<s>
Signal	B-Application
Transition	I-Application
Graphs	I-Application
(	O
STGs	O
)	O
are	O
typically	O
used	O
in	O
electronic	O
engineering	O
and	O
computer	O
engineering	O
to	O
describe	O
dynamic	O
behaviour	O
of	O
asynchronous	B-Application
circuits	I-Application
,	O
for	O
the	O
purposes	O
of	O
their	O
analysis	O
or	O
synthesis	O
.	O
</s>
<s>
Informally	O
,	O
an	O
STG	O
is	O
a	O
graphical	O
description	O
of	O
the	O
behaviour	O
of	O
an	O
asynchronous	B-Application
circuit	I-Application
in	O
the	O
form	O
where	O
information	O
about	O
causal	O
relations	O
between	O
signalling	O
events	O
is	O
represented	O
directly	O
,	O
as	O
opposed	O
to	O
descriptions	O
based	O
on	O
states	O
.	O
</s>
<s>
In	O
that	O
way	O
,	O
STGs	O
help	O
to	O
formalise	O
the	O
description	O
of	O
a	O
circuit	O
typically	O
represented	O
by	O
timing	B-Application
diagrams	I-Application
,	O
sometimes	O
also	O
called	O
waveforms	O
.	O
</s>
<s>
More	O
formally	O
,	O
an	O
STG	O
is	O
a	O
type	O
of	O
an	O
interpreted	O
(	O
or	O
labelled	O
)	O
Petri	B-Operating_System
net	I-Operating_System
whose	O
transitions	O
are	O
labelled	O
with	O
the	O
names	O
of	O
changes	O
in	O
the	O
values	O
of	O
signals	O
(	O
cf	O
.	O
</s>
<s>
STGs	O
usually	O
give	O
more	O
compact	O
descriptions	O
of	O
the	O
behaviour	O
of	O
asynchronous	B-Application
circuits	I-Application
than	O
state	O
graphs	O
.	O
</s>
<s>
The	O
complexity	O
of	O
an	O
STG	O
specification	O
of	O
a	O
circuit	O
is	O
typically	O
linear	O
in	O
the	O
number	O
of	O
signals	O
in	O
the	O
circuit	O
while	O
the	O
complexity	O
of	O
a	O
state	O
graph	O
can	O
grow	O
exponentially	O
,	O
due	O
to	O
the	O
fact	O
that	O
asynchronous	B-Application
circuits	I-Application
have	O
high	O
degree	O
of	O
concurrency	O
.	O
</s>
<s>
Amongst	O
the	O
various	O
examples	O
of	O
using	O
STGs	O
in	O
designing	O
asynchronous	B-Application
circuits	I-Application
,	O
the	O
most	O
well	O
known	O
are	O
those	O
in	O
the	O
domain	O
of	O
asynchronous	O
interfaces	O
,	O
controllers	O
,	O
arbiters	O
and	O
analog-mixed	O
signal	O
circuits	O
,	O
cf.	O
,	O
most	O
recently	O
STGs	O
have	O
been	O
extended	O
to	O
model	O
causal	O
behaviour	O
involving	O
causality	O
mediated	O
by	O
capacitive	O
coupling	O
,	O
such	O
as	O
those	O
used	O
in	O
switched	O
capacitor	O
converters	O
(	O
SCCs	O
)	O
.	O
</s>
<s>
Special	O
extensions	O
of	O
the	O
basic	O
underlying	O
Petri	B-Operating_System
net	I-Operating_System
models	O
,	O
to	O
capture	O
asynchrony	O
and	O
interrupts	O
in	O
a	O
compact	O
form	O
,	O
were	O
introduced	O
in	O
Place	O
Chart	O
Nets	O
.	O
</s>
<s>
An	O
important	O
connection	O
between	O
state-based	O
models	O
of	O
asynchronous	B-Application
circuits	I-Application
and	O
Petri	O
net-based	O
models	O
(	O
inc	O
.	O
STGs	O
)	O
has	O
been	O
established	O
in	O
using	O
Theory	B-General_Concept
of	I-General_Concept
Regions	I-General_Concept
(	O
cf	O
.	O
)	O
.	O
</s>
<s>
Theory	B-General_Concept
of	I-General_Concept
regions	I-General_Concept
was	O
used	O
to	O
derive	O
an	O
STG	O
model	O
and	O
its	O
circuit	O
implementation	O
in	O
for	O
Counterflow	O
Pipeline	O
Processor	O
due	O
to	O
Robert	O
Sproull	O
,	O
Ivan	O
Sutherland	O
and	O
Charles	O
Molnar	O
.	O
</s>
<s>
The	O
comparison	O
between	O
Petri	B-Operating_System
nets	I-Operating_System
and	O
change	O
diagrams	O
in	O
terms	O
of	O
their	O
descriptive	O
power	O
and	O
their	O
unification	O
in	O
the	O
form	O
of	O
Causal	O
Logic	O
Nets	O
has	O
been	O
presented	O
in	O
.	O
</s>
<s>
STGs	O
have	O
been	O
interfaced	O
with	O
various	O
HDLs	O
,	O
see	O
for	O
example	O
links	O
with	O
VHDL	O
(	O
1996	O
)	O
and	O
Verilog	O
(	O
2000	O
)	O
with	O
the	O
aim	O
to	O
support	O
asynchronous	B-Application
design	I-Application
.	O
</s>
<s>
Placed	O
into	O
the	O
synthesis	O
flow	O
from	O
VHDL	O
,	O
STGs	O
and	O
Petri	B-Operating_System
nets	I-Operating_System
have	O
been	O
shown	O
instrumental	O
,	O
and	O
likewise	O
with	O
Verilog	O
,	O
where	O
a	O
tool	O
VERISYN	O
was	O
developed	O
.	O
</s>
<s>
Likewise	O
,	O
realising	O
that	O
