Information Technology Reference
In-Depth Information
Q
1
×
Γ
1
Q
2
×
Γ
2
z
i
∈
Δ
∗
. Then it holds that z
i
h
i
=
for some
(
p
i
,
α
i
)
∈
,
(
p
i
,
β
i
)
∈
,
z
i
,
∈
Δ
±∗
,
=
,
,...,
hz
i
for some h
i
i
1
2
l.
(
,
α
)
=
ε
=
(b)
In case
p
is in
ε
-mode, let a
1
,l
1
,
z
1
===
⇒
ε
/
(
p
,
α
)
T
1
|
F
(
p
1
,
α
1
)
and
(
p
,
β
)=(
p
1
,
β
1
)
.
Then it holds that z
1
h
1
=
h.
(c)
In case
(
p
,
α
)
is not in
ε
-mode and
(
p
,
β
)
is in
ε
-mode, let a
1
=
ε
,l
=
1
,
z
1
===
⇒
ε
/
(
p
,
α
)=(
p
1
,
α
1
)
and
(
p
,
β
)
T
2
|
F
(
p
1
,
β
1
)
.
Then it holds that h
1
=
hz
1
.
(iii)
Concerning the above condition (ii), it holds that
(
p
i
,
α
i
)
≡
h
i
(
p
i
,
β
i
)
,
i
=
1
,
2
,...,
l.
Proof.
It follows from Proposition 1 (i).
The checking whether conditions (i) and (ii) in Lemma 3 hold or not is named
output branch checking
to the node labeled eq.(4) in question. When it is verified
to hold, the checking is said to be
successful
. Then we expand the above node to
have
l
sons in
unchecked
status labeled by (iii), with edges labeled
z
i
\
a
i
/
z
i
,
i
=
1
l
, and we turn the node in question to be
checked
. The step of developing
the comparison tree in this way is named
branching
to the node in question. If
condition (i) or (ii) does not hold, conclude that “
T
1
≡
,
2
,...,
T
2
”.
When the branching has been applied to the node in question, the number of sons
of it is at most
|
Σ
|
.
4.2
Stack Reduction
(
,
α
)
≡
(
,
β
)
(
,
α
)=
From Lemma 1, for any node labeled
p
h
p
,if
N
p
0, then there
α
∈
Γ
1
,
α
)=
,
α
)=
(
(
(
,
α
)
exists
such that
N
p
0 and TRANS
p
TRANS
p
(i.e.
,
α
)
≡
(
≤|
α
|≤|
β
∈
Γ
2
(
p
p
,
α
)
) with 1
Q
1
|
. Similarly, if
N
(
p
,
β
)=
0, there exists
,
β
)=
,
β
)=
,
β
)
≡
(
such that
N
(
p
0 and TRANS
(
p
TRANS
(
p
,
β
)
(i.e.
(
p
p
,
β
)
)
≤|
β
|≤|
with 1
Q
2
|
.
(
,
α
)
≡
(
,
β
)
(
,
α
)=
Definition 9.
Let
p
h
p
be a label of the node in question. If
N
p
0,
(
,
α
)=
=
α
let
α
0
be the
shortest
stack such that
N
p
0. Otherwise, let
α
. Moreover,
0
0
if
N
(
p
,
β
)=
0, let
β
0
be the
shortest
stack such that
N
(
p
,
β
0
)=
0. Otherwise, let
β
0
=
β
. If it holds that
|
α
0
| < |
α
|
or
|
β
0
| < |
β
|
, we say that the stack reduction is
applicable
to the node in question.
When stack reduction is applicable to the node in question, we expand it to have
only one son labeled by
(
p
,
α
0
)
≡
h
(
p
,
β
0
)
in
unchecked
status, with edge labeled
Σ
∗
, and then we turn the node in question to
be
checked
. The step of developing the comparison tree in this way is named
stack
reduction
to the node in question.
ε
\
λ
/
ε
,where
λ
is a new symbol not in
Search WWH ::
Custom Search