Probabilistic Graph Models WMCI 20154
90
Probabilistic Reasoning using Graph Models Vineet Sahula Dept. of ECE, MNIT Jaipur [email protected]
description
.
Transcript of Probabilistic Graph Models WMCI 20154
Prob
abili
stic
Rea
soni
ng u
sing
Gra
ph
Mod
els
Vine
etSa
hula
Dept.
of E
CE, M
NIT
Jaipu
rsa
hula@
acm
.org
Outli
ne
•Pr
obab
ilistic
reas
oning
unde
r unc
ertai
nty•
Infer
ence
s in g
raph
mod
els–
Baye
sian,
Marko
v Ran
dom
Fields
•Re
ason
ing w
ith tim
e•
Case
stud
y–
Nano
scale
memo
ry mo
delin
g: Re
liabil
ity &
Life
time
evalu
ation
in pr
esen
ce of
erro
rs•
Conc
lusion
s
WM
SC 1
6-Ja
n-20
15
ACKN
OWLE
DG
MEN
TS
Slide
s in p
arts
borro
wed f
rom
-Pr
of. W
elling
, UCI
rvine
, CA
-Pr
of. R
. J. M
oone
y, Ut
exas
, Aus
tin WM
SC 1
6-Ja
n-20
15
Perfo
rman
ce E
valu
atio
n of
SoC
Com
mun
icat
ion
Arch
itect
ures
-
Ulha
sD
eshm
ukh
•De
velop
ment
of eff
icien
t per
forma
nce e
valua
tion &
desig
n spa
ce
explo
ratio
n fra
mewo
rk for
comm
unica
tion
arch
itectu
re a
t Sys
tem
level
(ESL
)•
Cons
idere
d stoc
hasti
c natu
re of
Mod
el pa
rame
ters-
GENE
RALIZ
ED di
stribu
tions
for P
E’s c
ompu
tation
time,
and
comm
. Tim
e etc.
•Mo
del s
uppo
rts co
ncur
renc
y as w
ell as
hier
arch
y-Bu
s, mu
ltiple-
hiera
rchica
l bus
, NoC
•An
alytic
al me
thod
for ev
aluati
on o
f Gen
erali
zed
Semi
Mar
kov
Proc
ess (
GSMP
)-AF
OME,
HCF
G, S
AN•
BW, W
aiting
time,
utiliz
ation
facto
r, qu
eue l
ength
comp
uted
WM
SC 1
6-Ja
n-20
15
Effic
ient
Fac
ial f
eatu
re re
cogn
ition
-R. A
. Pat
il•
Prop
osed
an al
gorith
m for
facia
l featu
re ex
tracti
on w
ith tit
led fa
ces,
& no
n-ne
utra
l exp
ress
ion o
n ini
tial fr
ame
•Au
tomate
d wire
mod
el fitt
ing•
Topo
grap
hical
maps
(neu
ral c
ompu
tation
) for
map
ping o
f featu
re
vecto
rs, an
d mult
iclas
s SVM
•Ha
rdwa
re m
appin
g for
effic
ient, r
eal ti
me op
erati
on-s
ystol
ic ar
ray
comp
utatio
n, pa
rtial re
confi
gura
tion f
or lo
w po
wer
WM
SC 1
6-Ja
n-20
15
Nan
osca
le M
emor
y M
odel
ing
& S
ynth
esis
-R. K
umaw
at•
Molec
ular D
evice
Mod
el co
nside
red f
or su
b-sy
stem
mode
l gen
erati
on•
Nano
cellb
ased
Mole
cular
Mem
ory S
ynthe
sis in
pres
ence
of h
ard
erro
rs–
Omnip
otent
Train
ing–
Morta
l Tra
ining
•Pr
obab
ilistic
Ana
lysis
of Na
noce
llMole
cular
Mem
ory
–Re
liabil
ity P
redic
tion
of Na
noce
llin
•Sp
atial
Doma
in•
Time D
omain
–Ex
tende
d Con
tinuo
us P
aram
eter B
irth-D
eath
Mode
l for P
roba
bilist
ic An
alysis
of N
anoc
ellin
pres
ence
of T
rans
ient E
rrors.
WM
SC 1
6-Ja
n-20
15
Gra
phic
al M
odel
s
A ‘
mar
riag
e’ b
etw
een
prob
abili
ty th
eory
and
gra
ph th
eory
Why
pro
babi
litie
s?
•R
easo
ning
with
unc
erta
intie
s, co
nfid
ence
leve
ls•
Man
y pr
oces
ses a
re in
here
ntly
‘noi
sy’
robu
stne
ss is
sues
Why
gra
phs?
•Pr
ovid
e ne
cess
ary
stru
ctur
e in
larg
e m
odel
s:
-Des
igni
ng n
ew p
roba
bilis
tic m
odel
s.-R
eadi
ng o
ut (c
ondi
tiona
l) in
depe
nden
cies
.
•In
fere
nce
& o
ptim
izat
ion:
-Dyn
amic
al p
rogr
amm
ing
-Bel
ief P
ropa
gatio
n
Type
s of
Gra
phic
al M
odel
Und
irec
ted
grap
h (M
arko
v ra
ndom
fie
ld)
Dir
ecte
d gr
aph
(Bay
esia
n ne
twor
k)∏
∏=
iij
ji
iji
ix
xx
Zx
P)
()
()
,(
)(
1)
(ψ
ψ
i
j
)(
ii
xψ
),
( )(
ji
ijx
xψ
)|
()
()
(pa
rent
s∏
=i
ii
xx
Px
P
i
Pare
nts(
i)
fact
or g
raph
s
inte
ract
ions
vari
able
s
9
Gra
phic
al M
odel
s
•If n
o ass
umpti
on of
inde
pend
ence
is m
ade,
then a
n exp
onen
tial
numb
er of
para
meter
s mus
t be e
stima
ted fo
r sou
nd pr
obab
ilistic
inf
eren
ce.
•No
reali
stic a
moun
t of tr
aining
data
is su
fficien
t to e
stima
te so
man
y pa
rame
ters.
•If a
blan
ket a
ssum
ption
of co
nditio
nal in
depe
nden
ce is
mad
e, eff
icien
t train
ing an
d infe
renc
e is p
ossib
le, bu
t suc
h a st
rong
as
sump
tion
is ra
rely
warra
nted.
•Gr
aphi
cal m
odels
use d
irecte
d or u
ndire
cted g
raph
s ove
r a se
t of
rand
om va
riable
s to e
xplic
itly sp
ecify
varia
ble d
epen
denc
ies a
nd
allow
for le
ss re
strict
ive in
depe
nden
ce as
sump
tions
whil
e lim
iting
the nu
mber
of pa
rame
ters t
hat m
ust b
e esti
mated
.–
Baye
sian
Netw
orks
: Dire
cted a
cycli
c gra
phs t
hat in
dicate
caus
al str
uctur
e.–
Mark
ov N
etwo
rks:
Undir
ected
grap
hs th
at ca
pture
gene
ral d
epen
denc
ies.
