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MPRI 3 Dec 2007 Catuscia Palamidessi1
Why Probability and Nondeterminism?Concurrency Theory
• Nondeterminism– Scheduling within parallel composition– Unknown behavior of the environment– Underspecification
• Probability– Environment may be stochastic– Processes may flip coins
MPRI 3 Dec 2007 Catuscia Palamidessi2
Automata
A = (Q , q0 , E , H , D)
Transition relationD Q (EH) Q
Internal (hidden) actions
External actions: EH =
Initial state: q0 Q
States
MPRI 3 Dec 2007 Catuscia Palamidessi3
Example: Automata
A = (Q , q0 , E , H , D)
coffee
q0 q2
q1
q4
q3 q5
d
n
n
n
choc
ch
Execution: q0 n q1 n q2 ch q3 coffee q5
Trace: n n coffee
MPRI 3 Dec 2007 Catuscia Palamidessi4
Probabilistic Automata
PA = (Q , q0 , E , H , D)
Transition relationD Q (EH) Disc(Q)
Internal (hidden) actions
External actions: EH =
Initial state: q0 Q
States
MPRI 3 Dec 2007 Catuscia Palamidessi5
Example: Probabilistic Automata
q0
q1
q2
q3
q4
q5fair
unfair
flip
flip
1/2
1/2
2/3
1/3
beep
MPRI 3 Dec 2007 Catuscia Palamidessi6
Example: Probabilistic Automata
q0
qh
qt
qpflip
flip
1/2
1/2
2/3
1/3
beep
MPRI 3 Dec 2007 Catuscia Palamidessi7
Example: Probabilistic Automata
q0
q1
q2
q3
q4
q5fair
unfair
flip
flip
1/2
1/2
2/3
1/3
beep
What is the probability of beeping?
MPRI 3 Dec 2007 Catuscia Palamidessi8
Example: Probabilistic Executions
q0 q1 q3
q4
q5
1/2
1/2
q0 q2
q3
q4
q5
unfair flip
2/3
1/3
beep(beep) = 2/3
(beep) = 1/2
fair beepflip 1/2
2/3
MPRI 3 Dec 2007 Catuscia Palamidessi9
Example: Probabilistic Executions
q0
q1
q2
q3
q4
q3
q4
q5
q5
fair
unfair
flip
flip
beep
beep
1/21/2
2/3
1/3
1/2
1/2
1/4
2/6
7/12
MPRI 3 Dec 2007 Catuscia Palamidessi10
• Sample set– Set of objects
• Sigma-field (-field)– Subset F of 2 satisfying
• Inclusion of • Closure under complement• Closure under countable union• Closure under countable intersection
• Measure on (,F)– Function from F to
• For each countable collection {Xi}I of pairwise disjoint sets of F, (IXi) = I(Xi)
• (Sub-)probability measure– Measure such that () = 1 (() 1)
• Sigma-field generated by C 2
– Smallest -field that includes C
Measure Theory
Example: set of executions
Study probabilities of sets of executions which sets can I measure?
MPRI 3 Dec 2007 Catuscia Palamidessi11
Measure Theory
Why not F = 2?
