Approximate Simulations for Probabilistic I/O Automata. Sayan Mitra and Nancy Lynch TDS Seminar, CSAIL, MIT October 27 th 2006

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1 Approximate Simulations for Probabilistic I/O Automata Sayan Mitra and Nancy Lynch TDS Seminar CSAIL MIT October 7 th 006

2 Implementation Relations Timed automata probabilistic automata Executions record evolution of the system Traces or visible behaviors sequences of external actions sequences of external actions and intervals probability distribution over visible sequences Implementation as trace inclusion e.g. Lynch Vaandrager 995 A implements B if TracesA Œ TracesB written as A B A and B are equivalent if TracesA = TracesB written as A = B Other notions of implementation : strong and weak bisimulation reachable set inclusion

3 Simulation Relations for Proving Implementation A B automata Ris a relation on XäY Forward simulation a y 0 x 0 y x a y x Tracea = a

4 Fragility of Classical Implementation in Timed Systems dx = b dx = x 5 a x = 5 A dx = 0 dx = x 5 - ε a x = 5 - ε B dx = 0 TracesA = { [0t][05]a } TracesB = { [0t][05+ε]a [0t]b } A and B cannot be compared

5 and in Probabilistic Systems A B Suppose a p and a p are the only traces of A and A TracesA contains aa rp aa -rp TracesB contains aa r+εp aa -r- εp Again A B cannot be compared

6 Metrics on Traces Define a metric d on the space T containing traces of A and traces of B Td is a metric space A δ-implements B if for every µ A in TraceA there is µ B in TraceB s.t. d µ A µ B δ A and B are δ-equivalent if they δ-implement each other TracesB δ TracesA TracesA

7 Metrics on Traces Robust implementation relations Actual value of δ is unimportant Abstraction and model reduction through approximation Continuity of quantitative properties Universe of Traces TracesA f Property values

8 Define metrics for trace distributions of PIOA Simulation based techniques for proving A δ-implements B Outline Task-structured PIOA A simple approximate simulation Expanded approx. sim. Discounted approx. sim. Other metrics and new directions

9 Related Work I. Probabilistic CCS Giacalone Jou Smolka 994 Introduced metric in the study of bisimilarity Metric: ε-bisimilar. Probabilistic Concurrent -player Games Alfaro Henzinger Mazumdar 003 Discounted mu-calculus Fixpoint-based algorithms for checking discounted properties

10 Related Work II 3. Labelled Markov Processes LMPs Desharnais Gupta Jagadeesan Panagaden Metric on states defined based on a class dab = 0 implies A and B are bisimilar Van Breugel Mislove Ouaknine Worell 003 Intrinsic characterization of the above metric Topology induced by the above metric on the space of LMPs Polynomial time algorithm for the metric for finite LMPs 4. Generalized Semi Markov Processes Gupta Jagadeesan Panagaden 004 Pseudometric analogue of bisimulation Continuity of properties No nondeterminism based on bisimulations

11 Related Work III 5. Discrete/Continuous Metric Transition Systems Girard Pappas 005 Pseudometrics on: trace inclusion Reachable set inclusion bisimulation Algorithms for computing metrics 6. Probabilistic I/O Automata Cheung 006 Trace distributions are closed sets in [0] Traces* d Finite tests are sufficient to distinguish infinite processes

12 Task-Structured PTIOA Canetti Cheung Kaynar Liskov Lynch Pereira Segala Discrete probability measures on X me = S e є E m{e} and mx = DiscX Suppm A = QvIOHDR Q: countable set of states v: initial distribution IO H: countable pairwise disjoint sets of actions DŒ Q A DiscQ qaµ є D is written as q a m R: Equivalence relation on L; each equivalence class of R is a task T Axioms: Input enabled For any q & a there is at most one q a m For any q & T there is at most one a є T enabled at q. For this talk assume A is closed

13 Executions and Traces As usual Execution fragment α = q 0 a q a α is an execution if q 0 in Suppv Execs Execs* traceα captures the visible part of α delete all q s and the a s in H Traces Traces* But PIOA is probabilistic visible behavior = distribution over Traces a trace distribution Nondeterministic therefore set of trace distributions Canetti Cheung Kaynar Liskov Lynch Pereira Segala05 06

14 Task Scheduler Task scheduler for A is a finite or infinite sequence of tasks T T It interacts with A to give discrete distributions over execution fragments For this talk assume task scheduler is finite All distributions are finite we avoid limit arguments A finite measure can be viewed as a discrete measure on finite fragments In general σ-field on Execs generated by cones discrete σ-field of Execs* is contained in the above Likewise σ-fields for Traces Construct chains of measures and then take limits

