<s>
In	O
automata	B-Application
theory	I-Application
and	O
control	O
theory	O
,	O
branches	O
of	O
mathematics	O
,	O
theoretical	O
computer	O
science	O
and	O
systems	O
engineering	O
,	O
a	O
noncommutative	B-Application
signal-flow	I-Application
graph	I-Application
is	O
a	O
tool	O
for	O
modeling	O
interconnected	O
systems	O
and	O
state	B-Architecture
machines	I-Architecture
by	O
mapping	O
the	O
edges	O
of	O
a	O
directed	O
graph	O
to	O
a	O
ring	O
or	O
semiring	O
.	O
</s>
<s>
A	O
single	O
edge	O
weight	O
might	O
represent	O
an	O
array	O
of	O
impulse	O
responses	O
of	O
a	O
complex	O
system	O
(	O
see	O
figure	O
to	O
the	O
right	O
)	O
,	O
or	O
a	O
character	O
from	O
an	O
alphabet	O
picked	O
off	O
the	O
input	B-Architecture
tape	I-Architecture
of	O
a	O
finite	B-Architecture
automaton	I-Architecture
,	O
while	O
the	O
graph	O
might	O
represent	O
the	O
flow	O
of	O
information	O
or	O
state	O
transitions	O
.	O
</s>
<s>
They	O
are	O
known	O
to	O
hold	O
over	O
an	O
arbitrary	O
noncommutative	O
ring	O
and	O
over	O
the	O
semiring	O
of	O
regular	B-Language
expressions	I-Language
.	O
</s>
<s>
An	O
important	O
kind	O
of	O
noncommutative	B-Application
signal-flow	I-Application
graph	I-Application
is	O
a	O
finite	B-Architecture
state	I-Architecture
automaton	I-Architecture
over	O
an	O
alphabet	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
the	O
subsets	O
of	O
built	O
of	O
from	O
finitely	O
many	O
of	O
these	O
three	O
operations	O
can	O
be	O
identified	O
with	O
the	O
semiring	O
of	O
regular	B-Language
expressions	I-Language
.	O
</s>
<s>
Similarly	O
,	O
finite	O
graphs	O
whose	O
edges	O
are	O
weighted	O
by	O
subsets	O
of	O
can	O
be	O
identified	O
with	O
finite	B-Architecture
automata	I-Architecture
,	O
though	O
generally	O
that	O
theory	O
starts	O
with	O
singleton	O
sets	O
as	O
in	O
the	O
figure	O
.	O
</s>
