Fixed-point Characterization of Compositionality Properties of - - PowerPoint PPT Presentation

SLIDE 1

Fixed-point Characterization of Compositionality Properties of Probabilistic Processes Combinators

Daniel Gebler1, Simone Tini2

1VU University Amsterdam, The Netherlands 2University of Insubria, Italy

EXPRESS/SOS workshop September 1, 2014

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SLIDE 2

Motivation: Metric compositionality in a nutshell

An operation f ∈ Σ is congruent wrt. behavioral relation ∼ if: t1 ∼ t′

1

. . . tn ∼ t′

n

f(t1, . . . , tn) ∼ f(t′

1, . . . , t′ n)

An operation f is continuous wrt. behavioral distance d if: d t t d tn tn

n

d f t tn f t tn

f n

for some modulus of continuity

f.

Metric compositionality = congruence of kernel relation + modulus of continuity This talk: which process algebra operators are continuous? given any modulus of continuity, what are the syntactical properties s.t. a specified operator admits this modulus of continuity? given any operator specification, what is its modulus of continuity?

2 / 18

SLIDE 3

Motivation: Metric compositionality in a nutshell

An operation f ∈ Σ is congruent wrt. behavioral relation ∼ if: t1 ∼ t′

1

. . . tn ∼ t′

n

f(t1, . . . , tn) ∼ f(t′

1, . . . , t′ n)

An operation f ∈ Σ is continuous wrt. behavioral distance d if: d(t1, t′

1) ≤ ϵ1

. . . d(tn, t′

n) ≤ ϵn

d(f(t1, . . . , tn), f(t′

1, . . . , t′ n)) ≤ ωf(ϵ1, . . . , ϵn)

for some modulus of continuity ωf. Metric compositionality = congruence of kernel relation + modulus of continuity This talk: which process algebra operators are continuous? given any modulus of continuity, what are the syntactical properties s.t. a specified operator admits this modulus of continuity? given any operator specification, what is its modulus of continuity?

2 / 18

SLIDE 4

Motivation: Metric compositionality in a nutshell

An operation f ∈ Σ is congruent wrt. behavioral relation ∼ if: t1 ∼ t′

1

. . . tn ∼ t′

n

f(t1, . . . , tn) ∼ f(t′

1, . . . , t′ n)

An operation f ∈ Σ is continuous wrt. behavioral distance d if: d(t1, t′

1) ≤ ϵ1

. . . d(tn, t′

n) ≤ ϵn

d(f(t1, . . . , tn), f(t′

1, . . . , t′ n)) ≤ ωf(ϵ1, . . . , ϵn)

for some modulus of continuity ωf. Metric compositionality = congruence of kernel relation + modulus of continuity This talk: which process algebra operators are continuous? given any modulus of continuity, what are the syntactical properties s.t. a specified operator admits this modulus of continuity? given any operator specification, what is its modulus of continuity?

2 / 18

SLIDE 5

Motivation: Metric compositionality in a nutshell

An operation f ∈ Σ is congruent wrt. behavioral relation ∼ if: t1 ∼ t′

1

. . . tn ∼ t′

n

f(t1, . . . , tn) ∼ f(t′

1, . . . , t′ n)

An operation f ∈ Σ is continuous wrt. behavioral distance d if: d(t1, t′

1) ≤ ϵ1

. . . d(tn, t′

n) ≤ ϵn

d(f(t1, . . . , tn), f(t′

1, . . . , t′ n)) ≤ ωf(ϵ1, . . . , ϵn)

for some modulus of continuity ωf. Metric compositionality = congruence of kernel relation + modulus of continuity This talk: which process algebra operators are continuous? given any modulus of continuity, what are the syntactical properties s.t. a specified operator admits this modulus of continuity? given any operator specification, what is its modulus of continuity?

2 / 18

SLIDE 6

Probabilistic Transition Systems

A probabilistic transition system (S, A, − →) consists of a (countable) set of states S a (countable) set of actions A a transition relation − → ⊆ S × A × ∆(S) with ∆(S) the set of probability distributions over S.

3 / 18

SLIDE 7

Perturbation of probabilities and bisimulation equivalence

. p .

• .

p2 . p3 .

• .
• .

Specification . ̸∼ . d p q .

a

.

0.6

.

0.4

.

b

.

c

.

1.0

.

1.0

. q .

• .

q2 . q3 .

• .
• .

Implementation .

a

.

0.6 − ϵ

.

0.4 + ϵ

.

b

.

c

.

1.0

.

1.0 4 / 18

SLIDE 8

Perturbation of probabilities and bisimulation equivalence

. p .

• .

p2 . p3 .

• .
• .

Specification . ̸∼ . d p q .

a

.

0.6

.

0.4

.

b

.

c

.

1.0

.

1.0

. q .

• .

q2 . q3 .

• .
• .

Implementation .

a

.

0.6 − ϵ

.

0.4 + ϵ

.

b

.

c

.

1.0

.

1.0 4 / 18

SLIDE 9

Perturbation of probabilities and bisimulation equivalence

. p .

• .

p2 . p3 .

• .
• .

Specification . . d(p, q) = ϵ .

a

.

0.6

.

0.4

.

b

.

c

.

1.0

.

1.0

. q .

• .

q2 . q3 .

• .
• .

Implementation .

a

.

0.6 − ϵ

.

0.4 + ϵ

.

b

.

c

.

1.0

.

1.0 4 / 18

SLIDE 10

Bisimulation metrics between PTS

A pseudometric d: S × S → [0, 1] is a bisimulation metric if . s1 . s2 . π1 . π2 . ∀ .

a

.

a

. ∃ . d(s1, s2) ≤ ϵ . K(d)(π1, π2) ≤ ϵ with K(d): ∆(S) × ∆(S) → [0, 1] lifts state metric d to distributions.

