<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
graph	B-Application
transformation	I-Application
,	O
or	O
graph	B-Application
rewriting	I-Application
,	O
concerns	O
the	O
technique	O
of	O
creating	O
a	O
new	O
graph	O
out	O
of	O
an	O
original	O
graph	O
algorithmically	O
.	O
</s>
<s>
It	O
has	O
numerous	O
applications	O
,	O
ranging	O
from	O
software	B-General_Concept
engineering	I-General_Concept
(	O
software	B-General_Concept
construction	I-General_Concept
and	O
also	O
software	O
verification	O
)	O
to	O
layout	B-Algorithm
algorithms	I-Algorithm
and	O
picture	O
generation	O
.	O
</s>
<s>
Graph	B-Application
transformations	I-Application
can	O
be	O
used	O
as	O
a	O
computation	O
abstraction	O
.	O
</s>
<s>
The	O
basic	O
idea	O
is	O
that	O
if	O
the	O
state	O
of	O
a	O
computation	O
can	O
be	O
represented	O
as	O
a	O
graph	O
,	O
further	O
steps	O
in	O
that	O
computation	O
can	O
then	O
be	O
represented	O
as	O
transformation	B-Language
rules	O
on	O
that	O
graph	O
.	O
</s>
<s>
Formally	O
,	O
a	O
graph	B-Application
rewriting	I-Application
system	I-Application
usually	O
consists	O
of	O
a	O
set	O
of	O
graph	O
rewrite	O
rules	O
of	O
the	O
form	O
,	O
with	O
being	O
called	O
pattern	O
graph	O
(	O
or	O
left-hand	O
side	O
)	O
and	O
being	O
called	O
replacement	O
graph	O
(	O
or	O
right-hand	O
side	O
of	O
the	O
rule	O
)	O
.	O
</s>
<s>
A	O
graph	O
rewrite	O
rule	O
is	O
applied	O
to	O
the	O
host	O
graph	O
by	O
searching	O
for	O
an	O
occurrence	O
of	O
the	O
pattern	O
graph	O
(	O
pattern	B-Language
matching	I-Language
,	O
thus	O
solving	O
the	O
subgraph	O
isomorphism	O
problem	O
)	O
and	O
by	O
replacing	O
the	O
found	O
occurrence	O
by	O
an	O
instance	O
of	O
the	O
replacement	O
graph	O
.	O
</s>
<s>
Rewrite	O
rules	O
can	O
be	O
further	O
regulated	O
in	O
the	O
case	O
of	O
labeled	O
graphs	O
,	O
such	O
as	O
in	O
string-regulated	O
graph	B-Application
grammars	I-Application
.	O
</s>
<s>
Sometimes	O
graph	B-Application
grammar	I-Application
is	O
used	O
as	O
a	O
synonym	O
for	O
graph	B-Application
rewriting	I-Application
system	I-Application
,	O
especially	O
in	O
the	O
context	O
of	O
formal	O
languages	O
;	O
the	O
different	O
wording	O
is	O
used	O
to	O
emphasize	O
the	O
goal	O
of	O
constructions	O
,	O
like	O
the	O
enumeration	O
of	O
all	O
graphs	O
from	O
some	O
starting	O
graph	O
,	O
i.e.	O
</s>
<s>
The	O
algebraic	O
approach	O
to	O
graph	B-Application
rewriting	I-Application
is	O
based	O
upon	O
category	O
theory	O
.	O
</s>
<s>
From	O
the	O
perspective	O
of	O
the	O
DPO	O
approach	O
a	O
graph	B-Application
rewriting	I-Application
rule	O
is	O
a	O
pair	O
of	O
morphisms	O
in	O
the	O
category	O
of	O
graphs	O
and	O
graph	O
homomorphisms	O
between	O
them	O
:	O
,	O
also	O
written	O
,	O
where	O
is	O
injective	O
.	O
</s>
<s>
Another	O
graph	O
morphism	O
models	O
an	O
occurrence	O
of	O
L	O
in	O
G	O
and	O
is	O
called	O
a	O
match	B-Language
.	O
</s>
<s>
Practical	O
understanding	O
of	O
this	O
is	O
that	O
is	O
a	O
subgraph	O
that	O
is	O
matched	O
from	O
(	O
see	O
subgraph	O
isomorphism	O
problem	O
)	O
,	O
and	O
after	O
a	O
match	B-Language
is	O
found	O
,	O
is	O
replaced	O
with	O
in	O
host	O
graph	O
where	O
serves	O
as	O
an	O
interface	O
,	O
containing	O
the	O
nodes	O
and	O
edges	O
which	O
are	O
preserved	O
when	O
applying	O
the	O
rule	O
.	O
</s>
<s>
The	O
graph	O
is	O
needed	O
to	O
attach	O
the	O
pattern	O
being	O
matched	O
to	O
its	O
context	O