15 Applying a Schedule Given a distribution µ over Execs * a task schedule σ = T T T n applyµσ gives a probability distribution Execs by applying σ applyµ^ = µ applyµt = µ applyµ σt = applyapplyµ σ T applyµ σ = lim i applyµ σ i σ i is the length i prefix of σ Canetti Cheung Kaynar Liskov Lynch Pereira Segala05 06

16 applyµσ = probability distribution over fragments applyµ^ = µ applyµt = µ µ a = p a + p a p a = µa hq if a = a aq and a is in task T and lstatea a h p a = µa if T is not enabled in lstatea applyµ σt = applyapplyµ σ T

17 Applying Schedules σ a ^ a -p p aa -p p-p p aaa -p p-p p -p p 3 d ad 0 p -p aad -p p-p p aaad -p p-p p -p p 3 aada -p p-p p -p p 3 p a = µa p if a = a a R i+ lstatea =R i = µa -p if a = a a Y i lstatea =R i = µa if a = a d G i lstatea =Y i p a = µa if lstatea = R i = µa if lstatea = Y i

18 A task schedule σ defines µ=applyv σ is a probabilistic execution Corresponding trace distribution tdistµb = µtrace - b tdistsa = {tdistapplyv σ: σ is a task scheduler for A} set of all possible trace distributions A B if tdistsa Œ tdistsb

19 Metrics on Trace Distributions A metric d : DiscE*äDiscE* R dµ µ = 0 iff µ = µ dµ µ = dµ µ 3 dµ µ 3 dµ µ + dµ µ 3 A δ-implements B w.r.t metric d if for every trace dist µ of A there is a trace dist µ of B such that dµ µ δ. We write this as A δ B. A and B are δ-equivalent if they δ-implement each other. We write this as A = δ B.

20 Simple Approximate Simulation Relation SA * * Given ε δ > 0 a function φ : Disc Exec A Disc Exec B R is a Simple Approximate SA Simulation function if.start : φ ν ν ε.step : φ ε impliesφ ' ' ε 3. Trace: φ ε implies dtdist tdist δ 0 { } Simulation Relation: Segala R Œ DiscExecs*A DiscExecs*B. v Rv. µ R µ implies µ ER µ 3. µ R µ implies tdistµ = tdistµ

21 Simple Approximate Simulation Relation SA * * Given ε δ > 0 a function φ : Disc Exec A Disc Exec B R is a Simple Approximate SA Simulation function if.start : φ ν ν ε.step :There exists a function c : R if φ ε σ is a task schedule and T is a task for A and is consistent with σ and is consistent with full c σ then φ apply T apply c σ T ε 3. Trace: φ ε implies d u * R such that tdist tdist δ R * 0 { }

22 Soundness of SA Theorem. If there exists an εδ-sa simulation function from A to B then A δ B. Consider any µ A = applyv T T T n σ j = ct T j µ Aj = applyv T T T j µ Bj = applyv σ σ σ j µ B = µ Bn For all j fµ Aj µ Bj εby induction using For all j dtdistµ Aj tdistµ Bj δby 3 In particular dtdistµ A tdistµ B δ

23 For infinite task schedules take limit of a sequence of probability measures. Lim = and Lim = then Lim d = j Aj A j Bj B j Aj Bj d A B Step and Trace conditions are critical for the choice of the metric and the simulation function

24 Probabilistic Safety d u = C C sup C Traces* Define a function random variable X:Traces {0} Xb := if some bad action occurs in b 0 otherwise Suppose A is safe with probability at least p and B δ A Claim: B is safe with probability at least δ + p Let µ B be any trace distribution of B. There exits µ A such that for all C µ B C - µ A C δ. Then µ B [X=] δ+ µ A [X=] δ+ p

25 SA Simulation A simulation relation R tells us if two distributions are related or not An approximate simulation function f gives us a measure of how close two distributions are We want to capture internal branching So decompose µ A µ B into close components µ A a= S i l Ai µ Ai a where S i l Ai = µ B a= S i l Bi µ Bi a where S i l Bi = fµ Ai µ Bj is small More powerful simulation function

26 SA Simulation A simulation relation R tells us if two distributions are related or not An approximate simulation function f gives us a measure of how close two distributions are We want to capture internal branching So decompose µ A µ B into close components µ A a= S i h A µ Ai µ Ai a where h A is DiscDiscExec* A µ B a= S i h B µ Bi µ Bi a where h B is DiscDiscExec* B fµ Ai µ Bj is small More powerful simulation function

