News on Safety Properties for Timed Petri Nets Patrick Totzke - - PowerPoint PPT Presentation

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News on Safety Properties for Timed Petri Nets

Patrick Totzke

Edinburgh

2018-09-24

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Networks of Timed Automata

• a (finite-state) control program (S)
• K indistinguishable d-clock Timed Automata (C)

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SLIDE 4

Networks of Timed Automata

• a (finite-state) control program (S)
• K indistinguishable d-clock Timed Automata (C)
• global time ticks + handshake communication.

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SLIDE 5

Networks of Timed Automata

• a (finite-state) control program (S)
• K indistinguishable d-clock Timed Automata (C)
• global time ticks + handshake communication.

Q: Suppose all K clients are identically initialized. Can one of the clients reach a failure state?

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SLIDE 6

Networks of Timed Automata

• a (finite-state) control program (S)
• K indistinguishable d-clock Timed Automata (C)
• global time ticks + handshake communication.

Q: Suppose all K clients are identically initialized. Can one of the clients reach a failure state? A: Simple! It’s a big (K ∗ d)-clock TA! [AD94]

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SLIDE 7

Networks of Timed Automata

• a (finite-state) control program (S)
• K indistinguishable d-clock Timed Automata (C)
• global time ticks + handshake communication.

Q: Suppose all K clients are identically initialized. Can one of the clients reach a failure state? A: Simple! It’s a big (K ∗ d)-clock TA! [AD94] Q: What if the number K of clients is a parameter? Does there exist K such that S × CK can fail?

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SLIDE 8

Networks of Timed Automata

• a (finite-state) control program (S)
• K indistinguishable d-clock Timed Automata (C)
• global time ticks + handshake communication.

Q: Suppose all K clients are identically initialized. Can one of the clients reach a failure state? A: Simple! It’s a big (K ∗ d)-clock TA! [AD94] Q: What if the number K of clients is a parameter? Does there exist K such that S × CK can fail? A: Not so simple:

• decidable for d = 1 [AJ03];
• undecidable for d ≥ 2 [ADM04].

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SLIDE 9

Networks of Timed Automata

• a (finite-state) control program (S)
• K indistinguishable d-clock Timed Automata (C)
• global time ticks + handshake communication.

Q: Suppose all K clients are identically initialized. Can one of the clients reach a failure state? A: Simple! It’s a big (K ∗ d)-clock TA! [AD94] Q: What if the number K of clients is a parameter? Does there exist K such that S × CK can fail? A: Not so simple:

• decidable for d = 1 [AJ03];
• undecidable for d ≥ 2 [ADM04].

Q: What if there is no controller? Exists K so that CK can fail?

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SLIDE 10

Networks of Timed Automata

• a (finite-state) control program (S)
• K indistinguishable d-clock Timed Automata (C)
• global time ticks + handshake communication.

Q: Suppose all K clients are identically initialized. Can one of the clients reach a failure state? A: Simple! It’s a big (K ∗ d)-clock TA! [AD94] Q: What if the number K of clients is a parameter? Does there exist K such that S × CK can fail? A: Not so simple:

• decidable for d = 1 [AJ03];
• undecidable for d ≥ 2 [ADM04].

Q: What if there is no controller? Exists K so that CK can fail? A: PSPACE-complete [for d = 1, This paper]

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SLIDE 11

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.0 1.0 5.2

q

1.1 5.0

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

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SLIDE 12

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.0 1.0 5.2

q

1.1 5.0

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd

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SLIDE 13

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.0 1.0 5.2

q

1.1 5.0

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd

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SLIDE 14

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.0 1.0 5.2

q

1.1 5.0

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

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SLIDE 15

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.0 1.0 5.2

q

1.1 5.0

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

0.1

− − →

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SLIDE 16

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

0.1

− − →

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SLIDE 17

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

0.1

− − → and discrete steps

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SLIDE 18

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

0.1

− − → and discrete steps

t

− − →

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SLIDE 19

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

0.1

− − → and discrete steps

t

− − →

3 / 15

SLIDE 20

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

0.1

− − → and discrete steps

t

− − →

3 / 15

SLIDE 21

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

0.1

− − → and discrete steps

t

− − →

3 / 15

SLIDE 22

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

0.1

− − → and discrete steps

t

− − →

• Reachability, Coverability, Boundedness (. . . ) undecidable

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SLIDE 23

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

0.1

− − → and discrete steps

t

− − →

• Reachability, Coverability, Boundedness (. . . ) undecidable
• Coverability: M

∗

− − →

t

− − →?

