Probabilistic Automata and Logics Probabilistic Automata and Logics (PAuL) 2006 (PAuL) 2006 Approximate Simulations for Approximate Simulations for Task-Structured Task-Structured Probabilistic I/O Automata Probabilistic I/O Automata Sayan Mitra and Nancy Lynch Sayan Mitra and Nancy Lynch CSAIL, MIT CSAIL, MIT
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Approximate Simulations for Task-Structured Probabilistic I/O Automata
Approximate Simulations for Task-Structured Probabilistic I/O Automata. Sayan Mitra and Nancy Lynch CSAIL, MIT. Implementation. Implementation or simulation is a fundamental notion in concurrency theory “traces” or observable behavior, e.g. sequence of events, timing of events, probabilities - PowerPoint PPT Presentation
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Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006
Approximate Simulations for Task-Approximate Simulations for Task-Structured Probabilistic I/O AutomataStructured Probabilistic I/O Automata
Sayan Mitra and Nancy LynchSayan Mitra and Nancy Lynch
CSAIL, MITCSAIL, MIT
MIT, Computer Sc. and AI Lab
Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006Sayan Mitra
Implementation
Implementation or simulation is a fundamental notion in concurrency theory
“traces” or observable behavior, e.g. sequence of events, timing of events, probabilities
A implements B if traces(A) traces(B) A is equivalent to B if they implement each
other, i.e., traces(A) = traces(B)
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Fragility
waitd(x) = 1
x ≤ a
stopd(x) = 0
Jump
x = a
waitd(x) = 1x ≤ a + ε
stopd(x) = 0
Jump
x = a + ε
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Unequal, but similar
A metric d on the space T of traces of A (and B) (T,d) is a metric space A approximately implements B if the one-sided Hausdorff
distance from traces(A) to traces(B) is small.
A is approximately equivalent to B if the Hausdorff distance from traces(A) to traces(B) is small.
traces(A) traces(B)
traces(B)
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Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006Sayan Mitra
Previously
Metric-based approximate simulations and bisimultions PIOA [Jou and Smolka 1990] Labelled Markov Processes [Desharnais, et. al.
Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006Sayan Mitra
Outline
Background
Task PIOA vocabulary
Definitions: metrics and simulations
Soundness (sketch)
Discussions Generalization
Applications
Future directions
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Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006Sayan Mitra
Task PIOA
A = (Q,v,A,D,R) [Canetti, et. al. 2006] Countable set of states Q Initial distribution on states v Countable set of actions A = I O H
If I = then A is closed O H set of locally controlled actions
Set of (q,a,µ) transitions D An equivalence R relation on locally controlled actions
Each equivalence class of R is a task
Input enabled: for every state q and input action a, there exists (q,a,µ) Transition deterministic: for every state q and action a, there is at most one (q,a,µ) Action deterministic: for every state q and task T, there is at most one a in T enabled
at q
Nondeterministic choice over tasks.
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Task PIOA Vocabulary
Execution fragment α = q0a1q1a2… α is an execution if q0 in supp(v) trace(α) is obtained by deleting all q’s and the a’s in H.
trace is a measurable function Scheduler for resolving nondeterminism
In general a scheduler is a mapping from execution fragments to (sub-) distributions over transitions
Task scheduler σ is a sequence of tasks T1 T2 T3… apply(µ,σ) gives a probability distribution over fragments (sigma
algebra generated by cones of fragments) tdist(µ) is the corresponding measure on traces tdists(A) = {tdist(apply(v, σ)): σ is a task scheduler for A}
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Example: Consensus protocol
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Previously in PIOA: Exact implementations
Exact implementation for task-PIOAs tdists(A1) tdists(A2) Exact simulation relation A1 and A2 are comparable, closed task-PIOAs.
Let R Disc(Execs*(A1)) × Disc(Execs*(A1)). R is a simulation relation if: µ1R µ2 implies tdist(µ1) = tdist(µ2) v1Rv2 If µ1R µ2, there exists a function c:R1
* × R1 R2* such that for any task T
of A1 and any schedule σ of A1 if µ1 is consistent σ and µ2 is consistent with the sequence of tasks corresponding to σ then apply(µ1,T) E(R) apply(µ1,c(σ,T)).
E(R) is defined using lifting and flattening Needed for simulation proofs in the verification of OT protocol [Canetti, et. al. 2006]
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Approximate implementations
Uniform metric on traces
A1 δ-implements A2 if for every µ1 there is a µ2 with
du(µ1,µ2) ≤ δ
This implies for every µ1 of A1 there exists µ2 of A2
with
|)()(|),( 2121 sup CCTracesFC
u
d
|)()(| 21 CC FC Traces
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Chains and limits
µ1 ≤ µ2 if for every finite trace ß µ1(Cβ) ≤µ2(Cβ)
µ1 ≤ µ2 ≤ µ3 …≤ µn is a chain
µ(Cβ) := Ltn∞ µn(Cβ) limit of a chain
µ can be uniquely extended to a probability measure on the σ-algebra generated by the cones of finite traces
Lemma 1: If µ = Ltn∞ µn then tdist(µ) = Ltn∞ tdist(µn).