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SLIDE 11

Bisimulation metrics between PTS

A pseudometric d: S × S → [0, 1] is a bisimulation metric if . s1 . s2 . π1 . π2 . ∀ .

a

.

a

. ∃ . d(s1, s2) ≤ ϵ . K(d)(π1, π2) ≤ ϵ with K(d): ∆(S) × ∆(S) → [0, 1] lifts state metric d to distributions.

5 / 18

SLIDE 12

Bisimulation metric by example

. p .

• .

p2 .

• .
• .

p3 . p′

3

.

• .
• .

a

.

1/3

.

2/3

.

b

.

0.6

.

0.4

.

c

.

d

.

1.0

.

1.0

. q .

• .

q2 .

• .
• .

q3 . q′

3

.

• .
• .

.

a

.

1/3

.

2/3

.

b

.

0.6 − ϵ

.

0.4 + ϵ

.

c

.

d

.

1.0

.

1.0

. d p q . d p q . d(p3, q3) = 0 . d(p′

3, q′ 3) = 0

. d(p3, q′

3) = 1

. d(p′

3, q3) = 1

6 / 18

SLIDE 13

Bisimulation metric by example

. p .

• .

p2 .

• .
• .

p3 . p′

3

.

• .
• .

a

.

1/3

.

2/3

.

b

.

0.6

.

0.4

.

c

.

d

.

1.0

.

1.0

. q .

• .

q2 .

• .
• .

q3 . q′

3

.

• .
• .

.

a

.

1/3

.

2/3

.

b

.

0.6 − ϵ

.

0.4 + ϵ

.

c

.

d

.

1.0

.

1.0

. d p q . d(p2, q2) = ϵ . d(p3, q3) = 0 . d(p′

3, q′ 3) = 0

. d(p3, q′

3) = 1

. d(p′

3, q3) = 1

6 / 18

SLIDE 14

Bisimulation metric by example

. p .

• .

p2 .

• .
• .

p3 . p′

3

.

• .
• .

a

.

1/3

.

2/3

.

b

.

0.6

.

0.4

.

c

.

d

.

1.0

.

1.0

. q .

• .

q2 .

• .
• .

q3 . q′

3

.

• .
• .

.

a

.

1/3

.

2/3

.

b

.

0.6 − ϵ

.

0.4 + ϵ

.

c

.

d

.

1.0

.

1.0

. d(p, q) = 1/3 · ϵ . d(p2, q2) = ϵ . d(p3, q3) = 0 . d(p′

3, q′ 3) = 0

. d(p3, q′

3) = 1

. d(p′

3, q3) = 1

6 / 18

SLIDE 15

Continuity ensures robust composition semantics

. . f . t1 . t2 . . f . t′

1

. t′

2

. ϵ . δ1 . δ2 . Robustness: Given , which distances s.t. . if d ti ti

i for i

. then d f t t f t t

7 / 18

SLIDE 16

Continuity ensures robust composition semantics

. . f . t1 . t2 . . f . t′

1

. t′

2

. ϵ . δ1 . δ2 . Robustness: Given , which distances s.t. . if d ti ti

i for i

. then d f t t f t t

7 / 18

SLIDE 17

Continuity ensures robust composition semantics

. . f . t1 . t2 . . f . t′

1

. t′

2

. ϵ . δ1 . δ2 . Robustness: Given , which distances s.t. . if d ti ti

i for i

. then d f t t f t t

7 / 18

SLIDE 18

Continuity ensures robust composition semantics

. . f . t1 . t2 . . f . t′

1

. t′

2

. ϵ . δ1 . δ2 . Robustness: Given ϵ ∈ [0, 1], which distances (δ1, δ2) ∈ [0, 1]2 s.t. .

• if d(ti, t′

i) ≤ δi for i = 1, 2

.

• then d(f(t1, t2), f(t′

1, t′ 2)) ≤ ϵ

7 / 18

SLIDE 19

From GSOS to probabilistic GSOS

{xi

ai,m

− − → yi,m | i ∈ I, m ∈ Mi} {xj

bj,n

− − → ̸ | j ∈ J, n ∈ Nj} f(x1, . . . , xr(f))

a

− → t Two sorted signature (state and distribution terms) and each state operator is also available for distributions Distribution terms are defined as smallest set including distribution variables Vd instantiable Dirac distributions t for state term t

i I pi i if i are distribution terms and pi

with

i I pi

f

r f

if

i are distribution terms and f

Distribution term f

r f

represents the element-wise application of

• perator f to elements in

i, that is for closed substitution distribution

f

r f state term

f t tr f

r f i i

ti

8 / 18

SLIDE 20

From GSOS to probabilistic GSOS

{xi

ai,m

− − → µi,m | i ∈ I, m ∈ Mi} {xj

bj,n

− − → ̸ | j ∈ J, n ∈ Nj} f(x1, . . . , xr(f))

a

− → θ Two sorted signature (state and distribution terms) and each state operator is also available for distributions Distribution terms are defined as smallest set including distribution variables µ ∈ Vd instantiable Dirac distributions δ(t) for state term t ⊕

i∈I piθi if θi are distribution terms and pi ∈ (0, 1] with ∑ i∈I pi = 1

f(θ1, . . . , θr(f)) if θi are distribution terms and f ∈ Σ Distribution term f

r f

represents the element-wise application of

• perator f to elements in

i, that is for closed substitution distribution

f

r f state term

f t tr f

r f i i

ti

8 / 18

SLIDE 21

From GSOS to probabilistic GSOS

{xi

ai,m

− − → µi,m | i ∈ I, m ∈ Mi} {xj

bj,n

− − → ̸ | j ∈ J, n ∈ Nj} f(x1, . . . , xr(f))

