<s>
An	O
L-system	B-Language
or	O
Lindenmayer	B-Language
system	I-Language
is	O
a	O
parallel	O
rewriting	O
system	O
and	O
a	O
type	O
of	O
formal	O
grammar	O
.	O
</s>
<s>
An	O
L-system	B-Language
consists	O
of	O
an	O
alphabet	B-Application
of	O
symbols	O
that	O
can	O
be	O
used	O
to	O
make	O
strings	O
,	O
a	O
collection	O
of	O
production	O
rules	O
that	O
expand	O
each	O
symbol	O
into	O
some	O
larger	O
string	O
of	O
symbols	O
,	O
an	O
initial	O
"	O
axiom	B-Algorithm
"	O
string	O
from	O
which	O
to	O
begin	O
construction	O
,	O
and	O
a	O
mechanism	O
for	O
translating	O
the	O
generated	O
strings	O
into	O
geometric	O
structures	O
.	O
</s>
<s>
L-systems	B-Language
were	O
introduced	O
and	O
developed	O
in	O
1968	O
by	O
Aristid	O
Lindenmayer	O
,	O
a	O
Hungarian	O
theoretical	O
biologist	O
and	O
botanist	O
at	O
the	O
University	O
of	O
Utrecht	O
.	O
</s>
<s>
Lindenmayer	O
used	O
L-systems	B-Language
to	O
describe	O
the	O
behaviour	O
of	O
plant	O
cells	O
and	O
to	O
model	O
the	O
growth	O
processes	O
of	O
plant	O
development	O
.	O
</s>
<s>
L-systems	B-Language
have	O
also	O
been	O
used	O
to	O
model	O
the	O
morphology	O
of	O
a	O
variety	O
of	O
organisms	O
and	O
can	O
be	O
used	O
to	O
generate	O
self-similar	O
fractals	B-Application
.	O
</s>
<s>
Originally	O
,	O
the	O
L-systems	B-Language
were	O
devised	O
to	O
provide	O
a	O
formal	O
description	O
of	O
the	O
development	O
of	O
such	O
simple	O
multicellular	O
organisms	O
,	O
and	O
to	O
illustrate	O
the	O
neighbourhood	O
relationships	O
between	O
plant	O
cells	O
.	O
</s>
<s>
The	O
recursive	O
nature	O
of	O
the	O
L-system	B-Language
rules	O
leads	O
to	O
self-similarity	O
and	O
thereby	O
,	O
fractal-like	O
forms	O
are	O
easy	O
to	O
describe	O
with	O
an	O
L-system	B-Language
.	O
</s>
<s>
Lindenmayer	B-Language
systems	I-Language
are	O
also	O
popular	O
in	O
the	O
generation	O
of	O
artificial	O
life	O
.	O
</s>
<s>
L-system	B-Language
grammars	O
are	O
very	O
similar	O
to	O
the	O
semi-Thue	O
grammar	O
(	O
see	O
Chomsky	O
hierarchy	O
)	O
.	O
</s>
<s>
The	O
rules	O
of	O
the	O
L-system	B-Language
grammar	O
are	O
applied	O
iteratively	O
starting	O
from	O
the	O
initial	O
state	O
.	O
</s>
<s>
The	O
fact	O
that	O
each	O
iteration	O
employs	O
as	O
many	O
rules	O
as	O
possible	O
differentiates	O
an	O
L-system	B-Language
from	O
a	O
formal	O
language	O
generated	O
by	O
a	O
formal	O
grammar	O
,	O
which	O
applies	O
only	O
one	O
rule	O
per	O
iteration	O
.	O
</s>
<s>
There	O
are	O
some	O
strings	O
in	O
some	O
languages	O
,	O
however	O
,	O
that	O
cannot	O
be	O
generated	O
if	O
the	O
grammar	O
is	O
treated	O
as	O
an	O
L-system	B-Language
rather	O
than	O
a	O
language	O
specification	O
.	O
</s>
<s>
However	O
,	O
if	O
all	O
applicable	O
rules	O
are	O
applied	O
at	O
every	O
step	O
,	O
as	O
in	O
an	O
L-system	B-Language
,	O
then	O
we	O
cannot	O
get	O
this	O
sentential	O
form	O
.	O
</s>
<s>
Thus	O
,	O
the	O
set	O
of	O
strings	O
produced	O
by	O
an	O
L-systems	B-Language
from	O
a	O
given	O
grammar	O
is	O
a	O
subset	O
of	O
the	O
formal	O
language	O
defined	O
by	O
the	O
grammar	O
,	O
and	O
if	O
we	O
