Labeled Transition Systems 2IT70 Finite Automata and Process Theory - - PowerPoint PPT Presentation

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Labeled Transition Systems 2IT70 Finite Automata and Process Theory Technische Universiteit Eindhoven June 4, 2014 The lady or the tiger open open marry eat 2 IT70 (2014) Labeled Transition Systems 2 / 26 The lady or the tiger open open

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SLIDE 1

Labeled Transition Systems 2IT70 Finite Automata and Process Theory

Technische Universiteit Eindhoven June 4, 2014

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SLIDE 2

The lady or the tiger

  • pen
  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 2 / 26

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SLIDE 3

The lady or the tiger

  • pen
  • pen

eat marry

  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 2 / 26

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SLIDE 4

The lady or the tiger

  • pen
  • pen

eat marry

  • pen

eat marry Sleft ≈ Sright while Sleft / ↔ Sright

2 IT70 (2014) Labeled Transition Systems 2 / 26

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SLIDE 5

A testing machine

reset

2 IT70 (2014) Labeled Transition Systems 3 / 26

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SLIDE 6

A testing machine

  • pen
  • pen

eat marry

reset

Sleft

  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 3 / 26

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SLIDE 7

A testing machine

  • pen
  • pen

eat marry

reset

Sleft

  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 3 / 26

slide-8
SLIDE 8

A testing machine

  • pen
  • pen

eat marry

reset

Sleft

  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 3 / 26

slide-9
SLIDE 9

A testing machine

  • pen
  • pen

eat marry

reset

Sleft

  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 3 / 26

slide-10
SLIDE 10

A testing machine

  • pen
  • pen

eat marry

reset

Sleft

  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 3 / 26

slide-11
SLIDE 11

A testing machine

  • pen
  • pen

eat marry

reset

Sleft

  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 3 / 26

slide-12
SLIDE 12

A testing machine

  • pen

eat marry

reset

Sright

  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 3 / 26

slide-13
SLIDE 13

A testing machine

  • pen

eat marry

reset

Sright

  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 3 / 26

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SLIDE 14

A testing machine

  • pen

eat marry

reset

Sright

  • pen

eat marry

2 IT70 (2014) Labeled Transition Systems 3 / 26

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SLIDE 15

Labeled transition system

labeled transition system S = (Q, Σ, →S, q0 ) finite/infinite set of states Q finite/infinite set of actions Σ transition relation →S ⊆ Q × Στ × Q initial state q0 transitions q

α

  • →S q′ for action α ∈ Στ

2 IT70 (2014) Labeled Transition Systems 4 / 26

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SLIDE 16

Example LTS

a buffer of capacity 2

ε 1 00 11 10 01 in1 in0

  • ut1
  • ut0

in0 in1 in0

  • ut0

in1

  • ut1
  • ut0
  • ut1

2 IT70 (2014) Labeled Transition Systems 5 / 26

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SLIDE 17

An infinite LTS

a counter process

q0 q1 q2 q3 q4 p1 p2 p3 p4 up up up down down down down down up down

2 IT70 (2014) Labeled Transition Systems 6 / 26

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SLIDE 18

Bisimilarity of states

LTS S = (Q, Σ, →S, q0 ) bisimulation relation R ⊆ Q × Q: for all q,p ∈ Q and α ∈ Στ (i) if R(q,p) and q

α

  • →S q′ then p

α

  • →S p′ such that R(q′,p′)

(ii) if R(q,p) and p

α

  • →S p′ then q

α

  • →S q′ such that R(q′,p′)

states q,p ∈ Q bisimilar if R(q,p) for bisimulation R for S notation q ↔ p

2 IT70 (2014) Labeled Transition Systems 7 / 26

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SLIDE 19

Bisimilarity of LTS

LTS S1 = (Q1, Σ, →1, q0 ) and LTS S2 = (Q2, Σ, →2, p0 ) bisimulation relation R ⊆ Q1 × Q2: for all q,p ∈ Q and α ∈ Στ (i) if R(q,p) and q

α

  • →S q′ then p

α

  • →S p′ such that R(q′,p′)

(ii) if R(q,p) and p

α

  • →S p′ then q

α

  • →S q′ such that R(q′,p′)

LTS S1,S2 bisimilar if R(q0,p0) for bisimulation R for S1 and S2 notation S1 ↔ S2