Flip a fair coin infinitely many times
= {h,t}
() = 0 for each
(first coin h) = 1/2
Theorem: there is no probability measure on 2 such that () = 0 for each
MPRI 3 Dec 2007 Catuscia Palamidessi12
Theorem A measure on cones extends uniquely to a measure on the -field generated by cones
q0
q1
q2
q3
q4
q3
q4
q5
q5
fair
unfair
flip
flip
beep
beep
1/21/2
2/3
1/3
1/2
1/2
Cones and Measures
• Cone of – Set of executions with prefix – Represent event “ occurs”
• Measure of a cone– Product edges of
C
MPRI 3 Dec 2007 Catuscia Palamidessi13
Examples of Events
• Eventually action a occurs– Union of cones where action a occurs once
• Action a occurs at least n times– Union of cones where action a occurs n times
• Action a occurs at most n times– Complement of action a occurs at least n+1 times
• Action a occurs exactly n times– Intersection of previous two events
• Action a occurs infinitely many times– Intersection of action a occurs at least n times for all n
• Execution occurs and nothing is scheduled after– Set consisting of only– C intersected complement of cones that extend
MPRI 3 Dec 2007 Catuscia Palamidessi14
Schedulers - Resolution of nondeterminism
Scheduler
Function : exec*(A) Q x (E H) x Disc(Q)
if() = (q,a,) then q = lstate()
Probabilistic execution generated by from state r
Measure r(Cs) = 0 if r s
rr(Cr) = 1
r(Caq) = r(C)q) if() = (q,a,)
MPRI 3 Dec 2007 Catuscia Palamidessi15
Probabilistic CCS
P :: = 0 | P|P | .P | P + P | () P | X | let X = P in X | PpP
Probabilistic processes
.P (P)
Prefix
P1pP2 p1+ (1-p)2
P
Nondeterministic process
P + Q
MPRI 3 Dec 2007 Catuscia Palamidessi16
Probabilistic CCS
Communication
P
Interleaving
P1P2
P|Q |Q
P1 (P2) P2 (P2)
(P2 | P2)
âa
Hiding
(a) P
P
(a)
Recursion
P[ let X = P in X / X ]
X
a , â
let X = P in
MPRI 3 Dec 2007 Catuscia Palamidessi17
Bisimulation Relations
We have the following objectives
• They should extend the corresponding relations in the non probabilistic case
• Keep definitions simple
• Where are the key differences?
MPRI 3 Dec 2007 Catuscia Palamidessi18
Strong Bisimulation on Automata
Strong bisimulation between A1 and A2
Relation R Q x Q,
Q=Q1Q2, such that
q0
q1
q3
q2
q4
s0
s1
s3
a a
b
a
b b
s s
q qa
a
R R
q, s, a, q s
+
MPRI 3 Dec 2007 Catuscia Palamidessi19
Strong Bisimulation on Probabilistic Automata
Strong bisimulation between A1 and A2
Relation R Q x Q,
Q=Q1Q2, such that
q0
q1
q3
q2
q4
s0
s1
s3
a
b
a
b b11
C Q/R . (C ) = (C )
s
q a
a
R R
q, s, a,
1
1
R
+
[LS89]
MPRI 3 Dec 2007 Catuscia Palamidessi20
Probabilistic Bisimulations• These two Probabilistic Automata are not bisimilar
• Yet they satisfy the same formulas of a logic PCTL– The logic observes probability bounds on reachability properties
• Bisimilar if we match transitions with convex combinations of transitions
q1
q2 q3 q2 q3 q2 q3
.2 .8 .3 .7 .4 .6
s1
s2 s3 s2 s3
.2 .8 .4 .6
a a a a a
b c b c b c b c b c
~
~p
MPRI 3 Dec 2007 Catuscia Palamidessi21
Weak Bisimulation on Automata
Weak bisimulation between A1 and A2
Relation R Q x Q,
Q=Q1Q2, such that
q0
q1
q3
q2
q4
s1
s3
bb b
s s
q qa
a
R R
q, s, a, q s
+
s s
: trace()=a, fstate()=s, lstate()=s
a
MPRI 3 Dec 2007 Catuscia Palamidessi22
Weak bisimulation onProbabilistic Automata
q0
q1
q3
q2
q4
s1
s3
bb b
Weak bisimulation between A1 and A2
Relation R Q x Q,
Q=Q1Q2, such that
s
q a
a
R R
q, s, a,
11 1
C Q/R . (C ) = (C )
R
+
[LS89]
MPRI 3 Dec 2007 Catuscia Palamidessi23
Weak Transition
There is a probabilistic execution such that
– (exec*) = 1
– trace() = (a)
– fstate() = (q)
– lstate() =
q a
(it is finite)
(its trace is a)
(it starts from q)
(it leads to )
q siff: trace()=a, fstate()=q, lstate()=s
a
MPRI 3 Dec 2007 Catuscia Palamidessi24
Exercises
• Prove that the probabilistic CCS is an extension of CCS (to define what this means is part of the exercise)
• Prove that probabilistic bisimulation is an extension of bisimulation
• Write the Lehmann-Rabin algorithm in probabilistic CCS (without using guarded choice)