Post on 23-Feb-2016
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The Algorithmicsof
Probabilistic AutomataWeak Bisimulation
Andrea TurriniSaarland University
based on a joint work with
Holger Hermanns
Andrea TurriniSaarland University
Probabilistic Automata (PA)s
u
hf tf hu ntu
flip flip flip retry
coin
4/16 8/16 8/16 12/16
beepu beepf flashf flashu
sms
coin
f
vibrate
sms sms sms sms
Andrea TurriniSaarland University
Probabilistic Automata (PA)
PA = (Q, q0, E, H, D)Transition relation D Q (E ) (Q)
Internal (hidden) actions
External actions: E H =
Initial state: q0 Q
States
H Disc
Andrea TurriniSaarland University
Weak bisimulation
p 1-p
Weak bisimulation between A1 and A2
Equiv. Rel. R Q QQ = Q1 Q2, such that+
bbb
s0
s1 s2 t1
t3s3 s4
r
m
s, t, a,
s
t
R R
a
a
C Q/R . m (C) = r (C)
m R r [JL91]
t t
m r
Andrea TurriniSaarland University
Usual decision procedure
Bisimulation(A1, A2)
W := {Q1 Q2}
(C, a) := FindSplit(W)
while C do W := Refine(W, a, C)
(C, a) := FindSplit(W)
return W
+
FindSplit(W)
for each (s, a, ) D1 D2
for each t [s]W
if there does not exist such that t and W return ([s]W, a)
return (, a)
a r rmr
m
NN
N
N = max (| D1 D2 |, | Q1 Q2 |)+ +
+
Andrea TurriniSaarland University
Weak transition
{beep: 5/16, flash: 7/16, vibrate: 4/16} {beep: 2/16, flash: 2/16, vibrate: 12/16}
s
u
hf tf hu ntu
flip flip flip retry
coin
4/16 8/16 8/16 12/16
beepu beepf flashf flashu
coin
f
vibrate
sms sms sms sms sms
Andrea TurriniSaarland University
How many weak transitions?
C00: bbbbbC01: bbbbtC02: bbbtb
C30: ttttbC31: ttttt
t t t t t
t t t t t
Andrea TurriniSaarland University
Weak transition as Flow
{beep: 5/16, flash: 7/16, vibrate: 4/16}
s
u
hf tf hu ntu
flip flip flip retry
coin
4/16 8/16 8/16 12/16
beepu beepf flashf flashu
coin
f
vibrate
sms sms sms sms sms
Andrea TurriniSaarland University
Weak transition vs. Flow Problems
u
hf tf hu ntu
flip flip flip retry
coin
4/16 8/16 8/16 12/16
beepu beepf flashf flashu
coin
f
vibrate
{beep: 0/16, flash: 16/16, vibrate: 0/16}
16/16
16/16 16/16 16/16 16/16 16/16
16/16 16/16
16/16
sms sms sms sms sms
Andrea TurriniSaarland University
Flow as Linear Programming Problem
f0 = 1f9 + f12 = 5/16f10 + f11 = 7/16f13 = 4/16
s
u
hf tf hu ntu
8/16
beepu vibratebeepf flashf flashu
f
8/16 12/16 4/16
0 ≤ fi ≤ ci (< )f0 + f8 = f1 + f4
f1 = f2 + f3 f2 = f9
f3 = f10
f4 = f5 + f6 + f7 f5 = f8 + f11 f6 = f12 f7 = f13
f1 f4
f0
f2 f3 f5 f6 f7f8
f9 f10 f11 f12 f13
min S fi
under constraints
{beep: 5/16, flash: 7/16, vibrate: 4/16}f2 = 8/16(f2 + f3)f3 = 8/16(f2 + f3)f5 = 12/16(f5 + f6)f6 = 4/16(f5 + f6)
beep
flash
vibrate
Complexity:#variables O(|Q||D|)
#constraints O(|Q||D|)
Polynomial!s
1
8/16 8/16
4/16 4/16
4/16 4/16
4/16
4/161/16
1/16
3/16
3/16
0
✓
Andrea TurriniSaarland University
Usual decision procedure
Bisimulation(A1, A2)
W := {Q1 Q2}
(C, a) := FindSplit(W)
while C do W := Refine(W, a, C)
(C, a) := FindSplit(W)
return W
+
FindSplit(W)
for each (s, a, m) D1 D2
for each t [s]W
if there does not exist r such that t r and m W r
return ([s]W, a)
return (, a)
a
N
N
N
N = max (| D1 D2 |, | Q1 Q2 |)+ +
+
Polynomial
Polynomial!
