<s>
FD-DEVS	B-Application
(	O
Finite	B-Application
&	I-Application
Deterministic	I-Application
Discrete	I-Application
Event	I-Application
System	I-Application
Specification	I-Application
)	O
is	O
a	O
formalism	O
for	O
modeling	O
and	O
analyzing	O
discrete	O
event	O
dynamic	O
systems	O
in	O
both	O
simulation	O
and	O
verification	O
ways	O
.	O
</s>
<s>
FD-DEVS	B-Application
also	O
provides	O
modular	O
and	O
hierarchical	O
modeling	O
features	O
which	O
have	O
been	O
inherited	O
from	O
Classic	O
DEVS	B-Application
.	O
</s>
<s>
FD-DEVS	B-Application
was	O
originally	O
named	O
as	O
"	O
Schedule-Controllable	O
DEVS	B-Application
"	O
 [ Hwang05 ] 	O
and	O
designed	O
to	O
support	O
verification	O
analysis	O
of	O
its	O
networks	O
which	O
had	O
been	O
an	O
open	O
problem	O
of	O
DEVS	B-Application
formalism	O
for	O
30	O
years	O
.	O
</s>
<s>
In	O
addition	O
,	O
it	O
was	O
also	O
designated	O
to	O
resolve	O
the	O
so-called	O
"	O
OPNA	O
"	O
problem	O
of	O
SP-DEVS	B-Application
.	O
</s>
<s>
The	O
third	O
restriction	O
can	O
be	O
also	O
seen	O
as	O
a	O
relaxation	O
from	O
SP-DEVS	B-Application
where	O
the	O
schedule	O
is	O
always	O
preserved	O
by	O
any	O
input	O
events	O
.	O
</s>
<s>
Due	O
to	O
this	O
relaxation	O
there	O
is	O
no	O
longer	O
OPNA	O
problem	O
,	O
but	O
there	O
is	O
also	O
one	O
limitation	O
that	O
a	O
time-line	O
abstraction	O
which	O
can	O
be	O
used	O
for	O
abstracting	O
elapsed	O
times	O
of	O
SP-DEVS	B-Application
networks	O
is	O
no	O
longer	O
useful	O
for	O
FD-DEVS	B-Application
network	O
 [ Hwang05 ] 	O
.	O
</s>
<s>
But	O
another	O
time	O
abstraction	O
method	O
 [ Dill89 ] 	O
which	O
was	O
invented	O
by	O
Prof	O
.	O
D	O
.	O
Dill	O
can	O
be	O
applicable	O
to	O
obtain	O
a	O
finite-vertex	O
reachability	O
graph	O
for	O
FD-DEVS	B-Application
networks	O
.	O
</s>
<s>
Each	O
player	O
can	O
be	O
modeled	O
by	O
FD-DEVS	B-Application
such	O
that	O
the	O
player	O
model	O
has	O
an	O
input	O
event	O
?	O
receive	O
and	O
an	O
output	O
event	O
!	O
send	O
,	O
and	O
it	O
has	O
two	O
states	O
:	O
Send	O
and	O
Wait	O
.	O
</s>
<s>
One	O
can	O
model	O
it	O
with	O
FD-DEVS	B-Application
as	O
shown	O
in	O
Fig	O
.	O
</s>
<s>
Two	O
slots	O
are	O
modeled	O
as	O
atomic	O
FD-DEVS	B-Application
whose	O
input	O
event	O
is	O
"	O
?	O
push	O
"	O
and	O
output	O
event	O
is	O
"	O
!	O
pop	O
"	O
,	O
states	O
are	O
"	O
Idle	O
"	O
(	O
I	O
)	O
and	O
"	O
Toast	O
"	O
(	O
T	O
)	O
with	O
the	O
initial	O
state	O
is	O
"	O
idle	O
"	O
.	O
</s>
<s>
As	O
mentioned	O
above	O
,	O
FD-DEVS	B-Application
is	O
an	O
relaxation	O
of	O
SP-DEVS	B-Application
.	O
</s>
<s>
That	O
means	O
,	O