• decidable via WSTS if d = 1
• inter-reducible to Coverability for ordered data nets

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SLIDE 24

Timed Petri Nets

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• Configurations are finite multisets over P × Rd
• global time ticks

0.1

− − → and discrete steps

t

− − →

• Reachability, Coverability, Boundedness (. . . ) undecidable
• Coverability: M

∗

− − →

t

− − →?

• decidable via WSTS if d = 1
• inter-reducible to Coverability for ordered data nets
• Existential Coverability: ∃n ∈ N with M · n

∗

− − →

t

− − →?

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SLIDE 25

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

5.3

q

5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

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SLIDE 26

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

5.3

q

5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z The Region of this configuration is a sequence

s,0 q,5 r,1 r,1 r,1 p,5

.0 .1 .2 .3

• f multisets over P × {0, . . . , cmax}, one for each fractional value*

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SLIDE 27

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

5.3

q

5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z The Region of this configuration is a sequence

s,0 q,5 r,1 r,1 r,1 p,5

.0 .1 .2 .3

• f multisets over P × {0, . . . , cmax}, one for each fractional value*

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SLIDE 28

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

5.3

q

5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z The Region of this configuration is a sequence

s,0 q,5 r,1 r,1 r,1 p,5

.0 .1 .2 .3

• f multisets over P × {0, . . . , cmax}, one for each fractional value*

4 / 15

SLIDE 29

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

5.3

q

5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z The Region of this configuration is a sequence

s,0 q,5 r,1 r,1 r,1 p,5

.0 .1 .2 .3

• f multisets over P × {0, . . . , cmax}, one for each fractional value*

4 / 15

SLIDE 30

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

5.3

q

5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z The Region of this configuration is a sequence

s,0 q,5 r,1 r,1 r,1 p,5

.0 .1 .2 .3

• f multisets over P × {0, . . . , cmax}, one for each fractional value*

4 / 15

SLIDE 31

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

5.3

q

5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z The Region of this configuration is a sequence

s,0 q,5 r,1 r,1 r,1 p,5

.0 .1 .2 .3

• f multisets over P × {0, . . . , cmax}, one for each fractional value*

4 / 15

SLIDE 32

Coverability via Regions

5 / 15

SLIDE 33

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.0 1.0 5.2

q

1.1 5.0

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

5 / 15

SLIDE 34

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.0 1.0 5.2

q

1.1 5.0

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

p,1 p,1 q,5 q,1 p,5 5 / 15

SLIDE 35

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.0 1.0 5.2

q

1.1 5.0

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• short time ticks

p,1 p,1 q,5 q,1 p,5 5 / 15

SLIDE 36

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• short time ticks

p,1 p,1 q,5 q,1 p,5 p,1 p,1 q,5 q,1 p,5

ε

5 / 15

SLIDE 37

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• short time ticks
• discrete transition firing

p,1 p,1 q,5 q,1 p,5 p,1 p,1 q,5 q,1 p,5

ε

5 / 15

SLIDE 38

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• short time ticks
• discrete transition firing

p,1 p,1 q,5 q,1 p,5 p,1 p,1 q,5 q,1 p,5

ε

5 / 15

SLIDE 39

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• short time ticks
• discrete transition firing

p,1 p,1 q,5 q,1 p,5 p,1 p,1 q,5 q,1 p,5

ε

s,0 q,5 r,1 r,1 r,1 p,5

t

5 / 15

SLIDE 40

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• short time ticks
• discrete transition firing

p,1 p,1 q,5 q,1 p,5 p,1 p,1 q,5 q,1 p,5

ε

s,0 q,5 r,1 r,1 r,1 p,5

t

5 / 15

SLIDE 41

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• short time ticks
• discrete transition firing
• long time ticks

p,1 p,1 q,5 q,1 p,5 p,1 p,1 q,5 q,1 p,5

ε

s,0 q,5 r,1 r,1 r,1 p,5

t

5 / 15

SLIDE 42

Coverability via Regions

0 ≤ x ≤ 5 1 < y ≤ 2 0 ≤ z ≤ 0

t p

1.1 1.1 5.3

q

1.2 5.1

r

1.2 1.2 1.2

s

0.0

x2 y y3 z

• short time ticks
• discrete transition firing
• long time ticks

p,1 p,1 q,5 q,1 p,5 p,1 p,1 q,5 q,1 p,5

ε

s,0 q,5 r,1 r,1 r,1 p,5

t

p,6 s,0 q,5 r,1 r,1 r,1

t

5 / 15

SLIDE 43

Coverability via Regions

• Obs. 1
• region equality is a time-abstract bisimulation
• unlike for TA, it has infinite index