a

− → θ Two sorted signature (state and distribution terms) and each state operator is also available for distributions Distribution terms are defined as smallest set including distribution variables µ ∈ Vd instantiable Dirac distributions δ(t) for state term t ⊕

i∈I piθi if θi are distribution terms and pi ∈ (0, 1] with ∑ i∈I pi = 1

f(θ1, . . . , θr(f)) if θi are distribution terms and f ∈ Σ Distribution term f

r f

represents the element-wise application of

• perator f to elements in

i, that is for closed substitution distribution

f

r f state term

f t tr f

r f i i

ti

8 / 18

SLIDE 22

From GSOS to probabilistic GSOS

{xi

ai,m

− − → µi,m | i ∈ I, m ∈ Mi} {xj

bj,n

− − → ̸ | j ∈ J, n ∈ Nj} f(x1, . . . , xr(f))

a

− → θ Two sorted signature (state and distribution terms) and each state operator is also available for distributions Distribution terms are defined as smallest set including distribution variables µ ∈ Vd instantiable Dirac distributions δ(t) for state term t ⊕

i∈I piθi if θi are distribution terms and pi ∈ (0, 1] with ∑ i∈I pi = 1

f(θ1, . . . , θr(f)) if θi are distribution terms and f ∈ Σ Distribution term f(θ1, . . . , θr(f)) represents the element-wise application of

• perator f to elements in θi, that is for closed substitution σ

distribution

• σ(f(θ1, . . . , θr(f)))(

state term

• f(t1, . . . , tr(f))) =

r(f)

∏

i=1

σ(θi)(ti)

8 / 18

SLIDE 23

Parallel composition

. P .

• .

P1 . P2 .

a

.

0.4

.

0.6

. Q .

• .

Q1 . Q2 .

a

.

0.3

.

0.7

. x

a

− → µ y

a

− → ν x ∥ y

a

− → µ ∥ ν . P ∥ Q .

• .

P1 ∥ Q1 . P1 ∥ Q2 . P2 ∥ Q1 . P2 ∥ Q2 .

a

.

0.12

.

0.28

.

0.18

.

0.42 9 / 18

SLIDE 24

Parallel composition

. P .

• .

P1 . P2 .

a

.

0.4

.

0.6

. Q .

• .

Q1 . Q2 .

a

.

0.3

.

0.7

. x

a

− → µ y

a

− → ν x ∥ y

a

− → µ ∥ ν . P ∥ Q .

• .

P1 ∥ Q1 . P1 ∥ Q2 . P2 ∥ Q1 . P2 ∥ Q2 .

a

.

0.12

.

0.28

.

0.18

.

0.42 9 / 18

SLIDE 25

Parallel composition

. P .

• .

P1 . P2 .

a

.

0.4

.

0.6

. Q .

• .

Q1 . Q2 .

a

.

0.3

.

0.7

. x

a

− → µ y

a

− → ν x ∥ y

a

− → µ ∥ ν . P ∥ Q .

• .

P1 ∥ Q1 . P1 ∥ Q2 . P2 ∥ Q1 . P2 ∥ Q2 .

a

.

0.12

.

0.28

.

0.18

.

0.42 9 / 18

SLIDE 26

Compositionality properties

Various compositionality properties discussed in the literature: compositionality property modulus of continuity reference non-extensiveness ω(ϵ1, . . . , ϵn) =

n

max

i=1 ϵi

[BBLM13,GT14b] p-non-extensiveness ω(ϵ1, . . . , ϵn) = ( n ∑

i=1

ϵp

i

)1/p [BBLM13,GT14b] non-expansiveness ω(ϵ1, . . . , ϵn) =

n

∑

i=1

ϵi [DGJP04,GT13] K-Lipschitz continuity ω(ϵ1, . . . , ϵn) = K

n

∑

i=1

ϵi [GLT14,GT14b] uniform continuity ω(0, . . . , 0) = 0 and [GT14b] ω is continuous at (0, . . . , 0)

10 / 18

SLIDE 27

Parallel composition is non-expansive

. P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

. d(P, Q) = ϵ . P ∥ P .

• .

P′ ∥ P′ .

• .

a

.

1.0

.

b

.

1.0

. Q ∥ Q .

• .

Q′ ∥ Q′ . Q′ ∥ • .

• ∥ Q′

.

• ∥ •

.

• .

a

.

(1.0 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

b

.

1.0

. d(P ∥ P, Q ∥ Q) = 2ϵ − ϵ2 ≤ 2ϵ

11 / 18

SLIDE 28

Parallel composition is non-expansive

. P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

. d(P, Q) = ϵ . P ∥ P .

• .

P′ ∥ P′ .

• .

a

.

1.0

.

b

.

1.0

. Q ∥ Q .

• .

Q′ ∥ Q′ . Q′ ∥ • .

• ∥ Q′

.

• ∥ •

.

• .

a

.

(1.0 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

b

.

1.0

. d(P ∥ P, Q ∥ Q) = 2ϵ − ϵ2 ≤ 2ϵ

11 / 18

SLIDE 29

Parallel composition is non-expansive

. P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

. d(P, Q) = ϵ . P ∥ P .

• .

P′ ∥ P′ .

• .

a

.

1.0

.

b

.

1.0

. Q ∥ Q .

• .

Q′ ∥ Q′ . Q′ ∥ • .

• ∥ Q′

.

• ∥ •

.

• .

a

.

(1.0 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

b

.

1.0

. d(P ∥ P, Q ∥ Q) = 2ϵ − ϵ2 ≤ 2ϵ

11 / 18

SLIDE 30

Finite iteration and finite replication is continuous

. . P(r) .

• .