2 IT70 (2014) Labeled Transition Systems 8 / 26

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SLIDE 20

Example bisimilarity

a a b b

2 IT70 (2014) Labeled Transition Systems 9 / 26

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SLIDE 21

Example bisimilarity

a a b b bisimilarity of states

2 IT70 (2014) Labeled Transition Systems 9 / 26

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SLIDE 22

Example bisimilarity

a a b b a b

2 IT70 (2014) Labeled Transition Systems 9 / 26

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SLIDE 23

Example bisimilarity

a a b b a b bisimilarity of LTS

2 IT70 (2014) Labeled Transition Systems 9 / 26

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SLIDE 24

Clicker question L121

a a a Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 10 / 26

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SLIDE 25

Clicker question L121

a a a Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 10 / 26

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SLIDE 26

Clicker question L122

a b c c c a b Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 11 / 26

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SLIDE 27

Clicker question L122

a b c c c a b Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 11 / 26

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SLIDE 28

Clicker question L123

a a b b a Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 12 / 26

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SLIDE 29

Clicker question L123

a a b b a ? ? Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 12 / 26

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SLIDE 30

Clicker question L124

a b a b a b a b b a a b

Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 13 / 26

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SLIDE 31

Clicker question L124

a b a b a b a b b a a b

Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 13 / 26

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SLIDE 32

Clicker question L124

a b a b a b a b a b b a a b

Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 13 / 26

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SLIDE 33

Clicker question L124

a b a b a b a b a b b a a b

Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 13 / 26

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SLIDE 34

Clicker question L124

a b a b a b a b a b b a a b

Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 13 / 26

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SLIDE 35

Clicker question L124

a b a b a b a b a b b a a b

Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 13 / 26

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SLIDE 36

Coloring for bisimulation

coloring scheme (cn)∞

n=0 with cn ∶ Q → N satisfies

cn+1(q) = cn+1(p) ⇒ cn(q) = cn(p)

2 IT70 (2014) Labeled Transition Systems 14 / 26

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SLIDE 37

Coloring for bisimulation

coloring scheme (cn)∞

n=0 with cn ∶ Q → N satisfies

cn+1(q) = cn+1(p) ⇒ cn(q) = cn(p) finite LTS S = (Q, Σ, →S, q0 ), coloring scheme (cn)∞

n=0 such that

for all n ⩾ 0, p,q ∈ Q and α ∈ Στ cn+1(q) = cn+1(p) iff (i) if q

α

  • →S q′ then p

α

  • →S p′ such that cn(p′) = cn(q′)

(ii) if p

α

  • →S p′ then q

α

  • →S q′ such that cn(p′) = cn(q′)

2 IT70 (2014) Labeled Transition Systems 14 / 26

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SLIDE 38

Coloring for bisimulation

coloring scheme (cn)∞

n=0 with cn ∶ Q → N satisfies

cn+1(q) = cn+1(p) ⇒ cn(q) = cn(p) finite LTS S = (Q, Σ, →S, q0 ), coloring scheme (cn)∞

n=0 such that

for all n ⩾ 0, p,q ∈ Q and α ∈ Στ cn+1(q) = cn+1(p) iff (i) if q

α

  • →S q′ then p

α

  • →S p′ such that cn(p′) = cn(q′)

(ii) if p

α

  • →S p′ then q

α

  • →S q′ such that cn(p′) = cn(q′)

define R ⊆ Q × Q by R(q,p) ⇐ ⇒ ∀n ∶ cn(q) = cn(p) then R is a bisimulation relation for S