take	O
a	O
language	O
to	O
be	O
defined	O
as	O
a	O
set	O
of	O
strings	O
,	O
this	O
means	O
that	O
a	O
given	O
L-system	B-Language
is	O
effectively	O
a	O
subset	O
of	O
the	O
formal	O
language	O
defined	O
by	O
the	O
L-system	B-Language
'	O
s	O
grammar	O
.	O
</s>
<s>
An	O
L-system	B-Language
is	O
context-free	O
if	O
each	O
production	O
rule	O
refers	O
only	O
to	O
an	O
individual	O
symbol	O
and	O
not	O
to	O
its	O
neighbours	O
.	O
</s>
<s>
Context-free	O
L-systems	B-Language
are	O
thus	O
specified	O
by	O
a	O
context-free	O
grammar	O
.	O
</s>
<s>
If	O
a	O
rule	O
depends	O
not	O
only	O
on	O
a	O
single	O
symbol	O
but	O
also	O
on	O
its	O
neighbours	O
,	O
it	O
is	O
termed	O
a	O
context-sensitive	O
L-system	B-Language
.	O
</s>
<s>
If	O
there	O
is	O
exactly	O
one	O
production	O
for	O
each	O
symbol	O
,	O
then	O
the	O
L-system	B-Language
is	O
said	O
to	O
be	O
deterministic	O
(	O
a	O
deterministic	O
context-free	O
L-system	B-Language
is	O
popularly	O
called	O
a	O
D0L	O
system	O
)	O
.	O
</s>
<s>
If	O
there	O
are	O
several	O
,	O
and	O
each	O
is	O
chosen	O
with	O
a	O
certain	O
probability	O
during	O
each	O
iteration	O
,	O
then	O
it	O
is	O
a	O
stochastic	O
L-system	B-Language
.	O
</s>
<s>
Using	O
L-systems	B-Language
for	O
generating	O
graphical	O
images	O
requires	O
that	O
the	O
symbols	O
in	O
the	O
model	O
refer	O
to	O
elements	O
of	O
a	O
drawing	O
on	O
the	O
computer	O
screen	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
program	O
Fractint	B-Application
uses	O
turtle	B-Application
graphics	I-Application
(	O
similar	O
to	O
those	O
in	O
the	O
Logo	B-Device
programming	I-Device
language	I-Device
)	O
to	O
produce	O
screen	O
images	O
.	O
</s>
<s>
It	O
interprets	O
each	O
constant	O
in	O
an	O
L-system	B-Language
model	O
as	O
a	O
turtle	O
command	O
.	O
</s>
<s>
Lindenmayer	O
's	O
original	O
L-system	B-Language
for	O
modelling	O
the	O
growth	O
of	O
algae	O
.	O
</s>
<s>
The	O
result	O
is	O
the	O
sequence	O
of	O
Fibonacci	B-Algorithm
words	I-Algorithm
.	O
</s>
<s>
If	O
we	O
count	O
the	O
length	O
of	O
each	O
string	O
,	O
we	O
obtain	O
the	O
famous	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
of	O
numbers	O
(	O
skipping	O
the	O
first	O
1	O
,	O
due	O
to	O
our	O
choice	O
of	O
axiom	B-Algorithm
)	O
:	O
</s>
<s>
If	O
we	O
would	O
like	O
to	O
not	O
skip	O
the	O
first	O
1	O
,	O
we	O
can	O
use	O
axiom	B-Algorithm
B	O
.	O
</s>
<s>
The	O
shape	O
is	O
built	O
by	O
recursively	O
feeding	O
the	O
axiom	B-Algorithm
through	O
the	O
production	O
rules	O
.	O
</s>
<s>
Applying	O
this	O
to	O
the	O
axiom	B-Algorithm
of	O
'	O
0	O
 '	O
,	O
we	O
get	O
:	O
</s>
<s>
axiom	B-Algorithm
:	O
0	O
1st	O
recursion	O
:	O
1[0]0	O
2nd	O
recursion	O
:	O
11[1[0]0]1[0]0	O
3rd	O
recursion	O
:	O
1111[11[1[0]0]1[0]0]11[1[0]0]1[0]0	O
…	O
</s>
<s>
This	O
string	O
can	O
be	O
drawn	O
as	O
an	O
image	O
by	O
using	O
turtle	B-Application
graphics	I-Application
,	O
where	O
each	O
symbol	O
is	O
assigned	O
a	O
graphical	O
operation	O
for	O
the	O
turtle	O
to	O
perform	O
.	O
</s>
<s>
The	O
push	B-Application
and	I-Application
pop	I-Application
refer	O
to	O
a	O