P1 . P2 .

• .
• .

P′

1

. P′

2

.

• .

.

a

.

r

.

1 − r

.

b

.

c

. 1.0 . 1.0 .

√

.

√

. 1.0 . x

a

− → µ xn+1

a

− → µ; δ(xn) . x

a

− → µ xω

a

− → µ; δ(xω) . P(r) = a.(b.√ ⊕r c.√) . d(P(r), P(r′)) = |r − r′| = ϵ . d(P(r)n, P(r′)n) ≤ 1 − (1 − ϵ)n . d(P(r)ω, P(r′)ω) = 1 if ϵ > 0 . unbounded recursion is not continuous! . Similar results hold for the π-caluclus Bang-operator

12 / 18

SLIDE 31

Finite iteration and finite replication is continuous

. . P(r) .

• .

P1 . P2 .

• .
• .

P′

1

. P′

2

.

• .

.

a

.

r

.

1 − r

.

b

.

c

. 1.0 . 1.0 .

√

.

√

. 1.0 . x

a

− → µ xn+1

a

− → µ; δ(xn) . x

a

− → µ xω

a

− → µ; δ(xω) . P(r) = a.(b.√ ⊕r c.√) . d(P(r), P(r′)) = |r − r′| = ϵ . d(P(r)n, P(r′)n) ≤ 1 − (1 − ϵ)n . d(P(r)ω, P(r′)ω) = 1 if ϵ > 0 . unbounded recursion is not continuous! . Similar results hold for the π-caluclus Bang-operator

12 / 18

SLIDE 32

Finite iteration and finite replication is continuous

. . P(r) .

• .

P1 . P2 .

• .
• .

P′

1

. P′

2

.

• .

.

a

.

r

.

1 − r

.

b

.

c

. 1.0 . 1.0 .

√

.

√

. 1.0 . x

a

− → µ xn+1

a

− → µ; δ(xn) . x

a

− → µ xω

a

− → µ; δ(xω) . P(r) = a.(b.√ ⊕r c.√) . d(P(r), P(r′)) = |r − r′| = ϵ . d(P(r)n, P(r′)n) ≤ 1 − (1 − ϵ)n . d(P(r)ω, P(r′)ω) = 1 if ϵ > 0 . unbounded recursion is not continuous! . Similar results hold for the π-caluclus Bang-operator

12 / 18

SLIDE 33

Finite iteration and finite replication is continuous

. . P(r) .

• .

P1 . P2 .

• .
• .

P′

1

. P′

2

.

• .

.

a

.

r

.

1 − r

.

b

.

c

. 1.0 . 1.0 .

√

.

√

. 1.0 . x

a

− → µ xn+1

a

− → µ; δ(xn) . x

a

− → µ xω

a

− → µ; δ(xω) . P(r) = a.(b.√ ⊕r c.√) . d(P(r), P(r′)) = |r − r′| = ϵ . d(P(r)n, P(r′)n) ≤ 1 − (1 − ϵ)n . d(P(r)ω, P(r′)ω) = 1 if ϵ > 0 . unbounded recursion is not continuous! . Similar results hold for the π-caluclus Bang-operator

12 / 18

SLIDE 34

Finite iteration and finite replication is continuous

. . P(r) .

• .

P1 . P2 .

• .
• .

P′

1

. P′

2

.

• .

.

a

.

r

.

1 − r

.

b

.

c

. 1.0 . 1.0 .

√

.

√

. 1.0 . x

a

− → µ xn+1

a

− → µ; δ(xn) . x

a

− → µ xω

a

− → µ; δ(xω) . P(r) = a.(b.√ ⊕r c.√) . d(P(r), P(r′)) = |r − r′| = ϵ . d(P(r)n, P(r′)n) ≤ 1 − (1 − ϵ)n . d(P(r)ω, P(r′)ω) = 1 if ϵ > 0 . unbounded recursion is not continuous! . Similar results hold for the π-caluclus Bang-operator

12 / 18

SLIDE 35

Non-expansive and continuous PA operators

The following process combinators are non-expansive: action prefix sequential composition (prob. and nonprob.) choice (prob. and nonprob.) CCS and CSP like parallel composition hiding, renaming, priority, ... The following process combinators are continuous: finite iteration over sequential composition finite replication over parallel composition probabilistic infinite replication (bang) The following process combinators are not continuous: infinite iteration over sequential composition (Kleene star) infinite replication over parallel composition (Bang operator) probabilistic infinite iteration (probabilistic Kleene star)

13 / 18

SLIDE 36

Non-expansive and continuous PA operators

The following process combinators are non-expansive: action prefix sequential composition (prob. and nonprob.) choice (prob. and nonprob.) CCS and CSP like parallel composition hiding, renaming, priority, ... The following process combinators are continuous: finite iteration over sequential composition finite replication over parallel composition probabilistic infinite replication (bang) The following process combinators are not continuous: infinite iteration over sequential composition (Kleene star) infinite replication over parallel composition (Bang operator) probabilistic infinite iteration (probabilistic Kleene star)

13 / 18

SLIDE 37

Non-expansive and continuous PA operators

The following process combinators are non-expansive: action prefix sequential composition (prob. and nonprob.) choice (prob. and nonprob.) CCS and CSP like parallel composition hiding, renaming, priority, ... The following process combinators are continuous: finite iteration over sequential composition finite replication over parallel composition probabilistic infinite replication (bang) The following process combinators are not continuous: infinite iteration over sequential composition (Kleene star) infinite replication over parallel composition (Bang operator) probabilistic infinite iteration (probabilistic Kleene star)

13 / 18

SLIDE 38

Process replication determines modulus of continuity

. x

a

− → µ y

a

− → ν x ∥ y

a

− → µ ∥ ν . ω∥(ϵ1, ϵ2) ≤ 1 − (1 − ϵ1)(1 − ϵ2) . x

a

y

a

f x y

a

.

f

. x

a

y

a

f x y

a

.

f

. x

a

y

a

f x y

a

.

f

. x

a

y

a

f x y

a

.

f

.

