<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
double	B-Application
pushout	I-Application
graph	I-Application
rewriting	I-Application
(	O
or	O
DPO	O
graph	B-Application
rewriting	I-Application
)	O
refers	O
to	O
a	O
mathematical	O
framework	O
for	O
graph	B-Application
rewriting	I-Application
.	O
</s>
<s>
It	O
was	O
introduced	O
as	O
one	O
of	O
the	O
first	O
algebraic	O
approaches	O
to	O
graph	B-Application
rewriting	I-Application
in	O
the	O
article	O
"	O
Graph-grammars	O
:	O
An	O
algebraic	O
approach	O
"	O
(	O
1973	O
)	O
.	O
</s>
<s>
A	O
DPO	O
graph	B-Application
transformation	I-Application
system	O
(	O
or	O
graph	B-Application
grammar	I-Application
)	O
consists	O
of	O
a	O
finite	O
graph	O
,	O
which	O
is	O
the	O
starting	O
state	O
,	O
and	O
a	O
finite	O
or	O
countable	O
set	O
of	O
labeled	O
spans	O
in	O
the	O
category	O
of	O
finite	O
graphs	O
and	O
graph	O
homomorphisms	O
,	O
which	O
serve	O
as	O
derivation	O
rules	O
.	O
</s>
<s>
The	O
rule	O
spans	O
are	O
generally	O
taken	O
to	O
be	O
composed	O
of	O
monomorphisms	B-Application
,	O
but	O
the	O
details	O
can	O
vary	O
.	O
</s>
<s>
Double	B-Application
pushout	I-Application
graph	I-Application
rewriting	I-Application
allows	O
the	O
specification	O
of	O
graph	B-Application
transformations	I-Application
by	O
specifying	O
a	O
pattern	O
of	O
fixed	O
size	O
and	O
composition	O
to	O
be	O
found	O
and	O
replaced	O
,	O
where	O
part	O
of	O
the	O
pattern	O
can	O
be	O
preserved	O
.	O
</s>
<s>
These	O
can	O
be	O
non-overlapping	O
,	O
or	O
share	O
only	O
preserved	O
items	O
,	O
thus	O
showing	O
a	O
kind	O
of	O
concurrency	B-Operating_System
known	O
as	O
parallel	O
independence	O
,	O
or	O
they	O
may	O
be	O
incompatible	O
,	O
in	O
which	O
case	O
either	O
the	O
applications	O
can	O
sometimes	O
be	O
executed	O
sequentially	O
,	O
or	O
one	O
can	O
even	O
preclude	O
the	O
other	O
.	O
</s>
<s>
Termination	O
for	O
DPO	O
graph	B-Application
rewriting	I-Application
is	O
undecidable	O
because	O
the	O
Post	O
correspondence	O
problem	O
can	O
be	O
reduced	O
to	O
it	O
.	O
</s>
<s>
DPO	O
graph	B-Application
rewriting	I-Application
can	O
be	O
viewed	O
as	O
a	O
generalization	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
Hypergraph	O
,	O
typed	O
graph	O
and	O
attributed	B-Application
graph	I-Application
rewriting	O
,	O
for	O
example	O
,	O
can	O
be	O
handled	O
because	O
they	O
can	O
be	O
cast	O
as	O
adhesive	O
HLR	O
systems	O
.	O
</s>