10
Baye
sian
Net
wor
ks
•Di
recte
d Acy
clic G
raph
(DAG
)–
Node
s are
rand
om va
riable
s–
Edge
s ind
icate
caus
al inf
luenc
es
Bur
glar
yE
arth
quak
e
Ala
rm
John
Cal
lsM
aryC
alls
11
Cond
ition
al P
roba
bilit
y Ta
bles
•Ea
ch no
de ha
s a co
nditi
onal
prob
abilit
y tab
le(C
PT) t
hat g
ives t
he
prob
abilit
y of e
ach
of its
value
s give
n ev
ery p
ossib
le co
mbina
tion
of va
lues f
or its
pare
nts (c
ondit
ioning
case
).–
Roots
(sou
rces)
of the
DAG
that
have
no pa
rents
are g
iven p
rior p
roba
bilitie
s.
Bur
glar
yE
arth
quak
e
Ala
rm
John
Cal
lsM
aryC
alls
P(B)
.001
P(E)
.002
BE
P(A
)T
T.9
5T
F.9
4F
T.2
9F
F.0
01
AP(
M)
T.7
0F
.01
AP(
J)T
.90
F.0
5
12
CPT
Com
men
ts
•Pr
obab
ility o
f false
not g
iven s
ince r
ows m
ust a
dd to
1.•
Exam
ple re
quire
s 10 p
aram
eters
rathe
r tha
n 25 –
1 = 31
for s
pecif
ying t
he fu
ll join
t dis
tributi
on.
•Nu
mber
of pa
rame
ters i
n the
CPT
for a
node
is ex
pone
ntial
in the
numb
er of
pare
nts
(fan-
in).
13
Join
t Dis
tribu
tions
for B
ayes
Net
s
•A
Baye
sian N
etwor
k imp
licitly
defin
es a
joint
distrib
ution
.))
(Pa
rent
s|
()
,...
,(
12
1i
n ii
nX
xP
xx
xP
∏ =
=
•E
xam
ple
)(
EB
AM
JP
¬∧
¬∧
∧∧
)(
)(
)|
()
|(
)|
(E
PB
PE
BA
PA
MP
AJ
P¬
¬¬
∧¬
=00
062
.099
8.0
999
.000
1.0
7.09.0
=×
××
×=
•T
here
fore
an
inef
fici
ent a
ppro
ach
to in
fere
nce
is:
–1)
Com
pute
the
join
t dis
trib
utio
n us
ing
this
equ
atio
n.–
2) C
ompu
te a
ny d
esir
ed c
ondi
tiona
l pro
babi
lity
usin
g th
e jo
int d
istr
ibut
ion.
14
Naï
ve B
ayes
as
a Ba
yes
Net
•Na
ïve B
ayes
is a
simple
Bay
es N
et Y
X1
X2
…X
n
•Pr
iors
P(Y
) an
d co
nditi
onal
s P(
Xi|Y
) fo
r N
aïve
Bay
es p
rovi
de C
PTs
for
the
netw
ork.
Con
stru
ctin
g B
ayes
ian
Net
wor
ks
Cho
ose
the
right
ord
er fr
om c
ause
s to
effe
cts.
P(x1
,x2,
…,x
n) =
P(x
n|xn
-1,..
,x1)
P(xn
-1,…
,x1)
= Π
P(x
i|xi-1
,…,x
1)
--ch
ain
rule
Exam
ple:
P(x1
,x2,
x3) =
P(x
1|x2
,x3)
P(x2
|x3)
P(x3
)
How
to c
onst
ruct
BN
P(x1
,x2,
x3)
x3
x2
x1
root
cau
se
leaf
Cor
rect
ord
er: a
dd ro
ot c
ause
s firs
t, an
d th
en
“lea
ves”
, with
no
influ
ence
on
othe
r nod
es.
Com
pact
ness
BN
are
loca
lly st
ruct
ured
syst
ems.
They
repr
esen
t joi
nt d
istri
butio
ns c
ompa
ctly
.
Ass
ume
nra
ndom
var
iabl
es, e
ach
influ
ence
dby
k n
odes
. Si
ze B
BN
: n2k
F
ull s
ize:
2n
18
Inde
pend
enci
es in
Bay
es N
ets
•If r
emov
ing a
subs
et of
node
s Sfro
m the
netw
ork r
ende
rs no
des X
iand
Xjd
iscon
necte
d, the
n Xia
nd X
jare
ind
epen
dent
given
S, i.
e. P(
X i| X j, S
) = P
(Xi|
S)•
Howe
ver,
this i
s too
stric
t a cr
iteria
for c
ondit
ional
indep
ende
nce s
ince t
wo no
des w
ill sti
ll be c
onsid
ered
ind
epen
dent
if the
ir sim
ply ex
ists s
ome v
ariab
le tha
t de
pend
s on b
oth.
–Fo
r exa
mple,
Bur
glary
and E
arthq
uake
shou
ld be
cons
idere
d ind
epen
dent
since
they
both
caus
e Alar
m.
19
Inde
pend
enci
es in
Bay
es N
ets
(con
t.)
•Un
less w
e kno
w so
methi
ng ab
out a
comm
on ef
fect o
f two “
indep
ende
nt ca
uses
” or a
desc
ende
nt of
a com
mon e
ffect,
then
they
can b
e co
nside
red
indep
ende
nt.
–Fo
r exa
mple,
if we
know
nothi
ng el
se, E
arthq
uake
and B
urgla
ry ar
e ind
epen
dent.
•Ho
weve
r, if w
e hav
e info
rmati
on a
bout
a com
mon e
ffect
(or d
esce
nden
t the
reof)
then
the t
wo “in
depe
nden
t” ca
uses
bec
ome
prob
abilis
ticall
y lin
ked s
ince e
viden
ce fo
r one
caus
e can
“exp
lain a
way”
the ot
her.
–Fo
r exa
mple,
if we
know
the a
larm
went
off th
at so
meon
e call
ed ab
out th
e alar
m,
then i
t mak
es ea
rthqu
ake a
nd bu
rglar
y dep
ende
nt sin
ce ev
idenc
e for
earth
quak
e de
creas
es be
lief in
burg
lary.
and v
ice ve
rsa.
Why
do
we
need
it?
•A
nsw
er q
ueri
es :
-Giv
en p
ast p
urch
ases
, in
wha
t gen
re b
ooks
is a
clie
nt in
tere
sted
?-G
iven
a n
oisy
imag
e, w
hat w
as th
e or
igin
al im
age?
•L
earn
ing
prob
abili
stic
mod
els
from
exa
mpl
es
( exp
ecta
tion
max
imiz
atio
n, it
erat
ive
scal
ing
)
•Opt
imiz
atio
n pr
oble
ms:
min
-cut
, max
-flo
w, V
iterb
i, …
Infe
renc
e in
Gra
phic
al M
odel
s
Infe
renc
e:•
Ans
wer
que
ries
abo
ut u
nobs
erve
d ra
ndom
var
iabl
es, g
iven
val
ues
of o
bser
ved
rand
om v
aria
bles
.