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SLIDE 44

Coverability via Regions

• Obs. 1
• region equality is a time-abstract bisimulation
• unlike for TA, it has infinite index
• Obs. 2
• steps between regions are monotone wrt. region embedding
• embedding is a well-quasi-order

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SLIDE 45

Coverability via Regions

• Obs. 1
• region equality is a time-abstract bisimulation
• unlike for TA, it has infinite index
• Obs. 2
• steps between regions are monotone wrt. region embedding
• embedding is a well-quasi-order

Together, this yields decidability via the WSTS approach (and completeness for Fωω ).

6 / 15

SLIDE 46

Coverability via Regions

• Obs. 1
• region equality is a time-abstract bisimulation
• unlike for TA, it has infinite index
• Obs. 2
• steps between regions are monotone wrt. region embedding
• embedding is a well-quasi-order

Together, this yields decidability via the WSTS approach (and completeness for Fωω ). NB: this fails for d ≥ 2, for several reasons... Indeed we have undecidability in general.

6 / 15

SLIDE 47

Existential Coverability

In: A TPN, a marking M, a transition t Question: Does there exist ∃n ∈ N with M · n

∗

− − →

t

− − →?

7 / 15

SLIDE 48

Existential Coverability

In: A TPN, a marking M, a transition t Question: Does there exist ∃n ∈ N with M · n

∗

− − →

t

− − →? ≈ parametrized safety checking Networks of Timed Automata

7 / 15

SLIDE 49

Existential Coverability

In: A TPN, a marking M, a transition t Question: Does there exist ∃n ∈ N with M · n

∗

− − →

t

− − →? ≈ parametrized safety checking Networks of Timed Automata ≈ Coverability for TPN with continuous firing semantics ´ a la Haddad et al.’17.

7 / 15

SLIDE 50

Existential Coverability

In: A TPN, a marking M, a transition t Question: Does there exist ∃n ∈ N with M · n

∗

− − →

t

− − →? ≈ parametrized safety checking Networks of Timed Automata ≈ Coverability for TPN with continuous firing semantics ´ a la Haddad et al.’17. − (logspace) reduces to Coverability

7 / 15

SLIDE 51

Existential Coverability

In: A TPN, a marking M, a transition t Question: Does there exist ∃n ∈ N with M · n

∗

− − →

t

− − →? ≈ parametrized safety checking Networks of Timed Automata ≈ Coverability for TPN with continuous firing semantics ´ a la Haddad et al.’17. − (logspace) reduces to Coverability − We show PSPACE-completeness

7 / 15

SLIDE 52

Existential Coverability

In: A TPN, a marking M, a transition t Question: Does there exist ∃n ∈ N with M · n

∗

− − →

t

− − →? ≈ parametrized safety checking Networks of Timed Automata ≈ Coverability for TPN with continuous firing semantics ´ a la Haddad et al.’17. − (logspace) reduces to Coverability − We show PSPACE-completeness

7 / 15

SLIDE 53

Existential Coverability

In: A TPN, a marking M, a transition t Question: Does there exist ∃n ∈ N with M · n

∗

− − →

t

− − →? ≈ parametrized safety checking Networks of Timed Automata ≈ Coverability for TPN with continuous firing semantics ´ a la Haddad et al.’17. − (logspace) reduces to Coverability − We show PSPACE-completeness

LB: iterated monotone circuits

7 / 15

SLIDE 54

Existential Coverability

In: A TPN, a marking M, a transition t Question: Does there exist ∃n ∈ N with M · n

∗

− − →

t

− − →? ≈ parametrized safety checking Networks of Timed Automata ≈ Coverability for TPN with continuous firing semantics ´ a la Haddad et al.’17. − (logspace) reduces to Coverability − We show PSPACE-completeness

LB: iterated monotone circuits UB: Regions + forward acceleration

7 / 15

SLIDE 55

Existential Coverability

Q: What’s different compared to Coverability?