27 Need for Expansion /3 /3 /3 /3 - ε /3 + ε choose /3 - ε y=0 y= y= choose z=0 z= z= z=y+ 3 z=y+ 3 z=y+ 3 out out out0 out0 out out comp comp comp

28 Need for Expansion /3 /3 /3 /3 - ε choose /3 - ε /3 + ε y=0 y= y= choose z=0 z= z= z=y+ 3 z=y+ 3 z=y+ 3 out out out0 out0 out out comp comp comp

29 Need for Expansion choose /3 /3 /3 /3 - ε choose /3 - ε /3 + ε y=0 y= y= z=0 z= z= comp comp comp out0 out out z=y+ 3 z=y+ 3 z=y+ 3 out out out0 B A s u : support of lsateµ A lstateµ B max A α + B β α β φ A B = 0 max A α / 3 α if s u if s u s.z u.z s.z = u.z = and s.y = otherwise

30 max A α + B β α β φ A B = 0 max A α / 3 α if su if s u s.z u.z s.z = u.z = and s.y = otherwise fv A v B = 0 µ A = applyv A choose = {y=0 z=^/3 ε y= z=^/3 ε y= z=^/3 +ε} µ B = applyv B ^ = v B = {z=^} fµ A µ B = max{εε} = ε µ A = applyµ A comp = {y=0 z=/3 ε y= z=/3 ε y= z=0/3 +ε} µ B = applyµ B comp = {z=0/3 z=/3 z=/3} fµ A µ B = /3 as the z s are different Will return to this

31 Expanded Approximate Simulation Relation EA Expansion of φ X Y R 0 { }is φ x y = max D X Y xy supp x y min = E[ ] = E[ ] x y φ x y Expansion of: R Œ X Y x y are in ER if there exists wn n. wx y >0 x R y. S y wxy = n x 3. S x n x = x

32 Expanded Approximate Simulation Relation EA }is { of Expansion 0 R Y X φ y x y x y x y x Y X D = = = max min supp xy ] [ ] [ φ φ E E = = y x y x y y x y x y x x y x Y X D y x supp xy and such that joint distribution witnessing a ˆ max ε φ ε φ

33 Expansion φ x y ε One witnessing joint distribution is the delta at x y φ x y ε φ x y = δ x y x y x y φ x y ε

34 φ x y > ε φ x y φ x y ε x y x x y y φ x y ε

35 A small digression Finding the witness is an LP Optimal transportation problem Kantorovich 94 Let M be the set of all probability distributions over executions of Given φ ν the pth Wasserstein metric is given by w w p = = inf Γ M M inf φ ν sup Γ ν supp p d ν φ ν p A &B

36 Expanded Approximate Simulation Relation EA * * Given ε δ > 0 a function φ : Disc Exec A Disc Exec B R is a Expanded Approximate EA Simulation function if.start : φ ν ν ε.step :if φ ε then ˆ φ ' ' ε 3.Trace: φ ε implies d u tdist tdist δ 0 { }

37 Expanded Approximate Simulation Relation EA * * Given ε δ > 0 a function φ : Disc Exec A Disc Exec B R is a Expanded Approximate EA Simulation function if.start : φ ν ν ε.step :There exists a function c : R if φ ε σ is a task schedule and T is a task for A and is consistent with σ and is consistent with full c σ then ˆ φ apply T apply c σ T ε 3.Trace: φ ε implies d u * R such that tdist tdist δ R * 0 { }

38 Return to Example choose /3 /3 /3 /3 - ε choose /3 - ε /3 + ε y=0 y= y= z=0 z= z= comp comp comp out0 out out z=y+ 3 z=y+ 3 z=y+ 3 out out out0 B A s u : support of lsateµ A lstateµ B max A α + B β α β φ A B = 0 max A α / 3 α if s u if s u s.z u.z s.z = u.z = and s.y = otherwise

39 max A α + B β α β φ A B = 0 max A α / 3 α if s u if s u s.z u.z s.z = u.z = and s.y = otherwise fv A v B = 0 µ A = applyv A choose = {y=0 z=^/3 ε y= z=^/3 ε y= z=^/3 +ε} µ B = applyv B ^ = v B = {z=^} fµ A µ B = max{εε} = ε µ A = applyµ A comp = {y=0 z=/3 ε y= z=/3 ε y= z=0/3 +ε} µ B = applyµ B comp = {z=0/3 z=/3 z=/3} fµ A µ B = /3 as the z s are different For each r r in the support of y fr r ε µ A = S i h d i i+ 3 d y=0z= d y=z= d y=z=0 µ B = S i h d i y ˆ φ A B ε d z=0 0 0 /3 d z= /3 - ε 0 ε d z= 0 /3 - ε ε