•M
ore
gene
ral:
com
pute
thei
r jo
int p
oste
rior
dist
ribu
tion: (
|)
{(
|)}
iPu
oor
Pu
o
lear
ning
infe
renc
e
Appr
oxim
ate
Infe
renc
e
Infe
renc
e is
com
puta
tiona
lly in
trac
tabl
e fo
r la
rge
grap
hs (
with
cyc
les)
.
App
roxi
mat
e m
etho
ds:
•M
arko
v C
hain
Mon
te C
arlo
sam
plin
g.
•M
ean
fiel
d an
d m
ore
stru
ctur
ed v
aria
tiona
ltec
hniq
ues.
•B
elie
f Pr
opag
atio
n al
gori
thm
s.
Belie
f Pro
paga
tion
on tr
ees
ik
k
k
k
ij
k
k
kMki
∏∑
→→
∝k
ii
kx
ii
ji
ijj
ji
xM
xx
xx
Mi
)(
)(
),
()
(ψ
ψ
Com
patib
ilitie
s (i
nter
actio
ns)
exte
rnal
evi
denc
e
∏∝
kk
ki
ii
ix
Mx
xb
)(
)(
)(
ψ
mes
sage
belie
f (
appr
oxim
ate
mar
gina
l pro
babi
lity)
Belie
f Pro
paga
tion
on lo
opy
grap
hs
ik
k
k
k
ij
k
k
kMki
∏∑
→→
∝k
ii
kx
ii
ji
ijj
ji
xM
xx
xx
Mi
)(
)(
),
()
(ψ
ψ
Com
patib
ilitie
s (i
nter
actio
ns)
exte
rnal
evi
denc
e
∏∝
kk
ki
ii
ix
Mx
xb
)(
)(
)(
ψ
mes
sage
belie
f (
appr
oxim
ate
mar
gina
l pro
babi
lity)
Som
e fa
cts
abou
t BP
•B
P is
exa
ct o
n tr
ees.
•If
BP
conv
erge
s it
has
reac
hed
a lo
cal m
inim
um o
f an
obj
ectiv
e fu
nctio
n (t
he B
ethe
fre
e en
ergy
Yed
idia
et.a
l ‘00
, H
eske
s’0
2)of
ten
good
app
roxi
mat
ion
•If
it co
nver
ges,
con
verg
ence
is f
ast n
ear
the
fixe
d po
int.
•M
any
exci
ting
appl
icat
ions
: -
erro
r co
rrec
ting
deco
ding
(Mac
Kay
, Yed
idia
, McE
liece
, Fre
y)-
visi
on (
Fre
eman
, Wei
ss)
-bi
oinf
orm
atic
s (W
eiss
)-
cons
trai
nt s
atis
fact
ion
prob
lem
s (D
echt
er)
-ga
me
theo
ry (
Kea
rns)
-…
Gen
eral
ized
Bel
ief P
ropa
gatio
n
∏∑
→
→∝ k
ii
kx
ii
ji
ij
jj
i
xM
xx
x
xM
i
)(
)(
),
(
)(
ψψ
Idea
: To
gues
s th
e di
stri
butio
n of
one
of
your
nei
ghbo
rs, y
ou a
sk
your
oth
erne
ighb
ors
to g
uess
you
r di
stri
butio
n. O
pini
ons
get
com
bine
d m
ultip
licat
ivel
y.
∏∑
→
→∝ k
ii
kx
ii
ji
ij
jj
i
xM
xx
x
xM
i
)(
)(
),
(
)(
ψψ
BP
GB
P
26
Baye
s N
et In
fere
nce
•Gi
ven k
nown
value
s for
some
evid
ence
varia
bles
, dete
rmine
the
poste
rior p
roba
bility
of so
me q
uery
varia
bles
.•
Exam
ple: G
iven t
hat J
ohn c
alls,
what
is the
prob
abilit
y tha
t the
re is
a Bu
rglar
y?
Bur
glar
yE
arth
quak
e
Ala
rm
John
Cal
lsM
aryC
alls
???
John
cal
ls 9
0% o
f the
tim
e th
ere
is a
n A
larm
and
the A
larm
det
ects
94%
of B
urgl
arie
s so
peop
lege
nera
lly th
ink
it sh
ould
be
fairl
y hi
gh.
How
ever
, thi
s ign
ores
the
prio
rpr
obab
ility
of J
ohn
calli
ng.
27
Baye
s N
et In
fere
nce
•Ex
ample
: Give
n tha
t Joh
n call
s, wh
at is
the pr
obab
ility t
hat
there
is a
Burg
lary?
Bur
glar
yE
arth
quak
e
Ala
rm
John
Cal
lsM
aryC
alls
???
John
als
o ca
lls 5
% o
f the
tim
e w
hen
ther
eis
no
Ala
rm. S
o ov
er 1
,000
day
s we
expe
ct 1
Bur
glar
y an
d Jo
hn w
ill p
roba
bly
call.
How
ever
, he
will
als
o ca
ll w
ith a
fa
lse
repo
rt 50
tim
es o
n av
erag
e. S
o th
e ca
ll is
abo
ut 5
0 tim
es m
ore
likel
y a
fals
e re
port:
P(B
urgl
ary
| Joh
nCal
ls) ≈
0.0
2
P(B)
.001
AP(
J)T
.90
F.0
5
28
Baye
s N
et In
fere
nce
•Ex
ample
: Give
n tha
t Joh
n call
s, wh
at is
the pr
obab
ility t
hat
there
is a
Burg
lary?
Bur
glar
yE
arth
quak
e
Ala
rm
John
Cal
lsM
aryC
alls
???
Act
ual p
roba
bilit
y of
Bur
glar
y is
0.0
16
sinc
e th
e al
arm
is n
ot p
erfe
ct (a
n Ea
rthqu
ake
coul
d ha
ve se
t it o
ff or
it
coul
d ha
ve g
one
off o
n its
ow
n). O
n th
e ot
her s
ide,
eve
n if
ther
e w
as n
ot a
n al
arm
and
John
cal
led
inco
rrec
tly, t
here
co
uld
have
bee
n an
und
etec
ted
Bur
glar
y an
yway
, but
this
is u
nlik
ely.
P(B)
.001
AP(
J)T
.90
F.0
5
29
Type
s of
Infe
renc
e
30
Sam
ple
Infe
renc
es
•Di
agno
stic
(evid
entia
l, abd
uctiv
e): F
rom
effec
t to ca
use.
–P(
Burg
lary |
John
Calls
) = 0.
016
–P(
Burg
lary |
John
Calls
∧Ma
ryCall
s) =
0.29
–P(
Alar
m | J
ohnC
alls ∧
MaryC
alls)
= 0.7
6–
P(Ea
rthqu
ake |
John
Calls
∧Ma
ryCall
s) =
0.18
•Ca
usal
(pre
dict
ive):
From
caus
e to e
ffect
–P(
John
Calls
| Bur
glary)
= 0.
86–
P(Ma
ryCall
s | B
urgla
ry) =
0.67
•In
terc
ausa
l (ex
plain
ing
away
): Be
twee
n ca
uses
of a
comm
on e
ffect.