14 / 18

SLIDE 39

Process replication determines modulus of continuity

. x

a

− → µ y

a

− → ν x ∥ y

a

− → µ ∥ ν . ω∥(ϵ1, ϵ2) ≤ 1 − (1 − ϵ1)(1 − ϵ2) . x

a

− → µ y

a

− → ν f(x, y)

a

− → µ ∥ ν . ωf(ϵ1, ϵ2) ≤ 1 − (1 − ϵ1)(1 − ϵ2) . x

a

y

a

f x y

a

.

f

. x

a

y

a

f x y

a

.

f

. x

a

y

a

f x y

a

.

f

.

14 / 18

SLIDE 40

Process replication determines modulus of continuity

. x

a

− → µ y

a

− → ν x ∥ y

a

− → µ ∥ ν . ω∥(ϵ1, ϵ2) ≤ 1 − (1 − ϵ1)(1 − ϵ2) . x

a

y

a

f x y

a

.

f

. x

a

− → µ y

a

− → ν f(x, y)

a

− → µ ∥ µ ∥ ν . ωf(ϵ1, ϵ2) ≤ 1 − (1 − ϵ1)2(1 − ϵ2) . x

a

y

a

f x y

a

.

f

. x

a

y

a

f x y

a

.

f

.

14 / 18

SLIDE 41

Process replication determines modulus of continuity

. x

a

− → µ y

a

− → ν x ∥ y

a

− → µ ∥ ν . ω∥(ϵ1, ϵ2) ≤ 1 − (1 − ϵ1)(1 − ϵ2) . x

a

y

a

f x y

a

.

f

. x

a

y

a

f x y

a

.

f

. x

a

− → µ y

a

− → ν f(x, y)

a

− → µ ∥ µ ∥ µ ∥ ν ∥ ν . ωf(ϵ1, ϵ2) ≤ 1 − (1 − ϵ1)3(1 − ϵ2)2 . x

a

y

a

f x y

a

.

f

.

14 / 18

SLIDE 42

Process replication determines modulus of continuity

. x

a

− → µ y

a

− → ν x ∥ y

a

− → µ ∥ ν . ω∥(ϵ1, ϵ2) ≤ 1 − (1 − ϵ1)(1 − ϵ2) . x

a

y

a

f x y

a

.

f

. x

a

y

a

f x y

a

.

f

. x

a

y

a

f x y

a

.

f

. x

a

− → µ y

a

− → ν f(x, y)

a

− → [0.7]µ ∥ ν ⊕ [0.3]µ ∥ µ . ωf(ϵ1, ϵ2) ≤ 0.7(1 − (1 − ϵ1)(1 − ϵ2))+ . 0.3(1 − (1 − ϵ1)2)

14 / 18

SLIDE 43

From open term denotations to moduli of continuity

Basic idea: Assign a denotation to an open term that allows to derive an upper bound on the distance between closed instances of that term. Special case: Denotation of f(x1, . . . , xn) allows to derive the modulus of continuity of f. Closed substitutions σ1, σ2 induce a process distance e: V → [0, 1] defined by e(x) = d(σ1(x), σ2(x)) denotational model modulus of continuity

• det. process

M = V → N∞ D(m, e) = 1 − ∏

x∈V

(1 − e(x))m(x)

• prob. process

P = ∆(M) P(p, e) = ∑

m∈M

p(m) · D(m, e)

• nondet. prob.

D = {P ⊆ P | P ̸= ∅ ∧ A(P, e) = sup

p∈P

P(p, e) process ↓P = P}

15 / 18

SLIDE 44

From open term denotations to moduli of continuity

Basic idea: Assign a denotation to an open term that allows to derive an upper bound on the distance between closed instances of that term. Special case: Denotation of f(x1, . . . , xn) allows to derive the modulus of continuity of f. Closed substitutions σ1, σ2 induce a process distance e: V → [0, 1] defined by e(x) = d(σ1(x), σ2(x)) denotational model modulus of continuity

• det. process

M = V → N∞ D(m, e) = 1 − ∏

x∈V

(1 − e(x))m(x)

• prob. process

P = ∆(M) P(p, e) = ∑

m∈M

p(m) · D(m, e)

• nondet. prob.

D = {P ⊆ P | P ̸= ∅ ∧ A(P, e) = sup

p∈P

P(p, e) process ↓P = P}

15 / 18

SLIDE 45

From open term denotations to moduli of continuity

Basic idea: Assign a denotation to an open term that allows to derive an upper bound on the distance between closed instances of that term. Special case: Denotation of f(x1, . . . , xn) allows to derive the modulus of continuity of f. Closed substitutions σ1, σ2 induce a process distance e: V → [0, 1] defined by e(x) = d(σ1(x), σ2(x)) denotational model modulus of continuity

• det. process

M = V → N∞ D(m, e) = 1 − ∏

x∈V

(1 − e(x))m(x)

• prob. process

P = ∆(M) P(p, e) = ∑

m∈M

p(m) · D(m, e)

• nondet. prob.

D = {P ⊆ P | P ̸= ∅ ∧ A(P, e) = sup

p∈P

P(p, e) process ↓P = P}

15 / 18

SLIDE 46

From open term denotations to moduli of continuity

Basic idea: Assign a denotation to an open term that allows to derive an upper bound on the distance between closed instances of that term. Special case: Denotation of f(x1, . . . , xn) allows to derive the modulus of continuity of f. Closed substitutions σ1, σ2 induce a process distance e: V → [0, 1] defined by e(x) = d(σ1(x), σ2(x)) denotational model modulus of continuity

• det. process

M = V → N∞ D(m, e) = 1 − ∏

x∈V

(1 − e(x))m(x)

• prob. process

P = ∆(M) P(p, e) = ∑

m∈M

p(m) · D(m, e)

• nondet. prob.