40 Key Lemmas Lemma. Lemma. δ ε φ tdist tdist implies u d. then supp If are distributive functions.. with witness ε φ ε ρ ρ φ ρ ρ ε φ f f f f f f

41 sketch of proof for Lemma. joint for be the witnessing let supp each For ε ρ ρ φ ρ ρ ρ ρ f f ': new joint distribution a Define supp ρ ρ ρ ρ ρ ρ = i ' Show : = i f i ε φ implies ' supp and. then supp If are distributive functions.. with witness ε φ ε ρ ρ φ ρ ρ ε φ f f f f f f

42 sketch of proof for Lemma d be the witness Let E* u Sup = = = ε β tdist tdist tdist tdist tdist tdist δ β E* Sup tdist tdist δ ε φ tdist tdist implies u d

43 Soundness of SA Theorem. If there exists an εδ-sa simulation function from A to A then A δ A. Consider any µ = applyv T T T n Define: σ j = ct T j µ j = applyv T T T j µ j = applyv σ σ σ j µ = µ n For all j fµ j µ j εby induction For all j d u tdistµ j tdistµ j δby 3. In particular d u tdistµ tdistµ δ

44 Soundness of EA Theorem. If there exists an εδ-ea simulation function from A to A then A δ A. Consider any µ = applyv T T T n Define:σ j = ct T j µ j = applyv T T T j µ j = applyv σ σ σ j µ = µ n For all j fµ j µ j ε by Lemma φ j+ j+ φ 0 j = apply j Tj distributive functions = apply j σ j 0 j = φ ν ν ε by start condition ε implies that ˆ φ j+ j+ ε by step condition For all j d u tdistµ j tdistµ j δby Lemma In particular d u tdistµ tdistµ δ

45 Need for Discounting The error β β can be small because µ b and µ b are small Typically when b is long Discounted uniform metric: d k Given {δ k } A δ k -implements B if for every trace dist µ of A there is a trace dist µ of B such that d k µ µ δ k. Write this as A δk A. = β β sup β E * β = k

46 Discount Factors δ k k

47 Discounted Approximate Simulation Relation DA Given ε δ k.start : φ ν ν ε * * k > 0 a collection{ φk} φk : R 0 then for all k L ' ' φ ' ' ε 3.Trace:for all k L φ ε implies k for all k Ltdist tdist d tdist tdist δ k k k Disc k k Exec is a Discounted Approximate DA Simulation if 0.Step :if for all k L φ ε then 0 k A Disc k Exec B { }

48 Soundness of DA Theorem. If there exists an ε k δ k -DA simulation functions from A to A then A δk A.

49 Round-based Randomized Consensus σ a ^ a -p p aa -p p-p p aaa -p p-p p -p p 3 d ad 0 p -p aad -p p-p p aaad -p p-p p -p p 3 aada -p p-p p -p p 3 p a = µa hq if a = a aq and a є T and lstatea a h p a = µa if T is not enabled in lstatea

50 Example A: protocol with unbiased coins B: protocol biased coins; p p+e -p -p-e f k µ A µ B = max a =k µ A a-µ B a ε k = p+e k p k d k =ε k A= δk B

51 Step condition µ A ap+e - µ B ap =pµ A a - µ B a + eµ A a pp+e k -p k + ep+e k =p+e k+ -p k+ = ε k Trace condition If tracea = tracea = b then tdistµ A b - tdistµ B b = µ A a + µ A a - µ B a - µ B a = µ A a - µ B a where a is the common prefix of a a

52 Expanded and Discounted Simple Approx Sim Expanded Approx Sim Discounted Approx Sim Expanded Discounted Approx Sim

53 Generalization Approximate simulations for Probabilistic Times I/O Automata Q X DiscX PX Trajectories Expansion: µa= S i hµ i µ i a where h is DiscDiscExec* µa= Ûµ*a dhµ* where h is PPExec*

54 Metrics Duality sup given by the pth Wasserstein metric is Given A &B probability distributions over executions of all be the set of Let ] [ : supp sup inf inf β β β ν φ ν ν φ ν φ ν = = = Γ Γ M M f p M M p p d f w w d w M

55 Acknowledgement Dilsun Kaynar Sanjoy Mitter

Trace-based Semantics for Probabilistic Timed I/O Automata

Trace-based Semantics for Probabilistic Timed I/O Automata Sayan Mitra and Nancy Lynch Computer Science and AI Laboratory, Massachusetts Inst. of Technology, 32 Vassar Street, Cambridge, MA 02139 {lynch,