From Equivalences to Metrics Filippo Bonchi PACE meeting - - PowerPoint PPT Presentation

From Equivalences to Metrics

Filippo Bonchi PACE meeting (BOLOGNA)

From Equivalences to Metrics

• F. van Breugel, J. Worrell: A behavioural pseudometric for

probabilistic transition systems. TCS 2005

• F. van Breugel, James Worrell: Approximating and computing

behavioural distances in probabilistic transition systems. TCS 2006

• F. van Breugel, B. Sharma, J. Worrell: Approximating a

Behavioural Pseudometric Without Discount for Probabilistic

• Systems. LMCS 2008

Motivations

s and sε are NOT behaviorally equivalent … … but for small ε, they almost behave the same

s s2 s3

1

sε t2 t3

1/2-ε 1/2+ε 1

s s2 s3

1

s s2 s3

1/2 1/2 1

From Equivalences to Distances

• Behavioural Equivalences are the

foundations of qualitative reasoning

• Behavioural Distances are the

foundations of quantitative resoning

• A Behavioural Distance is a

pseudo-Metric d:SxS[0,1] that assigns to two systems the distance of their behaviours

• d(p,q)= 0 iff p is behaviourally

equivalent to q

From Equivalences to Distances

• Behavioural Equivalences are the

foundations of qualitative reasoning

• Behavioural Distances are the

foundations of quantitative resoning

• A Behavioural Distance is a

pseudo-Metric d:SxS[0,1] that assigns to two systems the distance of their behaviours

• d(p,q)= 0 iff p is behaviourally

equivalent to q

Does these systems behave the same? Does a system satify a certain property?

How far apart is the behaviour

• f these

systems? How much closely does as system come to satisfy a certain property?

Plan of the Talk

1.Coalgebras in a nutshell 2.Probabilistic Systems 3.Behavioural (Pseudo-)Metrics – Coalgebras – Coinduction – Refinement Algorithm – Modal Logic

Coalgebras in a nutshell

At the blackboard

A functor F induces:

1) Behavioural equivalence ≅F 2) Coinduction Proof Principle: x≅Fy iff xRy for some F-bisimulation R 3) Partition F-Refinement Algorithm 4) An Hennessy-Milner Logic: x≅Fy iff f(x)=f(y) for all F-formulas f

Functors

Set is the category of sets and functions F::= Id, A, F+F, FxF, FA, P(F), D(F)

Plan of the Talk

1.Coalgebras in a nutshell 2.Probabilistic Systems 3.Behavioural (Pseudo-)Metrics – Coalgebras – Coinduction – Refinement Algorithm – Modal Logic

A Hierarchy of Probabilistic Systems Types

• F. Bartels, A. Sokolova, E. de Vink: A hierarchy of probabilistic system types.

TCS 2004

Markov Chains

D(Id):SetSet

D(S)={µ:S[0,1] | µ[X]=1, spt(µ) finite} <S,α:SD(S)>

s1 s2 s3

1/3 1 1/3 1/3 1

Markov Chains

D(Id):SetSet

D(S)={µ:S[0,1] | µ[X]=1, spt(µ) finite} <S,α:SD(S)>

s1 s2 s3

1/3 1 1/3 1/3 1

s1 s2

2/3 1/3 1

Markov Chains

D(Id):SetSet

D(S)={µ:S[0,1] | µ[X]=1, spt(µ) finite} <S,α:SD(S)>

s1 s2 s3

1/3 1 1/3 1/3 1

s1 s2 Ο

2/3 1 1/3 1

D(AxId):SetSet A={a,b}

<S,α:SD(AxS)>

Generative Systems

a

s1 s2 s3

1/3 1/3 1/3 a b 1 b a 1

D(Id)A:SetSet A={a,b}

<S,α:SD(S)A>

Reactive Systems

a

s1 s2 s3

2/3 1 1/3 1 a b b 1 b a 1

D(Id)+P(A×Id): SetSet A={a,b}

<S,α:S D(S)+P(A×S)>

Alternating Systems

s1 s2 s3

2/3 1/3 b b

s4

1

P(A×D(Id)): SetSet A={a,b}

<S,α:SP (A×D(S))>

Simple Probabilistic Automata (Simple Segala Systems)

s1 s2 s4

1/2 a

s3

1/2 1/3 a

s5

1/3 1/3

PD(A×id): SetSet A={a,b}

<S,α:SP D(A×S)>

Probabilistic Automata (Segala Systems)