Lemma 2: If µ1i µ1 and µ2i µ2 then du(µ1i,,µ2i) du(µ1,,µ2).
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Approximate simulation, roughly
A function on pairs of distributions over execution fragments is an (ε, δ)-approximate simulation function if:
))tdist(),(tdist( implies ),( : Trace
),( implies ),( : Step
),( :Start
2121
2121
21
ud
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Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006Sayan Mitra
Given
Phi and Phi Hat
}{),( 0 YX
yxyx
y
xyx
YXD
),(),( maxmin)supp(y)(x,
][][
)(11
1
1
EE
yxyxyyxyxyxx
yx
YXD yx
,1
,1
)supp(yx,
11
),( and ),(
),(
such that
)(),(ˆ
max
witnessing distribution
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Expansion
),( yx
),( 11 yx
x
y
),( yx
11 , yx Witnessing joint distribution is the dirac mass at x1,y1
),( 11 yx
),( 11 yx
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),( yx
),( 11 yx
x
),( yx
x
y
),( yx
),( yx
y
Expansion
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2. There exists a function c:R1* × R1 R2
* such that for any task T of A1
and any schedule σ of A1 if µ1 is consistent σ and µ2 is consistent with full(c)(σ) then
Approximate simulation
),(, ),(
),(, ),(
yxyx
yxyx
),( ),( yxyx
Weaker requirement in the definition of approximate simulation.
Stronger soundness theorem.
is an (ε, δ)-approximate simulation function from A1 to A2 if:
),( .1 21
))),(,apply(),,(apply( implies ),( 2121 TcT
))tdist(),(tdist( implies ),( 3. 2121 ud
MIT, Computer Sc. and AI Lab
Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006Sayan Mitra
2. There exists a function c:R1* × R1 R2
* such that for any task T of A1
and any schedule σ of A1 if µ1 is consistent σ and µ2 is consistent with full(c)(σ) then
Approximate simulation
),(, ),(
),(, ),(
yxyx
yxyx
),( ),( yxyx
Weaker requirement in the definition of approximate simulation.
Stronger soundness theorem.
is an (ε, δ)-approximate simulation function from A1 to A2 if:
),( .1 21
))),(,apply(),,(apply( implies ),( 2121 TcT
))tdist(),(tdist( implies ),( 3. 2121 ud
MIT, Computer Sc. and AI Lab
Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006Sayan Mitra
Key Lemmas
Lemma 3:
.))(),((then
))(),(( ),supp(, If
functions. vedistributi are )Disc(X)Disc(X:
. ess with witn),( },{)Disc(X)Disc(X:
2211
221121
ii
1111
ff
ff
fi
.))(),((for joint g witnessin thebe let ),supp(,each For 2211,21 1 ff
),( : ' Define21
21
,21)supp(,
i21,
),(' )( :Show21
iif ),( ),'supp(, and 2121
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Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006Sayan Mitra
Key Lemmas
Lemma 4:
Lemma 1: If µ = Ltn∞ µn then tdist(µ) = Ltn∞ tdist(µn).
Lemma 2: If µ1i µ1 and µ2i µ2 then du(µ1i,,µ2i) du(µ1,,µ2).
))tdist(),(tdist( implies ),( 2121 ud
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Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006Sayan Mitra
Soundness
Theorem: Let A1 and A2 be two closed comparable task-PIOAs. If there exists an (ε, δ)-approximate simulation function from A1 to A2, then A1 δ-implements A2.
Construct a chain of distributions for A1 applying one task at a time. Construct the corresponding chain for A2.
Induction on the length of the chain Base case from start condition Induction step from Lemma 2
Show that f1 = apply( . ,Tj) is distributive and
Use Lemmas 2 & 4 for n∞
))(),(( )supp(, implies ess with witn),( 221121,2,1 ffjj
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Probabilistic Safety
X be a random variable on (T, FT). If A1 is δ-equivalent to A2 and for every trace distribution µ2 of A2 , µ2[X=x] = p then µ1[X=x]≤ p + δ
Xu: T {0,1} defined as Xu(β) :=1 if some unsafe action U occurs in β. If A2 is safe with probability p then A1 is safe with probability at least p + δ
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Task-PIOAs
An environment E for a task PIOA A is another task-PIOA such that E||A is closed
External behavior of A is a function mapping each environment E of A to the set of trace distributions of E||A
A1 δ-implements A2 if for every environment E, for every trace distribution µ1 in extbehA(E) there is a trace distribution µ2 in extbehA(E).
Suppose for every environment E, there exists a (εE, δ)-approximate simulation function from A1||E to A2||E, then A1 δ-implements A2.
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Applications: Consensus protocol
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Probabilistic Automata and Logics (PAuL) 2006Probabilistic Automata and Logics (PAuL) 2006Sayan Mitra
Future directions
Applications: randomized consensus protocols, Approximate implementations and simulation relations
for task-PIOAs with continuous state spaces. Simulations as functions of distributions over states (as
opposed to distributions over fragments). Explore the possibility of automating simulation proofs