:	O
if	O
it	O
is	O
empty	O
,	O
the	O
match	B-Language
can	O
only	O
designate	O
a	O
whole	O
connected	O
component	O
of	O
the	O
graph	O
.	O
</s>
<s>
In	O
contrast	O
a	O
graph	B-Application
rewriting	I-Application
rule	O
of	O
the	O
SPO	O
approach	O
is	O
a	O
single	O
morphism	O
in	O
the	O
category	O
of	O
labeled	O
multigraphs	O
and	O
partial	O
mappings	O
that	O
preserve	O
the	O
multigraph	O
structure	O
:	O
.	O
</s>
<s>
The	O
DPO	O
approach	O
only	O
deletes	O
a	O
node	O
when	O
the	O
rule	O
specifies	O
the	O
deletion	O
of	O
all	O
adjacent	O
edges	O
as	O
well	O
(	O
this	O
dangling	O
condition	O
can	O
be	O
checked	O
for	O
a	O
given	O
match	B-Language
)	O
,	O
whereas	O
the	O
SPO	O
approach	O
simply	O
disposes	O
the	O
adjacent	O
edges	O
,	O
without	O
requiring	O
an	O
explicit	O
specification	O
.	O
</s>
<s>
There	O
is	O
also	O
another	O
algebraic-like	O
approach	O
to	O
graph	B-Application
rewriting	I-Application
,	O
based	O
mainly	O
on	O
Boolean	O
algebra	O
and	O
an	O
algebra	O
of	O
matrices	O
,	O
called	O
matrix	O
graph	B-Application
grammars	I-Application
.	O
</s>
<s>
Yet	O
another	O
approach	O
to	O
graph	B-Application
rewriting	I-Application
,	O
known	O
as	O
determinate	O
graph	B-Application
rewriting	I-Application
,	O
came	O
out	O
of	O
logic	O
and	O
database	B-General_Concept
theory	I-General_Concept
.	O
</s>
<s>
Another	O
approach	O
to	O
graph	B-Application
rewriting	I-Application
is	O
term	B-Application
graph	B-Application
rewriting	I-Application
,	O
which	O
involves	O
the	O
processing	O
or	O
transformation	B-Language
of	O
term	B-Application
graphs	I-Application
(	O
also	O
known	O
as	O
abstract	O
semantic	O
graphs	O
)	O
by	O
a	O
set	O
of	O
syntactic	O
rewrite	O
rules	O
.	O
</s>
<s>
Term	B-Application
graphs	I-Application
are	O
a	O
prominent	O
topic	O
in	O
programming	O
language	O
research	O
since	O
term	B-Application
graph	I-Application
rewriting	O
rules	O
are	O
capable	O
of	O
formally	O
expressing	O
a	O
compiler	O
's	O
operational	O
semantics	O
.	O
</s>
<s>
Term	B-Application
graphs	I-Application
are	O
also	O
used	O
as	O
abstract	O
machines	O
capable	O
of	O
modelling	O
chemical	O
and	O
biological	O
computations	O
as	O
well	O
as	O
graphical	O
calculi	O
such	O
as	O
concurrency	O
models	O
.	O
</s>
<s>
Term	B-Application
graphs	I-Application
can	O
perform	O
automated	O
verification	O
and	O
logical	O
programming	O
since	O
they	O
are	O
well-suited	O
to	O
representing	O
quantified	O
statements	O
in	O
first	O
order	O
logic	O
.	O
</s>
<s>
Symbolic	O
programming	O
software	O
is	O
another	O
application	O
for	O
term	B-Application
graphs	I-Application
,	O
which	O
are	O
capable	O
of	O
representing	O
and	O
performing	O
computation	O
with	O
abstract	O
algebraic	O
structures	O
such	O
as	O
groups	O
,	O
fields	O
and	O
rings	O
.	O
</s>
<s>
The	O
TERMGRAPH	O
conference	O
focuses	O
entirely	O
on	O
research	O
into	O
term	B-Application
graph	I-Application
rewriting	O
and	O
its	O
applications	O
.	O
</s>
<s>
Graph	B-Application
rewriting	I-Application
systems	I-Application
naturally	O
group	O
into	O
classes	O
according	O
to	O
the	O
kind	O
of	O
representation	O
of	O
graphs	O
that	O
are	O
used	O
and	O
how	O
the	O
rewrites	O
are	O
expressed	O
.	O
</s>
<s>
The	O
term	B-Application
graph	I-Application
grammar	O
,	O
otherwise	O
equivalent	O
to	O
graph	B-Application
rewriting	I-Application
system	I-Application
or	O
graph	O
replacement	O