s1 s2 s4

1/2a

s3

1/2 1/3 a

s5

1/3 1/3 a a b

(Partial) Markov Chains

D(1+Id):SetSet

<S,π:SD(1+S)>

s1 s2 s3

1/3 1/3

(Partial) Markov Chains Bisimulation

D(1+Id):SetSet

<S,π:SD(1+S)> R is a Bisimulation iff whenever s1Rs2 then For all equivalence classes E of R

s1 s2 s3

1/3 1/3

(Partial) Markov Chains Partition Refinement

s3 s5 s6

1/2 1/2

s2 s4

1

s1

1/3 2/3

t

1

(Partial) Markov Chains Partition Refinement

s3 s5 s6

1/2 1/2

s2 s4

1

s1

1/3 2/3

t

1

(Partial) Markov Chains Partition Refinement

s3 s5 s6

1/2 1/2

s2 s4

1

s1

1/3 2/3

t

1

(Partial) Markov Chains Partition Refinement

s3 s5 s6

1/2 1/2

s2 s4

1

s1

1/3 2/3

t

1

(Partial) Markov Chains Partition Refinement

s3 s5 s6

1/2 1/2

s2 s4

1

s1

1/3 2/3

t

1 1 1 1

Plan of the Talk

1.Coalgebras in a nutshell 2.Probabilistic Systems 3.Behavioural (Pseudo-)Metrics – Coalgebras – Coinduction – Refinement Algorithm – Modal Logic

Functors

Ms is the category of metric spaces and non-expansive maps F::= Id, A, F+F, FxF, FA, P(F), D(F), δ(F) δ is in [0,1]

Plan of the Talk

1.Coalgebras in a nutshell 2.Probabilistic Systems 3.Behavioural (Pseudo-)Metrics – Coalgebras – Coinduction – Refinement Algorithm – Modal Logic

Example of Coinduction

s s2 s3 s2 s3 s2 s3

1/3 1/3 1 1/3

t

1/3+ε 1/3-ε 1/3

d s s2 s3 t s 1 1 2ε s2 1 1 1 s3 1 1 1 t 2ε 1 1 s s2 s3 t π(s) s 1/3 1/3 s2 1/3-ε ε 1/3 s3 1/3 1/3 t π(t) 1/3-ε 1/3+ε 1/3

Coupling of s and t

∆(d)(s,t) ≤ ε+2ε/3 ≤ 2ε = d(s,t)

Post-fix point

dF(s,t) ≤ 2e

Plan of the Talk

1.Coalgebras in a nutshell 2.Probabilistic Systems 3.Behavioural (Pseudo-)Metrics – Coalgebras – Coinduction – Refinement Algorithm – Modal Logic

Metric Refinement

s s2 s3 s2 s3 s2 s3

1/3 1/3 1 1/3

t

1/3+ε 1/3-ε 1/3

d0 s s2 s3 t s s2 s3 t

Metric Refinement

d0 s s2 s3 t s s2 s3 t d1 s s2 s3 t s 1 s2 1 s3 1 1 1 1 t 1 s s2 s3 s2 s3 s2 s3

1/3 1/3 1 1/3

t

1/3+ε 1/3-ε 1/3

Metric Refinement

s s2 s3 s2 s3 s2 s3

1/3 1/3 1 1/3

t

1/3+ε 1/3-ε 1/3

d2 s s2 s3 t s 1/3 1

ε

s2 1/3 1 1/3+ε s3 1 1 1 t

ε

1/3+ε 1

Metric Refinement

s s2 s3 s2 s3 s2 s3

1/3 1/3 1 1/3

t

1/3+ε 1/3-ε 1/3

d2 s s2 s3 t s 1/3 1

ε

s2 1/3 1 1/3+ε s3 1 1 1 t

ε

1/3+ε 1 d3 s s2 s3 t s 1/3+1/9 1

ε+ε/3

s2 1/3+1/9 1 1/3+ε+1

/9+ε/3

s3 1 1 1 t

ε+ε/3

1/3+ε+

1/9+ε/3

1

Plan of the Talk

1.Coalgebras in a nutshell 2.Probabilistic Systems 3.Behavioural (Pseudo-)Metrics – Coalgebras – Coinduction – Refinement Algorithm – Modal Logic

A Logical Characterization

• Modal Formulas f are functions

f:S{0,1}

• Quantitative Formulas f:S[0,1]

d(s1,s2)= supf |f(s1)-f(s2)|

• P. Panangaden et al.

Metrics for labeled Markov Processes, TCS 2004

A quantitative logic

D(1+Id):SetSet

<S,π:SD(1+S)>