LIFO	B-Application
stack	B-Application
(	O
more	O
technical	O
grammar	O
would	O
have	O
separate	O
symbols	O
for	O
"	O
push	O
position	O
"	O
and	O
"	O
turn	O
left	O
"	O
)	O
.	O
</s>
<s>
This	O
produces	O
the	O
famous	O
Cantor	O
's	O
fractal	B-Application
set	O
on	O
a	O
real	O
straight	O
line	O
R	O
.	O
</s>
<s>
A	O
variant	O
of	O
the	O
Koch	B-Algorithm
curve	I-Algorithm
which	O
uses	O
only	O
right	O
angles	O
.	O
</s>
<s>
Here	O
,	O
F	O
means	O
"	O
draw	O
forward	O
"	O
,	O
+	O
means	O
"	O
turn	O
left	O
90°	O
"	O
,	O
and	O
means	O
"	O
turn	O
right	O
90°	O
"	O
(	O
see	O
turtle	B-Application
graphics	I-Application
)	O
.	O
</s>
<s>
The	O
Sierpinski	O
triangle	O
drawn	O
using	O
an	O
L-system	B-Language
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
approximate	O
the	O
Sierpinski	O
triangle	O
using	O
a	O
Sierpiński	O
arrowhead	O
curve	O
L-system	B-Language
.	O
</s>
<s>
Here	O
,	O
A	O
and	O
B	O
both	O
mean	O
"	O
draw	O
forward	O
"	O
,	O
+	O
means	O
"	O
turn	O
left	O
by	O
angle	O
"	O
,	O
and	O
means	O
"	O
turn	O
right	O
by	O
angle	O
"	O
(	O
see	O
turtle	B-Application
graphics	I-Application
)	O
.	O
</s>
<s>
The	O
dragon	B-Algorithm
curve	I-Algorithm
drawn	O
using	O
an	O
L-system	B-Language
.	O
</s>
<s>
A	O
number	O
of	O
elaborations	O
on	O
this	O
basic	O
L-system	B-Language
technique	O
have	O
been	O
developed	O
which	O
can	O
be	O
used	O
in	O
conjunction	O
with	O
each	O
other	O
.	O
</s>
<s>
The	O
grammar	O
model	O
we	O
have	O
discussed	O
thus	O
far	O
has	O
been	O
deterministic	O
—	O
that	O
is	O
,	O
given	O
any	O
symbol	O
in	O
the	O
grammar	O
's	O
alphabet	B-Application
,	O
there	O
has	O
been	O
exactly	O
one	O
production	O
rule	O
,	O
which	O
is	O
always	O
chosen	O
,	O
and	O
always	O
performs	O
the	O
same	O
conversion	O
.	O
</s>
<s>
In	O
a	O
parametric	O
grammar	O
,	O
each	O
symbol	O
in	O
the	O
alphabet	B-Application
has	O
a	O
parameter	O
list	O
associated	O
with	O
it	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
string	O
rewriting	O
process	O
in	O
the	O
Example	O
2	O
(	O
Fractal	B-Application
tree	O
)	O
is	O
independent	O
on	O
how	O
graphical	O
operations	O
are	O
assigned	O
to	O
the	O
symbols	O
.	O
</s>
<s>
There	O
are	O
many	O
open	O
problems	O
involving	O
studies	O
of	O
L-systems	B-Language
.	O
</s>
<s>
Characterisation	O
of	O
all	O
the	O
deterministic	O
context-free	O
L-systems	B-Language
which	O
are	O
locally	O
catenative	O
.	O
</s>
<s>
Given	O
a	O
structure	O
,	O
find	O
an	O
L-system	B-Language
that	O
can	O
produce	O
that	O
structure	O
.	O
</s>
<s>
L-systems	B-Language
on	O
the	O
real	O
line	O
R	O
:	O
</s>
<s>
Well-known	O
L-systems	B-Language
on	O
a	O
plane	O
R2	O
are	O
:	O
</s>
<s>
space-filling	B-Algorithm
curves	I-Algorithm
(	O
Hilbert	B-Algorithm
curve	I-Algorithm
,	O
Peano	B-Algorithm
's	I-Algorithm
curves	I-Algorithm
,	O
Dekking	O
's	O
church	O
,	O
kolams	O
)	O
,	O
</s>
<s>
median	O
space-filling	B-Algorithm
curves	I-Algorithm
(	O
Lévy	B-Algorithm
C	I-Algorithm
curve	I-Algorithm
,	O
Harter-Heighway	B-Algorithm
dragon	I-Algorithm
curve	I-Algorithm
,	O
Davis-Knuth	O
terdragon	O
)	O
,	O
</s>