FD-DEVS	B-Application
is	O
a	O
supper	O
class	O
of	O
SP-DEVS	B-Application
.	O
</s>
<s>
We	O
would	O
give	O
a	O
model	O
of	O
FD-DEVS	B-Application
of	O
a	O
crosswalk	O
light	O
controller	O
which	O
is	O
used	O
for	O
SP-DEVS	B-Application
in	O
this	O
Wikipedia	O
.	O
</s>
<s>
The	O
property	O
of	O
non-negative	O
rational-valued	O
lifespans	O
which	O
can	O
be	O
preserved	O
or	O
changed	O
by	O
input	O
events	O
along	O
with	O
finite	O
numbers	O
of	O
states	O
and	O
events	O
guarantees	O
that	O
the	O
behavior	O
of	O
FD-DEVS	B-Application
networks	O
can	O
be	O
abstracted	O
as	O
an	O
equivalent	O
finite-vertex	O
reachability	O
graph	O
by	O
abstracting	O
the	O
infinitely-many	O
values	O
of	O
the	O
elapsed	O
times	O
using	O
the	O
time	O
abstracting	O
technique	O
introduced	O
by	O
Prof	O
.	O
D	O
.	O
Dill	O
 [ Dill89 ] 	O
.	O
</s>
<s>
As	O
a	O
qualitative	O
property	O
,	O
safety	O
of	O
a	O
FD-DEVS	B-Application
network	O
is	O
decidable	O
by	O
(	O
1	O
)	O
generating	O
RG	O
of	O
the	O
given	O
network	O
and	O
(	O
2	O
)	O
checking	O
whether	O
some	O
bad	O
states	O
are	O
reachable	O
or	O
not	O
 [ HZ06b ] 	O
.	O
</s>
<s>
As	O
a	O
qualitative	O
property	O
,	O
liveness	O
of	O
a	O
FD-DEVS	B-Application
network	O
is	O
decidable	O
by	O
(	O
1	O
)	O
generating	O
RG	O
of	O
the	O
given	O
network	O
,	O
(	O
2	O
)	O
from	O
RG	O
,	O
generating	O
kernel	O
directed	O
acyclic	O
graph	O
(	O
KDAG	O
)	O
in	O
which	O
a	O
vertex	O
is	O
strongly	O
connected	O
component	O
,	O
and	O
(	O
3	O
)	O
checking	O
if	O
a	O
vertex	O
of	O
KDAG	O
contains	O
a	O
state	O
transition	O
cycle	O
which	O
contains	O
a	O
set	O
of	O
liveness	O
states[HZ06b]	O
.	O
</s>
<s>
The	O
features	O
that	O
all	O
characteristic	O
functions	O
,	O
of	O
FD-DEVS	B-Application
are	O
deterministic	O
can	O
be	O
seen	O
as	O
somehow	O
a	O
limitation	O
to	O
model	O
the	O
system	O
that	O
has	O
non-deterministic	O
behaviors	O
.	O
</s>
<s>
1	O
.	O
has	O
a	O
stochastic	O
lifespan	O
at	O
"	O
Send	O
"	O
state	O
,	O
FD-DEVS	B-Application
does	O
n't	O
capture	O
the	O
non-determinism	O
effectively	O
.	O
</s>
<s>
There	O
are	O
two	O
open	O
source	O
libraries	O
DEVS#	O
written	O
in	O
C#	B-Application
at	O
and	O
XSY	O
written	O
in	O
Python	B-Language
at	O
that	O
support	O
some	O
reachability	O
graph-based	O
verification	O
algorithms	O
for	O
finding	O
safeness	O
and	O
liveness	O
.	O
</s>
<s>
For	O
standardization	O
of	O
DEVS	B-Application
,	O
especially	O
using	O
FDDEVS	B-Application
,	O
Dr.	O
Saurabh	O
Mittal	O
together	O
with	O
co-workers	O
has	O
worked	O
on	O
defining	O
of	O
XML	O
format	O
of	O
FDDEVS	B-Application
.	O
</s>