–P(
Burg
lary |
Alar
m) =
0.37
6–
P(Bu
rglar
y | A
larm ∧
Earth
quak
e) =
0.00
3•
Mixe
d: T
wo or
mor
e of th
e abo
ve co
mbine
d –
(diag
nosti
c and
caus
al) P
(Alar
m | J
ohnC
alls ∧
¬Ear
thqua
ke) =
0.03
–(d
iagno
stic a
nd in
terca
usal)
P(B
urgla
ry | J
ohnC
alls ∧
¬Ear
thqua
ke) =
0.01
7
31
Com
plex
ity o
f Bay
es N
et In
fere
nce
•In
gene
ral, t
he pr
oblem
of B
ayes
Net
infer
ence
is N
P-ha
rd
(exp
onen
tial in
the s
ize of
the g
raph
).•
For s
ingl
y-co
nnec
ted
netw
orks
or p
olyt
rees
in wh
ich
there
are n
o und
irecte
d loo
ps, th
ere a
re lin
ear-t
ime
algor
ithms
base
d on b
elief
pro
paga
tion.
–Ea
ch n
ode
send
s loc
al ev
idenc
e me
ssag
es to
their
child
ren
and
pare
nts.
–Ea
ch n
ode u
pdate
s beli
ef in
each
of its
poss
ible v
alues
base
d on
inco
ming
mes
sage
s fro
m it n
eighb
ors a
nd pr
opag
ates
evide
nce o
n to i
ts ne
ighbo
rs.•
Ther
e are
appr
oxim
ation
s to i
nfere
nce f
or ge
nera
l ne
twor
ks ba
sed o
n loo
py b
elief
pro
paga
tion
that
itera
tively
refin
es pr
obab
ilities
that
conv
erge
to ac
cura
te va
lues i
n the
limit.
32
Belie
f Pro
paga
tion
Exam
ple
•λm
essa
ges a
re se
nt fro
m ch
ildre
n to p
aren
ts re
pres
entin
g abd
uctiv
e evid
ence
for a
node
.•
πme
ssag
es ar
e sen
t from
pare
nts to
child
ren
repr
esen
ting c
ausa
l evid
ence
for a
node
.
Bur
glar
yE
arth
quak
e
Ala
rm
John
Cal
lsM
aryC
alls
λ
λλ
πA
larm
Bur
glar
yE
arth
quak
e
Mar
yCal
ls
33
Mul
tiply
Con
nect
ed N
etw
orks
•Ne
twor
ks w
ith un
direc
ted lo
ops,
more
than
one d
irecte
d pa
th be
twee
n som
e pair
of no
des.
•In
gen
eral
, inf
eren
ce in
suc
h ne
twor
ks is
NP-
hard
.
•So
me
met
hods
con
stru
ct a
pol
ytre
e(s)
fro
m g
iven
ne
twor
k an
d pe
rfor
m in
fere
nce
on tr
ansf
orm
ed g
raph
.
34
Nod
e Cl
uste
ring
•El
imina
te all
loop
s by m
ergin
g nod
es to
crea
te m
egan
odes
that
have
the c
ross
-pro
duct
of va
lues o
f the
mer
ged n
odes
.
•N
umbe
r of
val
ues
for
mer
ged
node
is e
xpon
entia
l in
the
num
ber
of n
odes
mer
ged.
•St
ill r
easo
nabl
y tr
acta
ble
for
man
y ne
twor
k to
polo
gies
req
uiri
ng r
elat
ivel
y lit
tle m
ergi
ng to
el
imin
ate
loop
s.
35
Stat
istic
al R
evol
utio
n
•Ac
ross
AI th
ere h
as be
en a
move
ment
from
logic-
base
d ap
proa
ches
to ap
proa
ches
base
d on p
roba
bility
and s
tatist
ics.
–St
atisti
cal n
atura
l lang
uage
pro
cess
ing–
Stati
stica
l com
puter
visio
n–
Stati
stica
l robo
t nav
igatio
n–
Stati
stica
l lear
ning
•Mo
st ap
proa
ches
are f
eatur
e-ba
sed a
nd “p
ropo
sition
al” an
d do
not h
andle
comp
lex re
lation
al de
scrip
tions
with
mult
iple e
ntitie
s lik
e tho
se ty
picall
y req
uiring
pred
icate
logic.
Marko
v Ran
dom
Field
(MRF
)Ba
yesia
n Netw
ork (
BN)
Prob
abilit
y Tra
nsfer
Matr
ix (P
TM)
Prob
abilit
y Dec
ision
Diag
ram
(PDD
)Pr
obab
ilistic
Mod
el Ev
aluati
ons:
a com
para
tive s
tudy
Prop
osed
exte
nded
birt
h-de
ath
mod
el
PROB
ABIL
ISTI
C M
ODEL
ING
AP
PROA
CHES
FOR
NAN
OSCA
LE
SUB-
SYST
EMS W
MSC
16-
Jan-
2015
Expe
rimen
tal S
etup
for O
mni
pote
nt T
rain
ing
of M
olec
ular
Mem
ory
WM
SC 1
6-Ja
n-20
15
Algo
rithm
for O
mni
pote
nt T
rain
ing
ALGO
RITH
M 1 :
Omn
ipoten
t train
ing of
n-bit
mem
ory u
sing G
eneti
c Algo
rithm
Inpu
t: untra
ined_
MEM
: an u
ntrain
ed sa
mple
of a n
anoc
ell;
Initia
l_P: in
itial p
opula
tion
of ‘M
’ mole
cules
with
each
eithe
r ‘ON
’ or ‘O
FF’;
fitnes
s_M
EM():
fitne
ss fu
nctio
n for
n-bit
mem
ory;
/*It d
efine
s the
desir
ed ou
tput v
oltag
e lev
els fo
r Write
/Rea
d mo
des o
f n-b
it mem
ory *
/nu
m_G
EN: n
umbe
r of g
ener
ation
s;m
ax_G
EN: m
axim
um n
umbe
r of g
ener
ation
s;Ou
tput
:tra
ined_
MEM
: A tr
ained
n-b
it mem
ory d
evice
;St
eps:
1: Ini
tializ
e Gen
etic A
lgorith
m (G
A) w
ith po
pulat
ion in
itial_P
;2:
for n
um_G
EN=1
to m
ax_G
ENdo
(i): M
odify
the s
tate
of ea
ch m
olecu
le to
‘ON’
or ‘O
FF’ in
unt
raine
d_M
EM;
(ii): S
imula
te the
mod
ified u
ntra
ined
_ M
EM u
sing H
SPIC
E;(iii
) Stor
e ou
tput v
oltag
e (O
ut) v
alues
for R
ead a
nd W
rite op
erati
ons f
or al
l n bi
ts in
volta
ge_V
AL;
(iii):
The G
A ev
aluate
s fitne
ss fu
nctio
n fitn
ess_
MEM
() us
ing vo
ltage
_VAL
;(iv
): It
quan
tifies
the
fitnes
s of e
ach i
ndivi
dual
and g
ener
ates n
ew po
pulat
ion;
end
3: Th
e GA
conv
erge
s and
prov
ides a
s outp
ut:(i)
:the
optim
al se
t of
molec
ules,
each
with
state
‘ON’
or ‘O
FF’;
(ii):th
e tra
ined_
MEM
by us
ing th
ese c
ontro
l volt
age v
alues
;re
turn
traine
d_M
EM;
WM
SC 1
6-Ja
n-20
15
Sim
ulat
ion
Resu
lt of
100
0 sa
mpl
es o
f 1-b
it m
emor
y
WM
SC 1
6-Ja
n-20
15
Prop
osed
Arc
hite
ctur
e of
Nan
ocel
l bas
ed
Mul
ti-bi
t Mem
ory
WM
SC 1
6-Ja
n-20
15
Expe
rimen
tal S
etup
for M
orta
l Tra
inin
g of
M
olec
ular
Mem
ory
WM
SC 1
6-Ja
n-20
15
Algo
rithm
-Mor
tal T
rain
ing
ALGO
RITH
M 2 :
Mor
tal tr
aining
of n-
bit m
emor
y usin
g Gen
etic A
lgorith
mIn
put: un
traine
d_M
EM: a
n untr
ained
samp
le of
a nan
ocell
;In
itial_C
: initia
l pop
ulatio
n of r
ando
mly g
ener
ated
kCon
trol V
oltag
e Sign
als (C
VS);
/* Th
ese C
VS ar
e den
oted a
s Ci ,∀i= 1
to k
/* Ea
ch C
ivalu
e is e
ither
low
(Vlow
=0.5
V) o
r high
(Vhig
h=2.