2 IT70 (2014) Labeled Transition Systems 14 / 26

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SLIDE 39

Coloring an LTS

q0 q1 q2 q3 q4 q5 q6 q7 a a c b b c c c d e

2 IT70 (2014) Labeled Transition Systems 15 / 26

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SLIDE 40

Coloring an LTS

q0 q1 q2 q3 q4 q5 q6 q7 a a c b b c c c d e

1 2 2 3 4 5 6 6

2 IT70 (2014) Labeled Transition Systems 15 / 26

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SLIDE 41

Coloring an LTS

q0 q1 q2 q3 q4 q5 q6 q7 a a c b b c c c d e

7 8 9 10 11 12 6 6

2 IT70 (2014) Labeled Transition Systems 15 / 26

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SLIDE 42

Dealing with silent steps

u v t S1 s u v s S0 u v t s S2 a τ a b a b a τ a b

2 IT70 (2014) Labeled Transition Systems 16 / 26

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SLIDE 43

A testing machine for τ-transitions

u v t s a τ a b t s u v a τ a b

reset

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 44

A testing machine for τ-transitions

u v t s a τ a b

reset

S1 a b

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 45

A testing machine for τ-transitions

u v t s a τ a b

reset

S1 a b

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 46

A testing machine for τ-transitions

u v t s a τ a b

reset

S1 a b

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 47

A testing machine for τ-transitions

u v t s a τ a b

reset

S1 a b

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 48

A testing machine for τ-transitions

u v t s a τ a b

reset

S1 a b

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 49

A testing machine for τ-transitions

t s u v a τ a b

reset

S2 a b

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 50

A testing machine for τ-transitions

t s u v a τ a b

reset

S2 a b

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 51

A testing machine for τ-transitions

t s u v a τ a b

reset

S2 a b

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 52

A testing machine for τ-transitions

t s u v a τ a b

reset

S2 a b

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 53

A testing machine for τ-transitions

t s u v a τ a b

reset

S2 a b

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 54

A testing machine for τ-transitions

u v t s a τ a b t s u v a τ a b

reset

2 IT70 (2014) Labeled Transition Systems 17 / 26

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SLIDE 55

Inert silent steps

τ τ τ a

two certainly inert τ-steps and one probably non-inert τ-step

2 IT70 (2014) Labeled Transition Systems 18 / 26

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SLIDE 56

Branching bisimulation

LTS S = (Q, Σ, →S, q0 ) branching bisimulation relation R ⊆ Q × Q for S (i) if R(q,p) and q

a

  • →S q′ then exist ¯

p,p′ ∈ Q such that p

τ

  • →∗

S ¯

p and ¯ p

a

  • →S p′ with R(q, ¯

p) and R(q′,p′) (ii) symmetric condition if R(q,p) and p

a

  • →S p′

2 IT70 (2014) Labeled Transition Systems 19 / 26

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SLIDE 57

Branching bisimulation

LTS S = (Q, Σ, →S, q0 ) branching bisimulation relation R ⊆ Q × Q for S (i) if R(q,p) and q

a

  • →S q′ then exist ¯

p,p′ ∈ Q such that p

τ

  • →∗

S ¯

p and ¯ p

a

  • →S p′ with R(q, ¯

p) and R(q′,p′) (ii) symmetric condition if R(q,p) and p

a

  • →S p′

(iii) if R(q,p) and q

τ

  • →S q′ then either exist ¯

p,p′ ∈ Q such that p

τ

  • →∗

S ¯

p and ¯ p

τ

  • →S p′ with R(q, ¯

p) and R(q′,p′) or exists p′ ∈ Q such that p

τ

  • →∗

S p′ with R(q,p′) and R(q′,p′)

(iv) symmetric condition if R(q,p) and p

τ

  • →S p′

states q,p ∈ Q branching bisimilar if R(q,p)

2 IT70 (2014) Labeled Transition Systems 19 / 26

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SLIDE 58

Branching bisimulation (cont.)

s1 t1 s2 s′

2

t2 a τ a s1 t1 s2 s′

2

t2 τ τ τ s1 t1 s2 τ

left-to-right transfer condition for visible actions left-to-right transfer condition for silent steps

2 IT70 (2014) Labeled Transition Systems 20 / 26

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SLIDE 59

Clicker question L125

a τ a Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 21 / 26

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SLIDE 60

Clicker question L125

a τ a ? Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 21 / 26

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SLIDE 61

Clicker question L126

a a τ b b a Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 22 / 26

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SLIDE 62

Clicker question L126

a a τ b b a Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 22 / 26

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SLIDE 63

Clicker question L127

a b a b τ τ Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 23 / 26

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SLIDE 64

Clicker question L127

a b a b τ τ Are these two LTS bisimilar?

  • A. Yes
  • B. No
  • C. Can’t tell

2 IT70 (2014) Labeled Transition Systems 23 / 26

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SLIDE 65

A negative example

τ τ a b S1 a τ b S2 τ b a S3 a b S4

2 IT70 (2014) Labeled Transition Systems 24 / 26

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SLIDE 66

A negative example

τ τ a b S1 a τ b S2 τ b a S3 a b S4 none branching bisimilar

2 IT70 (2014) Labeled Transition Systems 24 / 26

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SLIDE 67

A positive example

τ a b S′

1

a τ a b S′

2

τ b a b S′

3

a b S′

4

2 IT70 (2014) Labeled Transition Systems 25 / 26

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SLIDE 68

A positive example

τ a b S′

1

a τ a b S′

2

τ b a b S′

3

a b S′

4

all branching bisimilar

2 IT70 (2014) Labeled Transition Systems 25 / 26

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SLIDE 69

Coloring for branching bisimulation

q0 q4 q1 q5 q2 q6 q3 τ a τ b a b b τ

2 IT70 (2014) Labeled Transition Systems 26 / 26

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SLIDE 70

Coloring for branching bisimulation

q0 q1 q2 q3 q4 q5 q6 τ a τ b a b b τ

1 1 2 2 3 3 3

2 IT70 (2014) Labeled Transition Systems 26 / 26

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SLIDE 71

Coloring for branching bisimulation

q0 q1 q2 q3 q4 q5 q6 τ a τ b a b b τ

4 5 6 6 3 3 3

2 IT70 (2014) Labeled Transition Systems 26 / 26

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SLIDE 72

Coloring for branching bisimulation

q0 q1 q2 q3 q4 q5 q6 τ a τ b a b b τ

7 8 6 6 3 3 3

2 IT70 (2014) Labeled Transition Systems 26 / 26