D = {P ⊆ P | P ̸= ∅ ∧ A(P, e) = sup

p∈P

P(p, e) process ↓P = P}

15 / 18

SLIDE 47

From open term denotations to moduli of continuity

Basic idea: Assign a denotation to an open term that allows to derive an upper bound on the distance between closed instances of that term. Special case: Denotation of f(x1, . . . , xn) allows to derive the modulus of continuity of f. Closed substitutions σ1, σ2 induce a process distance e: V → [0, 1] defined by e(x) = d(σ1(x), σ2(x)) denotational model modulus of continuity

• det. process

M = V → N∞ D(m, e) = 1 − ∏

x∈V

(1 − e(x))m(x)

• prob. process

P = ∆(M) P(p, e) = ∑

m∈M

p(m) · D(m, e)

• nondet. prob.

D = {P ⊆ P | P ̸= ∅ ∧ A(P, e) = sup

p∈P

P(p, e) process ↓P = P}

15 / 18

SLIDE 48

Computation of modulus of continuity by example

Iteration over the evolution of the combined processes . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. f(x)

a

− → δ(g(x)) . x

a

− → µ g(x)

a

− → µ ∥ µ . f(P) .

• .

g(P) .

a

.

1.0

.

• .

P′ ∥ P′ .

• .

a

.

1.0

.

b

.

1.0

. f(Q) .

• .

g(Q) .

a

.

1.0

.

• .

Q′ ∥ Q′ . Q′ ∥ • .

• ∥ Q′

.

• ∥ •

.

• .

a

.

(1.0 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

b

.

1.0

. mg(x) = 2 .

d(g(P),g(Q))=1−(1−ϵ)2

. mf(x) = mg(x) = 2 .

d(f(P),f(Q))=1−(1−ϵ)2

16 / 18

SLIDE 49

Computation of modulus of continuity by example

Iteration over the evolution of the combined processes . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. f(x)

a

− → δ(g(x)) . x

a

− → µ g(x)

a

− → µ ∥ µ . f(P) .

• .

g(P) .

a

.

1.0

.

• .

P′ ∥ P′ .

• .

a

.

1.0

.

b

.

1.0

. f(Q) .

• .

g(Q) .

a

.

1.0

.

• .

Q′ ∥ Q′ . Q′ ∥ • .

• ∥ Q′

.

• ∥ •

.

• .

a

.

(1.0 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

b

.

1.0

. mg(x) = 2 .

d(g(P),g(Q))=1−(1−ϵ)2

. mf(x) = mg(x) = 2 .

d(f(P),f(Q))=1−(1−ϵ)2

16 / 18

SLIDE 50

Computation of modulus of continuity by example

Iteration over the evolution of the combined processes . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. f(x)

a

− → δ(g(x)) . x

a

− → µ g(x)

a

− → µ ∥ µ . f(P) .

• .

g(P) .

a

.

1.0

.

• .

P′ ∥ P′ .

• .

a

.

1.0

.

b

.

1.0

. f(Q) .

• .

g(Q) .

a

.

1.0

.

• .

Q′ ∥ Q′ . Q′ ∥ • .

• ∥ Q′

.

• ∥ •

.

• .

a

.

(1.0 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

b

.

1.0

. mg(x) = 2 .

d(g(P),g(Q))=1−(1−ϵ)2

. mf(x) = mg(x) = 2 .

d(f(P),f(Q))=1−(1−ϵ)2

16 / 18

SLIDE 51

Computation of modulus of continuity by example

Iteration over the evolution of the combined processes . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. f(x)

a

− → δ(g(x)) . x

a

− → µ g(x)

a

− → µ ∥ µ . f(P) .

• .

g(P) .

a

.

1.0

.

• .

P′ ∥ P′ .

• .

a

.

1.0

.

b

.

1.0

. f(Q) .

• .

g(Q) .

a

.

1.0

.

• .

Q′ ∥ Q′ . Q′ ∥ • .

• ∥ Q′

.

• ∥ •

.

• .

a

.

(1.0 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

b

.

1.0

. mg(x) = 2 .

d(g(P),g(Q))=1−(1−ϵ)2

. mf(x) = mg(x) = 2 .

d(f(P),f(Q))=1−(1−ϵ)2

16 / 18

SLIDE 52

Computation of modulus of continuity by example

Iteration over the evolution of the combined processes . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. f(x)

a

− → δ(g(x)) . x

a

− → µ g(x)

a

− → µ ∥ µ . f(P) .

• .

g(P) .

a

.

1.0

.

• .

P′ ∥ P′ .

• .

a

.

1.0

.

b

.

1.0

. f(Q) .

• .

g(Q) .

a

.

1.0

.

• .

Q′ ∥ Q′ . Q′ ∥ • .

• ∥ Q′

.

• ∥ •

.

• .

a

.

(1.0 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

b

.

1.0

. mg(x) = 2 .

d(g(P),g(Q))=1−(1−ϵ)2

. mf(x) = mg(x) = 2 .

d(f(P),f(Q))=1−(1−ϵ)2

16 / 18

SLIDE 53

Computation of modulus of continuity by example

Iteration over the evolution of the combined processes . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. f(x)

a

− → δ(g(x)) . x

a

− → µ g(x)

a

− → µ ∥ µ . f(P) .

• .

g(P) .

a

.

1.0

.

• .