8 / 15

SLIDE 56

Existential Coverability

Q: What’s different compared to Coverability? A: Token multiplicities do not matter. So,

8 / 15

SLIDE 57

Existential Coverability

Q: What’s different compared to Coverability? A: Token multiplicities do not matter. So,

• 1. A Region a sequence of multisets over Σ

def

= P × {0, . . . , cmax}

8 / 15

SLIDE 58

Existential Coverability

Q: What’s different compared to Coverability? A: Token multiplicities do not matter. So,

• 1. A Region a sequence of multisets over Σ

def

= P × {0, . . . , cmax} sets S ⊆ Σ

8 / 15

SLIDE 59

Existential Coverability

Q: What’s different compared to Coverability? A: Token multiplicities do not matter. So,

• 1. A Region a sequence of multisets over Σ

def

= P × {0, . . . , cmax} sets S ⊆ Σ This already improves the upper bound to Fω

8 / 15

SLIDE 60

Existential Coverability

Q: What’s different compared to Coverability? A: Token multiplicities do not matter. So,

• 1. A Region a sequence of multisets over Σ

def

= P × {0, . . . , cmax} sets S ⊆ Σ This already improves the upper bound to Fω

• 2. Wlog., the net is non-consuming: •t ⊆ t• for all transitions t.

8 / 15

SLIDE 61

Existential Coverability

Q: What’s different compared to Coverability? A: Token multiplicities do not matter. So,

• 1. A Region a sequence of multisets over Σ

def

= P × {0, . . . , cmax} sets S ⊆ Σ This already improves the upper bound to Fω

• 2. Wlog., the net is non-consuming: •t ⊆ t• for all transitions t.

This means that discrete transition firing is non-decreasing and for every region R

• there is a unique maximal region R′ with R

disc

− − →∗R′

• R′ is (Ptime) computable

8 / 15

SLIDE 62

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A

SLIDE 63

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

SLIDE 64

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε

SLIDE 65

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

SLIDE 66

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε

SLIDE 67

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

SLIDE 68

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε

SLIDE 69

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

SLIDE 70

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε

SLIDE 71

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

SLIDE 72

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 73

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 74

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 75

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 76

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 77

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 78

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 79

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 80

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 81

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 82

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

SLIDE 83

Existential Coverability: Key Observation

When forward exploring zeno behaviour regions “stabilize” and the limit is expressible as regular expression. In this example as ZY ∗A. A A

disc

A ∅ ε A B

disc

A B ∅ ε A B C

disc

A B C ∅ ε A B C D

disc

A B C D ∅ ε A B C D E

disc

A B C D E Y Z

9 / 15

SLIDE 84

Existential Coverability: Construction

• use regular expressions over 2Σ to represent (limit) regions
• careful forward exploration, using intermediate compression

steps that add Kleene *s

10 / 15

SLIDE 85

Forward Exploration

11 / 15

SLIDE 86

Forward Exploration

x∗ x1 start

SLIDE 87

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate

SLIDE 88

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate

SLIDE 89

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate

SLIDE 90

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate

SLIDE 91

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate

SLIDE 92

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate

SLIDE 93

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate

SLIDE 94

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate

SLIDE 95

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate

SLIDE 96

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate

SLIDE 97

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate

SLIDE 98

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate

SLIDE 99

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate (x4

0)∗

x4

1

(x4

3)∗

x4

4

collapse

SLIDE 100

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate (x4

0)∗

x4

1

(x4

3)∗

x4

4

collapse ∗

SLIDE 101

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate (x4

0)∗

x4

1

(x4

3)∗

x4

4

collapse ∗ (x4

3)∗

x4

4

(x4

0 + 1)∗

(x4

1 + 1)

rotate

SLIDE 102

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate (x4

0)∗

x4

1

(x4

3)∗

x4

4

collapse ∗ (x4

3)∗

x4

4

(x4

0 + 1)∗

(x4

1 + 1)

rotate

SLIDE 103

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate (x4

0)∗

x4

1

(x4

3)∗

x4

4

collapse ∗ (x4

3)∗

x4

4

(x4

0 + 1)∗

(x4

1 + 1)

rotate (x4

3)∗

(x4

1 + 1)

collapse

SLIDE 104

Forward Exploration

x∗ x1 start (x1

0)∗

x1

1

saturate (x1

0)∗

x1

1

(x1

0 + 1)

rotate (x2

0)∗

x2

1

x2

2

saturate (x2

0)∗

x2

1

x2

2

(x2

0 + 1)