0V)*/
fitnes
s_M
EM():
fitne
ss fu
nctio
n for
n-bit
mem
ory;
/*It d
efine
s the
desir
ed ou
tput v
oltag
e lev
els fo
r Write
/Rea
d mo
des o
f n-b
it mem
ory *
/nu
m_G
EN: n
umbe
r of g
ener
ation
s;m
ax_G
EN: m
axim
um n
umbe
r of g
ener
ation
s;Ou
tput
:tra
ined_
MEM
: A tr
ained
n-b
it mem
ory d
evice
;St
eps:
1: Ini
tializ
e Gen
etic A
lgorith
m (G
A) w
ith po
pulat
ion in
itial_C
;2:
for n
um_G
EN=1
to m
ax_G
ENdo
(i): M
odify
the v
alues
of C
ontro
l Volt
age s
ignals
in un
traine
d_M
EM;
(ii): S
imula
te the
mod
ified u
ntra
ined
_ M
EM u
sing H
SPIC
E;(iii
) Stor
e ou
tput v
oltag
e (O
ut) v
alues
for R
ead a
nd W
rite op
erati
ons f
or al
l n bi
ts in
volta
ge_V
AL;
(iii):
The G
A ev
aluate
s fitne
ss fu
nctio
n fitn
ess_
MEM
() us
ing vo
ltage
_VAL
;(iv
): It
quan
tifies
the
fitnes
s of e
ach i
ndivi
dual
and g
ener
ates n
ew po
pulat
ion;
end
3: Th
e GA
conv
erge
s and
prov
ides a
s outp
ut:(i)
:the
optim
al se
t of c
ontro
l volt
age
(Ci)
value
s;(ii)
:the
traine
d_M
EMby
using
thes
e con
trol v
oltag
e valu
es;
retur
n tra
ined_
MEM
;W
MSC
16-
Jan-
2015
Des
ign
Spac
e Ex
plor
atio
n fo
r Mor
tal T
rain
ing
ALGO
RITH
M 3 :
Des
ign S
pace
Exp
lorati
on fo
r Mor
tal tr
aining
of n
-bit m
olecu
lar m
emor
yIn
put: un
traine
d_M
EM: a
n untr
ained
samp
le of
a nan
ocell
cons
isting
of N
nano
partic
les;
num
_C: n
umbe
r of C
ontro
l Volt
age S
ignals
;nu
m_C
_ LO
C : n
umbe
r of n
anop
artic
les to
whic
h Con
trol V
oltag
e Sign
als ca
n be a
pplie
d;/*n
um_C
_ LO
C ⊂N */
/*num
_C_L
OC≤
(N−5
) */
/*num
_C_L
OC≥
num
C*/
Outp
ut:
traine
d_M
EM: s
et of
all su
cces
sfully
train
ed n
-bit m
emor
y dev
ices;
num
_MEM
: num
ber o
f train
ed n-
bit m
emor
y sam
ples i
n the
set tr
ained
_MEM
Step
s:
1: Ge
nera
te_C
(num
_C);
/*Gen
erate
a se
t Swh
ich co
nsist
s of a
ll bipa
rtition
s of n
um_C
Contr
ol Vo
ltage
Sign
als, C
i, ∀i=1 to n
um_C
.*/
/*Ass
ign {V
low=0
.5V
or V
high=
2.0
V} to
each
Ci j,∀i= 1
to n
um_C
, j =
1 to
size
of (
S). *
/ /*S
ave
these
in C
VS_V
ALUE
S.*/
;2:
Gene
rate
_C_L
oc(n
umC
LOC,
num
C);
/*Gen
erate
all p
ossib
le co
mbina
tions
of n
um_C
_LOC
nano
partic
les to
whic
h num
_Cco
ntrol
volta
ge si
gnals
can b
e con
necte
d. */
/*Sav
e the
se in
LOCA
TION
S.*/
;3:
Gene
rate
_Mem
(unt
raine
d_M
EM, L
OCAT
IONS
);/*M
odify
unt
raine
d_ M
EMby
using
Con
trol V
oltag
e Sign
al loc
ation
spec
ified i
n LOC
ATIO
NS */
/*Sav
e all
mem
ory s
ample
s in M
EM_M
ODIF
IED
*/ ;
4: Ge
nera
te_S
pfiles
(MEM
_MOD
IFIE
D, C
VS_V
ALUE
S);
/*For
eac
h mem
ory s
ample
in M
EM_M
ODIF
IED,
ass
ign C
ontro
l Volt
age
value
s fro
m CV
S_VA
LUES
.*//*S
ave
all ne
wly m
odifie
dmem
ory s
ample
s in S
P_FI
LES.
*/ ;
5: Si
mul
ate_
Mem
(SP_
FILE
S);
/*Sim
ulate
all m
emor
y sam
ples i
n SP_
FIL
ESus
ing H
SPIC
E. */
/*Sav
e ou
tput in
OUT
_FIL
ES.*/
;6:
Fetc
h_Me
m2b
it (O
UT_F
ILES
);/*S
elect
all th
ose
memo
ry sa
mples
, for w
hich n
-bit m
emor
y rea
d and
write
is do
ne in
acce
ptable
noise
mar
gin, u
sing t
he
OUT_
FILE
S.*/
/*Cop
y all s
electe
d mem
ory s
ample
s to
traine
d_M
EM. *
/;re
turn
traine
d_M
EM, n
um_M
EM;
WM
SC 1
6-Ja
n-20
15
Num
ber o
fna
nopa
rticle
s per
Na
noce
ll (N)
Num
ber o
f Mo
lecul
es(M
)
No. o
fsuc
cess
fully
train
ed n
anoc
ell in
stan
ces w
ith iC
VS
6 CVS
8C
VS
1
0 CVS
1
2 CVS
2097
20
00
2010
41
00
0
3022
327
135
0
3021
433
194
0
4039
910
355
132
4039
298
4310
1
Numb
er of
Mem
ory S
ample
s (c x
p)31
x 92
412
7 x 49
551
1 x 66
2047
x 1
Tabl
e 1.