P′ ∥ P′ .

• .

a

.

1.0

.

b

.

1.0

. f(Q) .

• .

g(Q) .

a

.

1.0

.

• .

Q′ ∥ Q′ . Q′ ∥ • .

• ∥ Q′

.

• ∥ •

.

• .

a

.

(1.0 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

b

.

1.0

. mg(x) = 2 .

d(g(P),g(Q))=1−(1−ϵ)2

. mf(x) = mg(x) = 2 .

d(f(P),f(Q))=1−(1−ϵ)2

16 / 18

SLIDE 54

Computation of modulus of continuity by example

Iteration over the evolution of the combined processes . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. f(x)

a

− → δ(g(x)) . x

a

− → µ g(x)

a

− → µ ∥ µ . f(P) .

• .

g(P) .

a

.

1.0

.

• .

P′ ∥ P′ .

• .

a

.

1.0

.

b

.

1.0

. f(Q) .

• .

g(Q) .

a

.

1.0

.

• .

Q′ ∥ Q′ . Q′ ∥ • .

• ∥ Q′

.

• ∥ •

.

• .

a

.

(1.0 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

b

.

1.0

. mg(x) = 2 .

d(g(P),g(Q))=1−(1−ϵ)2

. mf(x) = mg(x) = 2 .

d(f(P),f(Q))=1−(1−ϵ)2

16 / 18

SLIDE 55

Computation of modulus of continuity by example

Reactive testing of non-evolving process instances . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. x

a

− → µ f(x)

a

− → g(µ, µ) . x1

b

− → µ1 x2

b

− → µ2 g(x1, x2)

a

− → δ(•) . f(P) .

• .

g(P′, P′) .

• .
• .

a

.

1.0

.

a

.

1.0

. f(Q) .

• .

g(Q′, Q′) . g(Q′, •) . g(•, Q′) . g(•, •) .

• .
• .

a

.

(1 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

a

.

1.0

.

mg(x1)=mg(x2)=0

.

d(g(P′,P′),g(Q′,Q′))=0

.

sup(mg,1{x1,x2})(x1)=1

.

mf(x)=∑2

i=1 sup(mg,1{x1,x2})(xi)=2

.

d(f(P),f(Q))=1−(1−ϵ)2

17 / 18

SLIDE 56

Computation of modulus of continuity by example

Reactive testing of non-evolving process instances . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. x

a

− → µ f(x)

a

− → g(µ, µ) . x1

b

− → µ1 x2

b

− → µ2 g(x1, x2)

a

− → δ(•) . f(P) .

• .

g(P′, P′) .

• .
• .

a

.

1.0

.

a

.

1.0

. f(Q) .

• .

g(Q′, Q′) . g(Q′, •) . g(•, Q′) . g(•, •) .

• .
• .

a

.

(1 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

a

.

1.0

.

mg(x1)=mg(x2)=0

.

d(g(P′,P′),g(Q′,Q′))=0

.

sup(mg,1{x1,x2})(x1)=1

.

mf(x)=∑2

i=1 sup(mg,1{x1,x2})(xi)=2

.

d(f(P),f(Q))=1−(1−ϵ)2

17 / 18

SLIDE 57

Computation of modulus of continuity by example

Reactive testing of non-evolving process instances . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. x

a

− → µ f(x)

a

− → g(µ, µ) . x1

b

− → µ1 x2

b

− → µ2 g(x1, x2)

a

− → δ(•) . f(P) .

• .

g(P′, P′) .

• .
• .

a

.

1.0

.

a

.

1.0

. f(Q) .

• .

g(Q′, Q′) . g(Q′, •) . g(•, Q′) . g(•, •) .

• .
• .

a

.

(1 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

a

.

1.0

.

mg(x1)=mg(x2)=0

.

d(g(P′,P′),g(Q′,Q′))=0

.

sup(mg,1{x1,x2})(x1)=1

.

mf(x)=∑2

i=1 sup(mg,1{x1,x2})(xi)=2

.

d(f(P),f(Q))=1−(1−ϵ)2

17 / 18

SLIDE 58

Computation of modulus of continuity by example

Reactive testing of non-evolving process instances . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. x

a

− → µ f(x)

a

− → g(µ, µ) . x1

b

− → µ1 x2

b

− → µ2 g(x1, x2)

a

− → δ(•) . f(P) .

• .

g(P′, P′) .

• .
• .

a

.

1.0

.

a

.

1.0

. f(Q) .

• .

g(Q′, Q′) . g(Q′, •) . g(•, Q′) . g(•, •) .

• .
• .

a

.

(1 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

a

.

1.0

.

mg(x1)=mg(x2)=0

.

d(g(P′,P′),g(Q′,Q′))=0

.

sup(mg,1{x1,x2})(x1)=1

.

mf(x)=∑2

i=1 sup(mg,1{x1,x2})(xi)=2

.

d(f(P),f(Q))=1−(1−ϵ)2

17 / 18

SLIDE 59

Computation of modulus of continuity by example

Reactive testing of non-evolving process instances . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. x

a

− → µ f(x)

a

− → g(µ, µ) . x1

b

− → µ1 x2

b

− → µ2 g(x1, x2)

a

− → δ(•) . f(P) .

• .

g(P′, P′) .

• .
• .

a

.

1.0

.

a

.

1.0

. f(Q) .

• .

g(Q′, Q′) . g(Q′, •) . g(•, Q′) . g(•, •) .

• .
• .

a

.

(1 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

a

.