rotate (x3

0)∗

x3

1

x3

2

x3

3

saturate (x3

0)∗

x3

1

x3

2

x3

3

(x3

0 + 1)

rotate (x4

0)∗

x4

1

x4

2

x4

3

x4

4

saturate (x4

0)∗

x4

1

(x4

3)∗

x4

4

collapse ∗ (x4

3)∗

x4

4

(x4

0 + 1)∗

(x4

1 + 1)

rotate (x4

3)∗

(x4

1 + 1)

collapse ∗

11 / 15

SLIDE 105

Existential Coverability: Construction

• use regular expressions over 2Σ to represent (limit) regions
• careful forward exploration, using intermediate compression

steps that add Kleene *s

12 / 15

SLIDE 106

Existential Coverability: Construction

• use regular expressions over 2Σ to represent (limit) regions
• careful forward exploration, using intermediate compression

steps that add Kleene *s

Properties

• Computes the set of coverable regions

12 / 15

SLIDE 107

Existential Coverability: Construction

• use regular expressions over 2Σ to represent (limit) regions
• careful forward exploration, using intermediate compression

steps that add Kleene *s

Properties

• Computes the set of coverable regions
• does not need nondeterministic branching

12 / 15

SLIDE 108

Existential Coverability: Construction

• use regular expressions over 2Σ to represent (limit) regions
• careful forward exploration, using intermediate compression

steps that add Kleene *s

Properties

• Computes the set of coverable regions
• does not need nondeterministic branching
• every explored RE has length ≤ 5 .

12 / 15

SLIDE 109

Existential Coverability: Construction

• use regular expressions over 2Σ to represent (limit) regions
• careful forward exploration, using intermediate compression

steps that add Kleene *s

Properties

• Computes the set of coverable regions
• does not need nondeterministic branching
• every explored RE has length ≤ 5 .

12 / 15

SLIDE 110

Existential Coverability: Construction

• use regular expressions over 2Σ to represent (limit) regions
• careful forward exploration, using intermediate compression

steps that add Kleene *s

Properties

• Computes the set of coverable regions
• does not need nondeterministic branching
• every explored RE has length ≤ 5 .

Corollary

• the sequence is singly exponential
• checking Existential Coverability is in PSPACE.

12 / 15

SLIDE 111

WIP 1: multi-dimensional ECOVER

13 / 15

SLIDE 112

WIP 1: multi-dimensional ECOVER

Conjecture

ECOVER is PSPACE-completeness for any fixed dimension d.

13 / 15

SLIDE 113

WIP 1: multi-dimensional ECOVER

Conjecture

ECOVER is PSPACE-completeness for any fixed dimension d.

• Regions become directed (hyper) graphs with edges labelled

by subsets of P × [0 . . . cmax + 1]d

13 / 15

SLIDE 114

WIP 1: multi-dimensional ECOVER

Conjecture

ECOVER is PSPACE-completeness for any fixed dimension d.

• Regions become directed (hyper) graphs with edges labelled

by subsets of P × [0 . . . cmax + 1]d

• Semantics of Kleene stars?

13 / 15

SLIDE 115

WIP 2: Coverability for TBPP

BPP nets: every transition consumes at most one token

14 / 15

SLIDE 116

WIP 2: Coverability for TBPP

BPP nets: every transition consumes at most one token

• ∼ context-free controlled TPN

14 / 15

SLIDE 117

WIP 2: Coverability for TBPP

BPP nets: every transition consumes at most one token

• ∼ context-free controlled TPN
• Coverability
• in PSPACE for all d

14 / 15

SLIDE 118

WIP 2: Coverability for TBPP

BPP nets: every transition consumes at most one token

• ∼ context-free controlled TPN
• Coverability
• in PSPACE for all d

(lossy semantics witnesses visit only small regions)

14 / 15

SLIDE 119

WIP 2: Coverability for TBPP

BPP nets: every transition consumes at most one token

• ∼ context-free controlled TPN
• Coverability
• in PSPACE for all d

(lossy semantics witnesses visit only small regions)

• conjecture: NP-complete for d = 1

14 / 15

SLIDE 120

WIP 2: Coverability for TBPP

BPP nets: every transition consumes at most one token

• ∼ context-free controlled TPN
• Coverability
• in PSPACE for all d

(lossy semantics witnesses visit only small regions)

• conjecture: NP-complete for d = 1
• Reachability:
• 1st step: bound ”time to kill” a region?

14 / 15

SLIDE 121

thank you.

15 / 15