system	O
,	O
is	O
most	O
often	O
used	O
in	O
classifications	O
.	O
</s>
<s>
Attributed	B-Application
graph	I-Application
grammars	I-Application
,	O
typically	O
formalised	O
using	O
either	O
the	O
single-pushout	B-Application
approach	I-Application
or	O
the	O
double-pushout	B-Application
approach	I-Application
to	O
characterising	O
replacements	O
,	O
mentioned	O
in	O
the	O
above	O
section	O
on	O
the	O
algebraic	O
approach	O
to	O
graph	B-Application
rewriting	I-Application
.	O
</s>
<s>
Hypergraph	B-Application
grammars	I-Application
,	O
including	O
as	O
more	O
restrictive	O
subclasses	O
port	O
graph	B-Application
grammars	I-Application
,	O
linear	B-Application
graph	I-Application
grammars	I-Application
and	O
interaction	O
nets	O
.	O
</s>
<s>
GP	O
2	O
is	O
a	O
programming	O
language	O
for	O
computing	O
on	O
graphs	O
by	O
the	O
directed	O
application	O
of	O
graph	B-Application
transformation	I-Application
rules	O
.	O
</s>
<s>
GMTE	O
,	O
the	O
Graph	B-General_Concept
Matching	I-General_Concept
and	O
Transformation	B-Language
Engine	O
for	O
graph	B-General_Concept
matching	I-General_Concept
and	O
transformation	B-Language
.	O
</s>
<s>
It	O
is	O
an	O
implementation	O
of	O
an	O
extension	O
of	O
Messmer	O
’s	O
algorithm	O
using	O
C++	B-Language
.	O
</s>
<s>
GROOVE	O
,	O
a	O
Java-based	O
tool	O
set	O
for	O
editing	O
graphs	O
and	O
graph	B-Application
transformation	I-Application
rules	O
,	O
exploring	O
the	O
state	O
spaces	O
of	O
graph	B-Application
grammars	I-Application
,	O
and	O
model	B-Application
checking	I-Application
those	O
state	O
spaces	O
;	O
can	O
also	O
be	O
used	O
as	O
a	O
graph	B-Application
transformation	I-Application
engine	O
.	O
</s>
<s>
Verigraph	O
,	O
a	O
software	O
specification	O
and	O
verification	O
system	O
based	O
on	O
graph	B-Application
rewriting	I-Application
(	O
Haskell	B-Language
)	O
.	O
</s>
<s>
Tools	O
that	O
solve	O
software	B-General_Concept
engineering	I-General_Concept
tasks	O
(	O
mainly	O
MDA	B-Architecture
)	O
with	O
graph	B-Application
rewriting	I-Application
:	O
</s>
<s>
GraphSynth	O
is	O
an	O
interpreter	O
and	O
UI	O
environment	O
for	O
creating	O
unrestricted	O
graph	B-Application
grammars	I-Application
as	O
well	O
as	O
testing	O
and	O
searching	O
the	O
resultant	O
language	O
variant	O
.	O
</s>
<s>
It	O
saves	O
graphs	O
and	O
graph	B-Application
grammar	I-Application
rules	O
as	O
XML	B-Protocol
files	O
and	O
is	O
written	O
in	O
C#	B-Application
.	O
</s>
<s>
Soley	O
Studio	O
,	O
is	O
an	O
integrated	B-Application
development	I-Application
environment	I-Application
for	O
graph	B-Application
transformation	I-Application
systems	O
.	O
</s>
<s>
OpenCog	B-Application
provides	O
a	O
basic	O
pattern	O
matcher	O
(	O
on	O
hypergraphs	O
)	O
which	O
is	O
used	O
to	O
implement	O
various	O
AI	O
algorithms	O
.	O
</s>
<s>
RelEx	O
is	O
an	O
English-language	O
parser	O
that	O
employs	O
graph	O
re-writing	O
to	O
convert	O
a	O
link	B-General_Concept
parse	I-General_Concept
into	O
a	O
dependency	O
parse	O
.	O
</s>
<s>
The	O
Clean	B-Operating_System
programming	I-Operating_System
language	I-Operating_System
is	O
implemented	O
using	O
graph	B-Application
rewriting	I-Application
.	O
</s>
<s>
Graph	B-Application
transformation	I-Application
in	O
a	O
nutshell	O
.	O
</s>
<s>
Electronic	O
Notes	O
in	O
Theoretical	O
Computer	B-General_Concept
Science	I-General_Concept
148	O
(	O
1	O
SPEC	O
.	O
</s>