Num
ber o
f suc
cess
fully
trai
ned
nano
cell
conf
igur
atio
n fo
r 2-b
it m
emor
y re
ad a
nd w
rite
oper
atio
n.
Her
e no
ise
mar
gin
for D
ata
read
ope
ratio
n on
out
put
volta
ge n
ode
is c
onsi
dere
d as
0.4
V.
WM
SC 1
6-Ja
n-20
15
Prob
abili
stic
Ana
lysi
s of
Nan
ocel
l in
Spat
ial
Dom
ain
Forc
orre
ctfu
nctio
ning
atle
asto
neof
the
min
imal
path
mus
tbe
pres
ent
betw
een
Inpu
tand
Out
putP
orts
.is
ane
cess
ary
cond
ition
.Th
epr
esen
ceof
mul
tiple
path
sw
illin
trodu
cere
dund
ancy
and
incr
ease
prob
abili
tyof
getti
ngco
rrec
tout
put.
Let
usm
odel
the
nano
cell
asa
plan
argr
aph
G(V
,E),
whe
rena
nopa
rticl
esar
eth
eno
des
and
mol
ecul
arsw
itche
sin
‘ON
’sta
tear
eth
eed
ges.
The
grap
hG
(V,E
)is
assu
med
tobe
adi
rect
iona
lgra
phsu
chth
atal
lthe
mol
ecul
esar
eor
ient
edin
sam
edi
rect
ion,
i.e.f
rom
inpu
tto
the
outp
ut.
WM
SC 1
6-Ja
n-20
15
•Th
epr
imar
yin
putp
orti
sth
ero
otno
dean
dth
epr
imar
you
tput
port
isth
ele
afve
rtex
ofth
egr
aph
G(V
,E).
•C
onsi
dera
nano
cell
with
‘N’n
anop
artic
les
and
‘M’m
olec
ular
switc
hes.
•A
ssum
eth
atth
ese
nano
parti
cles
are
alw
ays
pres
ent
and
mol
ecul
arsw
itche
sar
edi
strib
uted
byG
auss
ian
dist
ribut
ion
with
inth
ena
noce
ll.•
Am
olec
ular
switc
hii
spr
esen
tin
‘ON
’sta
tew
ithpr
obab
ility
p i.
•Th
ispr
obab
ility
that
ithmo
lecule
ispr
esen
tin
ONsta
teis
calle
dre
liabil
ityof
that
molec
uleat
thatin
stant
oftim
e,de
noted
asr(p
i)
WM
SC 1
6-Ja
n-20
15
•So
,the
prob
abilit
ytha
tthe
reis
noed
gein
betw
een
two
nano
partic
lesis
(1−
pi).A
gain,
thepr
obab
ilityt
hata
tleas
tone
edge
ispr
esen
tbetw
een
twon
odes
isgiv
enby
:
•Le
tuss
uppo
setha
tsma
llnan
ocell
cons
istso
fthr
eemo
lecula
rswi
tches
(m1,
m2an
dm3
)and
three
nano
partic
les(A
,Ban
dC)
,as
show
nin
Figur
ea.
•He
re,in
putv
oltag
eis
appli
edon
Aan
drec
eived
onC.
•Th
isca
nbed
one
viatw
omini
malp
aths:-
WM
SC 1
6-Ja
n-20
15
•Co
rrect
outpu
twill
bere
ceive
don
Cvia
path
ifboth
molec
ulesm
1an
dm2
are
pres
ent.
Simi
larly,
data
willb
eco
rrectl
yre
ceive
don
Cvia
path
ACif
themo
lecule
m3is
inON
state.
Thes
ear
ethe
two
redu
ndan
tpath
sand
atlea
ston
eof
them
shou
ldbe
worki
ngco
rrectl
yto
obtai
nco
rrect
outpu
t.Th
us,th
e•
Prob
abilit
yofr
eceiv
ingco
rrect
data
onno
deC
isgiv
enby
•P Pa
th=
(Pro
babil
itytha
tboth
m1an
dm2
are
pres
enti
n‘O
N’sta
teon
path
ABC)
or(P
roba
bility
thatm
3isp
rese
ntin
‘ON’
state
onpa
thAC
)
WM
SC 1
6-Ja
n-20
15
Eval
uatin
g re
liabi
lity
boun
ds fo
r exa
mpl
e ci
rcui
t
Miss
ing
Molec
ules
V(lo
w)V(
High
)Pa
th P
roba
bilit
y ( A
toC)
Uppe
r Bou
nd
Low
erBo
und
None
0.493
51.9
935
0.625
m10.4
951
1.995
10.5
00m2
0.495
11.9
951
0.500
m30.4
902
1.990
20.2
50m1
, m2
0.495
11.9
951
0.500
m2, m
30
00.0
00m1
, m3
00
0.000
m1, m
2, m3
00
0.000
Table
3. S
imula
tion R
esult
s for
an ex
ample
nano
cell c
onfig
urati
on w
hen
none
or so
me of
the m
olecu
les ar
e miss
ing
WM
SC 1
6-Ja
n-20
15
•In
other
word
s,5/8
=0.6
25or
there
are
62.5%
chan
ceso
fgett
ingco
rrect
outpu
tvolt
age.
Theo
retic
ally
•Su
bstitu
ting
•Th
us,
theor
etica
lan
dex
perim
ental
resu
ltsar
ema
tching
.
•Th
elas
tthr
eeca
ses
inthe
Table
deno
testhe
minim
alcu
tsets
forthi
snan
ocell
,den
oted
by,
•As
depic
tedfro
mTa
bleIII,
outpu
tvo
ltage
isno
tre
ceive
dfor
these
case
s. WM
SC16
-Jan
-201
5
•Fu
rther,
toev
aluate
there
liabil
ityof
atra
ined
nano
cell,
weas
sume
thatt
here
are
kred
unda
ntpa
thsfro
minp
utto
outpu
t.Ea
chof
these
paths
may
vary
inlen
gth.
The
length
ofan
ypa
thi
can
bere
pres
ented
byva
riable
•Th
atis,
each
path
cons
ists
ofl im
olecu
lesco
nnec
tedin
serie
san
dsu
chkp
athsa
rewo
rking
inpa
ralle
l.
•Fo
rcor
rect
functi
oning
ofthe
syste
m,at
least
one
ofthe
ipath
smu
stfun
ction
corre
ctly.