Andrea TurriniSaarland University
From bisimulation to minimal automaton
A = (Q, q0, E, H, D)
QA = (W, [q0]W, E, H, DW)
mA = (W, [q0]W, E, H, DW)
Quotienting
Transition reduction
zeroconf-nt: |Q| = 670, |D| = 827
zeroconf-nt: |W| = 41, |DW| = 55
zeroconf-nt: |W| = 41, |DW| = 52
Andrea TurriniSaarland University
Adding costs to PAc
h0 h1 hn-1 hnp1-p
tr …p
1-p
tr
p1-p
tr
ds 1 1
r2 r2 r2 r2
Overall cost: 1 + nr2/p + 1
p1-p
tr
C(csh0) = 1 x 1C(h0trh1) = r2 x pC(h0trh0trh1) = (r2 + r2) x (1-p) x pC(h0trh0trh0trh1) = (r2 + r2 + r2) x (1-p) x (1-p) x pC((h0tr)ih1) = (i + 1) x r2 x (1-p)i x p
r2 / p
Andrea TurriniSaarland University
Adding costs to PAc
h0 h1 hn-1 hnp1-p
tr …p
1-p
tr
p1-p
tr
ds 1 1
r2 r2 r2 r2
Minimum cost equals optimal LP solution
p1-p
tr
c
h0 h1 hn-1 hnp1-p
…p
1-pp
1-p
1
1r2 r2 r2 r2
p1-p
r2r2r2r2
c
1
1
1
1
(1-p) / p
1
(1-p) / p
1
(1-p) / p
1
(1-p) / p1
0
0
Polynomial!
Andrea TurriniSaarland University
Nondeterminism and Cost PA
2 + 89/p cost 2 + 100/p
c
h0 h1 h3 h4p1-p
t5 h2 p1-p
t5
p1-p
t5
ds 1 1
52 52 52 52
p1-p
t5
q1 q3 q4p1-p
t4 q2 p1-p
t4
p1-p
t4
42 42 42
q5p1-p
t4
42
o o00 o
0o
0
Andrea TurriniSaarland University
Comparing Cost PA
2 + 89/p
2 + 80/pw
k0 k1 k3 k4p1-p
t4 k2 p1-p
t4
p1-p
t4
ds 1 1
42 42 42 42
p1-p
t4 k5p1-p
t4
42
c
h0 h1 h3 h4p1-p
t5 h2 p1-p
t5
p1-p
t5
ds 1 1
52 52 52 52
p1-p
t5
q1 q3 q4p1-p
t4 q2 p1-p
t4
p1-p
t4
42 42 42
q5p1-p
t4
42
o o00
o0
o0
Andrea TurriniSaarland University
Minor Cost Weak Bisimulation
Andrea TurriniSaarland University
Comparing Cost PA
2 + 89/p
2 + 80/pw
k0 k1 k3 k4p1-p
t4 k2 p1-p
t4
p1-p
t4
ds 1 1
42 42 42 42
p1-p
t4 k5p1-p
t4
42
c
h0 h1 h3 h4p1-p
t5 h2 p1-p
t5
p1-p
t5
ds 1 1
52 52 52 52
p1-p
t5
q1 q3 q4p1-p
t4 q2 p1-p
t4
p1-p
t4
42 42 42
q5p1-p
t4
42
o o00
o0
o0
Andrea TurriniSaarland University
Conclusion• Weak transitions as LP problems• Polynomial decision procedure for PA
weak probabilistic bisimulation• Minimal automaton• Cost decorated PA• Prototypical implementation
• Extensions of LP approach– Specific weak transitions, hyper-transitions,
restricted transitions, equivalence matching, …– MDP multi-objective reachability– Interval multi-objective reachability– Multi-objective reachability in interval models
• Base for other models: Markov Automata
Andrea TurriniSaarland University
Decision procedure in practice