1.0

.

mg(x1)=mg(x2)=0

.

d(g(P′,P′),g(Q′,Q′))=0

.

sup(mg,1{x1,x2})(x1)=1

.

mf(x)=∑2

i=1 sup(mg,1{x1,x2})(xi)=2

.

d(f(P),f(Q))=1−(1−ϵ)2

17 / 18

SLIDE 60

Computation of modulus of continuity by example

Reactive testing of non-evolving process instances . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. x

a

− → µ f(x)

a

− → g(µ, µ) . x1

b

− → µ1 x2

b

− → µ2 g(x1, x2)

a

− → δ(•) . f(P) .

• .

g(P′, P′) .

• .
• .

a

.

1.0

.

a

.

1.0

. f(Q) .

• .

g(Q′, Q′) . g(Q′, •) . g(•, Q′) . g(•, •) .

• .
• .

a

.

(1 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

a

.

1.0

.

mg(x1)=mg(x2)=0

.

d(g(P′,P′),g(Q′,Q′))=0

.

sup(mg,1{x1,x2})(x1)=1

.

mf(x)=∑2

i=1 sup(mg,1{x1,x2})(xi)=2

.

d(f(P),f(Q))=1−(1−ϵ)2

17 / 18

SLIDE 61

Computation of modulus of continuity by example

Reactive testing of non-evolving process instances . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. x

a

− → µ f(x)

a

− → g(µ, µ) . x1

b

− → µ1 x2

b

− → µ2 g(x1, x2)

a

− → δ(•) . f(P) .

• .

g(P′, P′) .

• .
• .

a

.

1.0

.

a

.

1.0

. f(Q) .

• .

g(Q′, Q′) . g(Q′, •) . g(•, Q′) . g(•, •) .

• .
• .

a

.

(1 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

a

.

1.0

.

mg(x1)=mg(x2)=0

.

d(g(P′,P′),g(Q′,Q′))=0

.

sup(mg,1{x1,x2})(x1)=1

.

mf(x)=∑2

i=1 sup(mg,1{x1,x2})(xi)=2

.

d(f(P),f(Q))=1−(1−ϵ)2

17 / 18

SLIDE 62

Computation of modulus of continuity by example

Reactive testing of non-evolving process instances . P .

• .

P′ .

• .

a

.

1.0

.

b

.

1.0

. Q .

• .

Q′ .

• .
• .

a

.

1.0 − ϵ

.

ϵ

.

b

.

1.0

.

d(P,Q)=ϵ

. x

a

− → µ f(x)

a

− → g(µ, µ) . x1

b

− → µ1 x2

b

− → µ2 g(x1, x2)

a

− → δ(•) . f(P) .

• .

g(P′, P′) .

• .
• .

a

.

1.0

.

a

.

1.0

. f(Q) .

• .

g(Q′, Q′) . g(Q′, •) . g(•, Q′) . g(•, •) .

• .
• .

a

.

(1 − ϵ)2

.

ϵ − ϵ2

.

ϵ − ϵ2

.

ϵ2

.

a

.

1.0

.

mg(x1)=mg(x2)=0

.

d(g(P′,P′),g(Q′,Q′))=0

.

sup(mg,1{x1,x2})(x1)=1

.

mf(x)=∑2

i=1 sup(mg,1{x1,x2})(xi)=2

.

d(f(P),f(Q))=1−(1−ϵ)2

17 / 18

SLIDE 63

Conclusion

Summary SOS meta theory: novel approach to

▶ derive from a given PGSOS specification the compositionality property of each

• perator, and

▶ derive from a given compositionality property the syntactic requirements for

matching operator specifications

Process algebra: All non-recursive PA operators are continuous Process algebra: Recursive operators are continuous if the combined processes are only finitely many times replicated (finite iteration, finite bang) Future work Compositional reasoning wrt. discounted bisimulation metric Compositional reasoning wrt. other behavioral metrics (trace, testing, TV&mK bisim) Rule formats from the logical characterization of behavioral metrics Interpretation of behavioral distances as performance measure

18 / 18

SLIDE 64

Conclusion

Summary SOS meta theory: novel approach to

▶ derive from a given PGSOS specification the compositionality property of each

• perator, and

▶ derive from a given compositionality property the syntactic requirements for

matching operator specifications

Process algebra: All non-recursive PA operators are continuous Process algebra: Recursive operators are continuous if the combined processes are only finitely many times replicated (finite iteration, finite bang) Future work Compositional reasoning wrt. discounted bisimulation metric Compositional reasoning wrt. other behavioral metrics (trace, testing, TV&mK bisim) Rule formats from the logical characterization of behavioral metrics Interpretation of behavioral distances as performance measure

18 / 18

SLIDE 65

Conclusion

Summary SOS meta theory: novel approach to

▶ derive from a given PGSOS specification the compositionality property of each

• perator, and

▶ derive from a given compositionality property the syntactic requirements for

matching operator specifications

Process algebra: All non-recursive PA operators are continuous Process algebra: Recursive operators are continuous if the combined processes are only finitely many times replicated (finite iteration, finite bang) Future work Compositional reasoning wrt. discounted bisimulation metric Compositional reasoning wrt. other behavioral metrics (trace, testing, TV&mK bisim) Rule formats from the logical characterization of behavioral metrics Interpretation of behavioral distances as performance measure

18 / 18

SLIDE 66

Conclusion

Summary SOS meta theory: novel approach to

▶ derive from a given PGSOS specification the compositionality property of each

• perator, and

▶ derive from a given compositionality property the syntactic requirements for

matching operator specifications

Process algebra: All non-recursive PA operators are continuous Process algebra: Recursive operators are continuous if the combined processes are only finitely many times replicated (finite iteration, finite bang) Future work Compositional reasoning wrt. discounted bisimulation metric Compositional reasoning wrt. other behavioral metrics (trace, testing, TV&mK bisim) Rule formats from the logical characterization of behavioral metrics Interpretation of behavioral distances as performance measure

18 / 18