•Co
nside
ran
indica
torva
riable
x ijwh
ichde
notes
thesta
teof
molec
ulars
witch
jonp
athi.
WM
SC 1
6-Ja
n-20
15
•W
edefi
neas
tructu
refun
ction
φ(x) ifo
rpath
ias
•Th
enstr
uctur
efun
ction
ofthe
whole
syste
mca
nbe
given
as:
•He
nce,
there
liabil
ityr(p
)oft
hewh
olesy
stem
atan
yins
tanto
ftime
isgiv
enas WM
SC 1
6-Ja
n-20
15
•Le
t’sco
nside
rano
there
xamp
leha
ving
multip
lepa
thsfro
minp
utto
outpu
tinfig
ureb
.
•Th
enod
eAis
input
andn
ode
Fis
outpu
t
•Th
e stru
cture
func
tion
for al
l mini
mal p
aths f
rom
A to
F ar
e
WM
SC 1
6-Ja
n-20
15
•R
elia
bilit
y is
com
pute
d as
:-
WM
SC 1
6-Ja
n-20
15
Boun
ds o
n re
liabi
lity
of a
nan
ocel
l
•Le
tA1
,A2,…
……
.,As
deno
temi
nimal
path
sets
conn
ectin
ginp
utno
deto
theou
tputn
ode
andw
edefi
neFi,
i=1…
…..s
as
•By
failin
gof
themo
lecula
rcon
necti
on,w
eme
anto
sayt
hat,
molec
uleis
in‘O
FF’s
tate.
Ifat
least
one
ofthe
molec
ules
inthe
minim
alpa
thse
thas
failed
,the
syste
mwi
llfai
leve
ntuall
y.Ma
thema
ticall
y,it
isde
noted
as:
•He
ncefo
rth, it
can b
e eas
ily de
rived
that
failur
e of a
t leas
t one
mo
lecule
in m
inima
l path
Ai c
an in
creas
e the
prob
abilit
y of fa
ilure
of at
lea
st on
e mole
cule
in pa
th Aj
WM
SC 1
6-Ja
n-20
15
•Th
iswo
uldbe
theca
seifb
othpa
thsAi
andA
jove
rlap.
So-
•Si
milar
ly
•Su
bstitu
ting
ineq
uatio
nsta
tedab
ove
wege
t
WM
SC 1
6-Ja
n-20
15
•Ag
ain,l
etC1
...C
rden
otethe
minim
alcu
tsets
.We
defin
ethe
even
tsE1
,...
,Erb
y
•Si
nce,
thena
noce
llwill
functi
oniff
allof
theev
ents
Eioc
cur,w
esay
•He
nce
WM
SC 1
6-Ja
n-20
15
•So
thebo
unds
onre
liabil
ityfun
ction
given
as
•Co
nside
rthe
same
exam
plesh
own
infig
ure
a,the
relia
bility
boun
dsar
eexp
ress
edas
:
WM
SC 1
6-Ja
n-20
15
•Su
bstitu
ting
•W
eget
–
•Th
ese
relia
bility
boun
dsma
tchthe
value
sco
mpute
din
colum
n4
ofTa
ble.
•Th
isex
ample
can
bege
nera
lized
forna
noce
llsco
nsist
ingof
more
than
three
nano
partic
lesan
dmole
cules
ands
imila
rres
ultsc
anbe
obtai
ned.
WM
SC 1
6-Ja
n-20
15
EXPE
RIM
ENTA
L SE
TUP
FOR
RELI
ABIL
ITY
EVAL
UATI
ON
WM
SC 1
6-Ja
n-20
15
ALGO
RITH
M 3:
Nan
ocell
Relia
bility
Pre
dictio
n Algo
rithm
(NRP
A)In
put: N
:= N
umbe
r of N
anop
artic
les;
IP:=
Prim
ary I
nput
Node
;OP
:= P
rimar
y Outp
ut No
de ;
P mol
:= P
roba
bility
of a
molec
ule be
ing pr
esen
t betw
een t
wo na
nopa
rticles
and f
ound
in ‘O
N’sta
te ;
Outp
ut:
P pat
h:=
Pro
babil
ity th
at at
least
one p
ath ex
ists b
etwee
n IP
and O
P;St
eps:
1: Ini
tializ
ation
;2:
Adj_M
atrix
= Ge
nera
te M
atrix
(N,N
,µ,σ
) ;/*
Gene
rate
a NxN
rand
om a
rray (
Adjac
ency
Matr
ix) w
ith G
auss
ian D
istrib
ution
*/ ;
3: Ed
ge_M
atrix
= Co
nver
t_to
_Edg
eMat
rix (A
dj_M
atrix
) ;/*
Conv
ert th
is Ad
jacen
cy M
atrix
to Ed
ge M
atrix
*/ ;
4: Fw
d_po
inting
_ ar
ray =
Gen
erat
e Dag
(Edg
e_M
atrix
) ;/*
Remo
ve se
lf loo
ps an
d bac
kwar
d poin
ting e
dges
from
this
matrix
*/ ;
5:M
= si
ze (F
wd_p
ointin
g_ar
ray,1
) ;/*
Calcu
late N
umbe
r of M
olecu
les, d
enote
d as
’M’ *
/ ;6:
Nano
cell_
Mat
[N][N
] = W
eight
ed M
atrix
(Fwd
_poin
ting_
arra
y,Pm
ol) ;
/* Ad
d weig
ht to
each
conn
ected
edg
e = P
mol
and u
ncon
necte
d ed
ge =
∞ of
Fwd
_poin
ting_
arra
y*/ ;
7: sh
orte
st_pa
th=
Dijks
tra_k
(Nan
ocell
_Mat
,IP,O
P,k) ;
/* Ca
lculat
e ksh
ortes
t path
s fro
m IP
to O
P us
ing m
odifie
d Di
jkstra
algor
ithm
*/ ;
8: P M
= As
sign_
Prob
(Pm
ol);
/* As
sign p
roba
bilitie
s to e
ach m
olecu
le */
;9:
P pat
h=
Com
pute
_Pro
b(sh
orte
st_pa
th, P
M);
/* Ca
lculat
e pro
babil
ity P
path
, as e
xplai
ned i
n Sec
tion 4
*/ ;
retu
rn P
path
;W
MSC
16-
Jan-
2015
Sim
ulat
ion
Resu
lts
WM
SC 1
6-Ja
n-20
15
WM
SC 1
6-Ja
n-20
15
Low
er B
ound
WM
SC 1
6-Ja
n-20
15
Uppe
r Bou
nd
WM
SC 1
6-Ja
n-20
15
Dis
tribu
tion
func
tion
for n
umbe
r of
conn
ecte
d pa
ths
WM
SC 1
6-Ja
n-20
15
PROB
ABIL
ITY
ANAL
YSIS
OF
NAN
OCEL
L IN
TIM
E D
OMAI
N
WM
SC 1
6-Ja
n-20
15
Augm
ente
d Co
ntin
uous
Par
amet
er B
irth
Dea
th
Mod
el•
Cons
ider a
nano
cell h
aving
N na
nopa
rticles
conn
ected
via M
mole
cular
switc
hes.
