if it has no
-rules.
Definition 2.
A dpda
M
(respectively, a dpdt
T
)issaidtobea
deterministic re-
stricted one-counter automaton
(
transducer
,resp.)(
droca
(
droct
, resp.) for short) if
Γ
=
{
ε
. When the droca
M
is the associated dpda for the droct
T
,
M
is called the
associated droca
for
T
.
Definition 3.
A
configuration
Z
0
}
(
p
,
α
)
of the dpdt
T
, with its associated dpda
M
,isan
×
Γ
∗
,wherethe
leftmost
symbol of
element of
Q
α
is the
top
symbol on the stack.
In particular,
(
q
0
,
Z
0
)
is called the
initial configuration
.
α
∈
Γ
+
and
A configuration
(
p
,
α
)
is said to be in
reading mode
if
α
=
A
z
−→
(
a
/
∈
Δ
∗
and
×
Γ
∗
, while it is said
(
p
,
A
)
q
,
θ
)
∈
μ
for some
a
∈
Σ
,
z
(
q
,
θ
)
∈
Q
ε
/
z
−→
(
α
∈
Γ
+
and
∈
Δ
∗
and
to be in
ε
-
mode
if
α
=
A
(
p
,
A
)
q
,
θ
)
∈
μ
for some
z
×
Γ
∗
.
The
height
of a configuration
(
q
,
θ
)
∈
Q
(
p
,
α
)
is
|
α
|
. Here, for a string
α
,
|
α
|
denotes the
length of
denotes the cardinality of
S
.
Definition 4.
The dpdt
T
, with its associated dpda
M
, makes a
move
α
. Moreover, for a set
S
,
|
S
|
z
−−−→
T
a
/
(
p
,
A
ω
)
(
q
,
θω
)
z
−→
(
a
/
ω
∈
Γ
∗
iff
for any
μ
contains a rule
(
p
,
A
)
q
,
θ
)
with
a
∈
Σ
∪{
ε
}
. A sequence
of such moves through successive configurations as
a
i
/
z
i
−−−→
T
p
i
,
α
i
α
)
p
i
+
1
,
α
i
+
1
α
)
,
(
(
1
≤
i
≤
m
,
is called a
derivation
, and is written as
,
α
1
α
)
,
α
m
+
1
α
)
,
y
===
⇒
T
x
/
(
m
)
(
p
1
(
p
m
+
1
where
x
=
a
1
a
2
···
a
m
and
y
=
z
1
z
2
···
z
m
,orsimply
x
/
y
===
⇒
T
p
1
,
α
1
α
)
p
m
+
1
,
α
m
+
1
α
)
,
(
(
p
1
,
α
1
α
)
p
m
+
1
,
α
m
+
1
α
)
x
===
⇒
M
(
m
)
if
(
(
as in Ref. [5], Definition 2.3, pp.191-
ε
/
ε
===
⇒
T
×
Γ
∗
.
192. By convention, we let
(
p
,
α
)
(
p
,
α
)
for any
(
p
,
α
)
∈
Q
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