the	O
model	O
of	O
finite	O
state	O
machine	O
(	O
FSM	O
)	O
can	O
be	O
easier	O
for	O
designers	O
to	O
handle	O
than	O
,	O
for	O
example	O
,	O
Petri	B-Operating_System
nets	I-Operating_System
or	O
STGs	O
,	O
a	O
link	O
with	O
Burst	O
Mode	O
FSMs	O
as	O
a	O
front-end	O
has	O
been	O
developed	O
.	O
</s>
<s>
At	O
the	O
moment	O
,	O
arguably	O
the	O
most	O
efficient	O
methods	O
for	O
analysis	O
and	O
synthesis	O
of	O
asynchronous	B-Application
circuits	I-Application
are	O
based	O
on	O
Petri	B-Application
net	I-Application
unfoldings	I-Application
-	O
they	O
were	O
studied	O
by	O
Victor	O
Khomenko	O
in	O
his	O
PhD	O
thesis	O
.	O
</s>
<s>
Performance	O
analysis	O
of	O
certain	O
subclasses	O
of	O
Petri	B-Operating_System
net	I-Operating_System
models	O
of	O
asynchronous	B-Application
circuits	I-Application
has	O
been	O
investigated	O
by	O
Aiguo	O
Xie	O
and	O
Peter	O
Beerel	O
in	O
.	O
</s>
<s>
Various	O
problems	O
in	O
the	O
synthesis	O
of	O
asynchronous	B-Application
circuits	I-Application
from	O
STG	O
specification	O
have	O
been	O
investigated	O
.	O
</s>
<s>
One	O
of	O
the	O
ways	O
for	O
their	O
classification	O
is	O
based	O
on	O
the	O
analysis	O
approaches	O
used	O
to	O
represent	O
the	O
state	O
space	O
of	O
the	O
STG	O
specification	O
,	O
such	O
as	O
explicit	O
state	O
spaces	O
,	O
unfolding	O
of	O
the	O
underlying	O
Petri	B-Operating_System
net	I-Operating_System
,	O
structural	O
analysis	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
and	O
direct	O
mapping	O
(	O
syntax-direct	O
translation	O
)	O
of	O
STGs	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
some	O
of	O
these	O
techniques	O
impose	O
certain	O
constraints	O
on	O
the	O
class	O
of	O
the	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
For	O
example	O
,	O
explicit	O
state	O
space	O
based	O
methods	O
typically	O
work	O
for	O
an	O
arbitrary	O
Petri	B-Operating_System
net	I-Operating_System
type	O
,	O
whereas	O
some	O
structural	O
methods	O
require	O
that	O
the	O
underlying	O
Petri	B-Operating_System
net	I-Operating_System
is	O
a	O
marked	O
graph	O
or	O
a	O
free-choice	O
net	O
.	O
</s>
<s>
Another	O
method	O
exploited	O
theory	B-General_Concept
of	I-General_Concept
regions	I-General_Concept
which	O
connects	O
elements	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
with	O
regions	O
of	O
states	O
in	O
state	O
graphs	O
.	O
</s>
<s>
Synthesis	O
methods	O
for	O
CSC	O
detection	O
and	O
resolution	O
based	O
on	O
partial	O
orders	O
and	O
Petri	B-Application
net	I-Application
unfoldings	I-Application
have	O
been	O
developed	O
by	O
Alex	O
Semenov	O
and	O
Victor	O
Khomenko	O
.	O
</s>
<s>
The	O
problem	O
of	O
scalability	O
of	O
synthesis	O
for	O
large	O
size	O
STGs	O
,	O
and	O
needs	O
to	O
alleviate	O
state	O
space	O
explosion	O
have	O
been	O
tackled	O
in	O
methods	O
based	O
on	O
contraction	O
of	O
STGs	O
with	O
respect	O
to	O
structural	O
properties	O
of	O
the	O
underlying	O
Petri	B-Operating_System
net	I-Operating_System
-	O
such	O
as	O
ways	O
of	O
partitioning	O
a	O
free-choice	O
Petri	B-Operating_System
net	I-Operating_System
into	O
state	O
machines	O
or	O
marked	O
graphs	O
-	O
as	O
well	O
as	O
fan-in	O
signal	O
subsets	O
(	O
trigger	O
events	O
for	O
a	O
signal	O
)	O
.	O
</s>
<s>
Another	O
approach	O
to	O
deal	O
with	O
scalability	O
is	O
via	O
a	O
direct	O
mapping	O
of	O
STGs	O
to	O
asynchronous	B-Application
circuits	I-Application
that	O
has	O
been	O
investigated	O
by	O
Danil	O
Sokolov	O
.	O
</s>
<s>
A	O
particularly	O
challenging	O
problem	O
is	O
to	O
automatically	O
synthesise	O
asynchronous	B-Application
circuits	I-Application
for	O
arbiters	O
,	O
as	O
their	O
STG	O
specification	O
would	O
contain	O
behavioural	O
conflicts	O
in	O
their	O
underlying	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
Special	O
techniques	O
such	O
as	O
semi-automated	O
insertion	O
of	O
mutex	B-Operating_System
signal	O
transitions	O
,	O
preserving	O
the	O
original	O
specification	O
,	O
have	O
been	O
developed	O
and	O
implemented	O
in	O
Workcraft	O
.	O
</s>
