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In-Depth Information
Ta b l e 2 . 9
A string theory associated to a Chomsky grammar
G
Derive
(
S
)
Terminal
(
a
)
for every
a
∈
T
Rule
(
α
,
β
)
for every α
→
β
∈
R
∀
x
((
Terminal
(
x
)
∧
Terminal
(
y
))
→
Terminal
(
xy
))
∀
x
,
y
,
z
((
Derive
(
xyz
)
∧
Rule
(
y
,
w
))
→
Derive
(
xwz
))
∀
x
((
Derive
(
x
)
∧
Terminal
(
x
))
→
Generate
(
x
))
Cells of this organism are indicated by the symbol
F
, cells inside brackets are
“branches”, inclined (alternatively in opposite verses) with respect to a main grow-
ing axis. For
n
called a basal part
and an apical part. Every second cell in the basal part carries a non-branching fila-
ment. These filaments develop linearly in time, they add at each stage one new cell.
At stage 6 the lengths of these filaments are 3, 2, 1, respectively. The apical part
at stage 6 consists of four cells without any branches. In the following stages, the
apical part is a repeat of the apical part at the previous stage, together with two new
cells at the end of the apical part. The second of these new cells carries a branch,
identical to the whole organism six stages before.
>
5 we distinguish in
R
(
n
)
two parts
B
(
n
)
,
A
(
n
)
•
R
(
0
)=
F
•
R
(
1
)=
FF
•
R
(
2
)=
FFFF
•
R
(
3
)=
FF
[
F
]
FF
•
R
(
4
)=
FF
[
FF
]
FF
[
F
]
FF
•
R
(
5
)=
FF
[
FFF
]
FF
[
FF
]
FF
[
F
]
FFFF
•
R
(
x
+
6
)=
B
(
x
+
6
)
A
(
x
+
6
)
•
A
(
5
)=
FF
•
A
(
x
+
6
)=
FF
[
R
(
x
)]
A
(
x
+
5
)
•
B
(
x
+
6
)=
FF
[
L
(
x
+
3
)]
FF
[
L
(
x
+
2
)]
FF
[
L
(
x
+
1
)]
•
L
(
0
)=
λ
•
L
(
x
+
1
)=
FL
(
x
)
.
The following derivation is based on the equations defining Red Algae structure:
•
R
(
10
)=
B
(
10
)
A
(
10
)
•
A
(
10
)=
FF
[
R
(
4
)]
A
(
5
)=
FF
[
R
(
4
)]
FF
=
•
FF
[
FF
[
FF
]
FF
[
F
]]
FF
•
B
(
10
)=
FF
[
L
(
7
)]
FF
[
L
(
6
)]
FFL
(
5
)=
•
FF
[
FL
(
6
)]
FF
[
FL
(
5
)]
FFL
(
5
)=
F
7
F
6
FFF
5
•
FF
[
]
FF
[
]
F
7
F
6
FFF
5
.
•
R
(
10
)=
FF
[
FF
[
FF
]
FF
[
F
]]
FFFF
[
]
FF
[
]