•Le
t (t) deno
te the
numb
er of
‘ON’
mole
cules
in th
e nan
ocell
, at ti
me t
•Le
t Sj(t)
=
j(t) de
notes
the s
uper
-state
of th
e sys
tem.
•Le
t the s
tate s
pace
of th
e pro
cess
be I =
{0, 1
, 2,…
,M} a
nd T
= [0
;1) be
its pa
rame
ter
spac
e. Th
at is,
each
supe
r-stat
e den
otes n
umbe
r of ‘
ON’ m
olecu
les an
d it c
onsis
ts of
one o
r mor
e sub
-state
s. •
The t
rans
ition f
rom
a give
n stat
e to a
nothe
r stat
e can
take
plac
e at a
ny in
stant
of tim
e. •
Thus
, as s
hown
in F
ig. , a
t any
time t
, one
of th
e sub
-state
Sk j,
k = 1
to M C
jof a
supe
r-sta
te S j
, can
mak
e1)
a do
wn tr
ansit
ion to
one o
f the (
M -j)
sub-
states
of S
j+1 s
uper
-state
, with
failu
re
rate
j.2)
an up
tran
sition
to on
e of th
e j su
b-sta
tes of
Sj-1
supe
r-stat
e, wi
th re
pair r
ate
j.
WM
SC 1
6-Ja
n-20
15
Hie
rarc
hica
l CTM
C m
odel
WM
SC 1
6-Ja
n-20
15
Exam
ple
Nan
ocel
l
WM
SC 1
6-Ja
n-20
15
Algo
rithm
WM
SC 1
6-Ja
n-20
15
Sim
ulat
ion
Resu
lts
WM
SC 1
6-Ja
n-20
15
Sim
ulat
ion
Resu
lt
No. of Success per case
No. of Success per case
WM
SC 1
6-Ja
n-20
15
Prob
abili
ty o
f Sta
te R
each
abili
ty
WM
SC 1
6-Ja
n-20
15
Expe
cted
Nan
ocel
l Life
time
WM
SC 1
6-Ja
n-20
15
Sim
ulat
ion
Resu
lt
WM
SC 1
6-Ja
n-20
15
Post-
fabric
ation
Tra
ining
of N
anoc
ell M
olecu
lar M
emor
yRe
liabil
ity A
nalys
is of
Nano
cell i
n pre
senc
e of T
rans
ient E
rrors
Futur
e Wor
k
CON
CLUS
ION
S AN
D F
UTUR
E W
ORK
WM
SC 1
6-Ja
n-20
15
•Th
e Nan
ocell
base
d mole
cular
mem
ory h
as be
en m
odele
d & tr
ained
using
•
Omni
pote
nt T
rain
ing
•Mo
rtal T
rain
ing
•Th
e sim
ulatio
n res
ults o
f 1-b
it mole
cular
mem
ory a
re co
mpar
ed ag
ainst
analy
tical
prob
abilis
tic m
odeli
ng ap
proa
ch.
•It i
s obs
erve
d tha
t 20 o
r mor
e mole
cules
mus
t be p
rese
nt in
the na
noce
ll for
relia
ble
molec
ular m
emor
y buff
er.
•Th
e con
verg
ence
time i
s high
in pr
opos
ed m
ethod
ology
for t
raini
ng m
emor
y. So
me ne
w ad
aptiv
e meth
od is
to be
appli
ed to
redu
ce th
e tra
ining
time.
•Th
e pro
babil
istic
mode
ling a
nd an
alysis
of na
noce
ll is d
one i
n pre
senc
e of s
oft tr
ansie
nt er
rors.
•
An ex
tende
d Con
tinuo
us P
aram
eter B
irth D
eath
Mode
l for a
Nan
ocell
is pr
opos
ed.
•It i
s obs
erve
d tha
t the n
anoc
ell w
ill fun
ction
relia
bly as
long
as fa
ilure
rate
is les
s tha
n re
pair r
ate.
Conc
lusi
ons
and
Furth
er w
orks
WM
SC 1
6-Ja
n-20
15
•Th
e des
ign of
high
dens
ity M
olecu
lar M
emor
y is u
nder
prog
ress
.•
Reali
zatio
n of N
anoc
ell M
olecu
lar M
emor
y.•
The M
olecu
lar M
odel
need
be im
prov
ed
•Th
e effe
ct on
nano
cell d
evice
beha
vior w
hen m
ore t
han o
ne m
olecu
le ar
e pre
sent
betw
een t
wo na
nopa
rticles
. Furth
er w
orks
cont
d.
WM
SC 1
6-Ja
n-20
15
1.R.
Kuma
wat,V
.Sah
ula,a
ndM.
S.Ga
ur,“
Prob
abilis
ticmo
delin
gand
analy
sisof
molec
ular
memo
ryce
ll”,AC
MJo
urna
lon
Emer
ging
Tech
nolog
iesin
Com
putin
gSy
stem
(ACM
JETC
)2.
Renu
Kuma
wat,
Vine
etSa
hula,
Mano
jSi
ngh
Gaur
,“A
Prob
abilis
ticMo
del
forRe
liabil
ityEv
aluati
onof
Nano
cell
inpr
esen
ceof
Tran
sient
Erro
rs",(
Acce
pted
inIE
TDi
gital
&Co
mpute
r)
Publ
icat
ions
In P
eer-r
evie
wed
Jour
nals
WM
SC 1
6-Ja
n-20
15
[1]Int
erna
tiona
ltech
nolog
yroa
dmap
forse
mico
nduc
tors,.
2009
,201
0.[2]
Y.Ch
en,D
.A.A
.Jun
g,G.
and
Ohlbe
rg,X
.Li,
D.R.
Stew
art,
J.O.
Jepp
esen
,K.A
.Niel
sen,
J.Fr
aser
Stod
dart,
and
R.S.
Willi
ams,
“Nan
osca
lemo
lecula
r-swi
tchcro
ssba
rcirc
uits,”
Nano
tech
nolog
y,vo
l.14,
no.4
,pp.
462.4
68.,
2003
[3]W
.Wu,
G.-Y
.Jun
g,D.
Olyn
ick,J
.Stra
znick
y,Z.
Li,X.
Li,D.
Ohlbe
rg,Y
.Che
n,S.
-Y.W
ang,
J.Lid
dle,W
.Ton
g,an
dR.
S.W
illiam
s,“O
ne-ki
lobitc
ross
-bar
molec
ularm
emor
ycirc
uitsa
t30-
nmha
lf-pitc
hfab
ricate
dby
nano
impr
intlith
ogra
phy,.
”App
lied
Phys
icsA:
Mat
erial
sScie
nce
amp;
Proc
essin
g,vo
l.80,
pp.1
173.
1178
,200
5,10
.100
7/s0
0339
-004
-317
6-y.
[4]J.
E.Gr
een,
J.W
ookC
hoi,
A.Bo
ukai,
Y.Bu
nimov
ich,E
.Joh
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WM
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WM
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WM
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