On Boundedness Problems for Pushdown Vector Addition Systems J er - - PowerPoint PPT Presentation

On Boundedness Problems for Pushdown Vector Addition Systems

J´ erˆ

• me Leroux

Gr´ egoire Sutre Patrick Totzke September 21, 2015

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Vector Addition Systems – Recap

Definition

A VAS is a finite set of vectors a ∈ Zd. For v, v′ : Nd it has a step v

a

− − → v′ if v′ = v + a.

◮ Equivalent to Petri Nets

(concurrency, weak counters, event systems)

◮ Reachability: decidable Mayr’81,Kosaraju’82, . . . Leroux and Schmitz’15 ◮ Coverability, Boundedness: ExpSpace-complete Lipton’76, Rackoff’78 ◮ Most Games/Equivalences undecidable (e.g. Bisimulation) Janˇ car’95

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Pushdown Vector Addition Systems

. . . are products of VAS with pushdown automata. s q

push(A), −1

• pop(A),

2

• nop,

−1

• 3 / 12

Pushdown Vector Addition Systems

. . . are products of VAS with pushdown automata. s q

push(A), −1

• pop(A),

2

• nop,

−1

• s, ⊥,
• 2

1

• 3 / 12

Pushdown Vector Addition Systems

. . . are products of VAS with pushdown automata. s q

push(A), −1

• pop(A),

2

• nop,

−1

• s, ⊥,
• 2

1

• −

− →− − → s, AA⊥, 1

• 3 / 12

Pushdown Vector Addition Systems

. . . are products of VAS with pushdown automata. s q

push(A), −1

• pop(A),

2

• nop,

−1

• s, ⊥,
• 2

1

• −

− →− − → s, AA⊥, 1

• −

− → q, AA⊥,

• 3 / 12

Pushdown Vector Addition Systems

. . . are products of VAS with pushdown automata. s q

push(A), −1

• pop(A),

2

• nop,

−1

• s, ⊥,
• 2

1

• −

− →− − → s, AA⊥, 1

• −

− → q, AA⊥,

• −

− →− − → q, ⊥, 4

• 3 / 12

Pushdown Vector Addition Systems

. . . are products of VAS with pushdown automata. They can for example model recursive prorams with variables over N.

1: x ← n 2: procedure DoubleX 3:

if (⋆ ∧ x > 0) then

4:

x ← (x − 1)

5:

DoubleX

6:

end if

7:

x ← (x + 2)

8: end procedure

2 start 3 5 6 7 8 −1 push(A) +2 pop(A)

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Pushdown Vector Addition Systems

◮ Reachability = Coverability (= State-Reachability)

Tower-hard Lazic’13

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Pushdown Vector Addition Systems

◮ Reachability d dim. = Coverability d + 1 dim.

Tower-hard Lazic’13

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Pushdown Vector Addition Systems

◮ Reachability d dim. = Coverability d + 1 dim.

Tower-hard Lazic’13

◮ Coverability in 1 dim. is decidable Leroux, Sutre, and T.’15

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Pushdown Vector Addition Systems

◮ Reachability d dim. = Coverability d + 1 dim.

Tower-hard Lazic’13

◮ Coverability in 1 dim. is decidable Leroux, Sutre, and T.’15 ◮ Boundedness: decidable with Hyper-Ackermannian bounds Leroux, Praveen, and Sutre’14

Theorem [LSP’14]

If a PVAS configuration (p, ⊥, n) is bounded then the cardinality of the reachability set is at most Fωd·|Q|(d · n).

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Pushdown Vector Addition Systems

◮ Reachability d dim. = Coverability d + 1 dim.

Tower-hard Lazic’13

◮ Coverability in 1 dim. is decidable Leroux, Sutre, and T.’15 ◮ Boundedness: decidable with Hyper-Ackermannian bounds Leroux, Praveen, and Sutre’14 ◮ Counter-, Stack-, and Combined Boundedness Problems.

Combined Stack Counter

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Pushdown Vector Addition Systems

◮ Reachability d dim. = Coverability d + 1 dim.

Tower-hard Lazic’13

◮ Coverability in 1 dim. is decidable Leroux, Sutre, and T.’15 ◮ Boundedness: decidable with Hyper-Ackermannian bounds Leroux, Praveen, and Sutre’14 ◮ Counter-, Stack-, and Combined Boundedness Problems.

Combined Stack Counter The following is in ExpTime.

1-PVAS Counter-Boundedness

Given: 1-dim. PVAS, initial configuration (p, w, a). Question: is {b | (p, w, a)

∗

− − → (p′, w′, b)} infinite?

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Another Perspective

Definition (Context-free Controlled VAS)

a VAS A ⊆ Zd together with a context-free language L ⊆ A∗. There is a step s − − → t between s, t ∈ Nd if a1a2 . . . ak ∈ L and s

a1

− − →

a2

− − → · · ·

ak

− − → t.

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Another Perspective

Definition (Context-free Controlled VAS)

a VAS A ⊆ Zd together with a context-free language L ⊆ A∗. There is a step s − − → t between s, t ∈ Nd if a1a2 . . . ak ∈ L and s

a1

− − →

a2

− − → · · ·

ak

− − → t.

Theorem

For Cf-Controlled VAS, Coverability (and Reachability) logspace reduces to Boundedness.

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Another Perspective

Definition (Context-free Controlled VAS)

a VAS A ⊆ Zd together with a context-free language L ⊆ A∗. There is a step s − − → t between s, t ∈ Nd if a1a2 . . . ak ∈ L and s

a1

− − →

a2

− − → · · ·

ak

− − → t.

Theorem

For Cf-Controlled VAS, Coverability (and Reachability) logspace reduces to Boundedness.

Observation

Relevant for the PVAS boundedness problem is the trace language {w ∈ A∗ | (p0, ⊥)

w

− − →} defined by the PDA.

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Another Perspective

Definition (Context-free Controlled VAS)

a VAS A ⊆ Zd together with a context-free language L ⊆ A∗. There is a step s − − → t between s, t ∈ Nd if a1a2 . . . ak ∈ L and s

a1

− − →

a2

− − → · · ·

ak

− − → t.

Theorem

For Cf-Controlled VAS, Coverability (and Reachability) logspace reduces to Boundedness.

Observation

Relevant for the PVAS boundedness problem is the trace language {w ∈ A∗ | (p0, ⊥)

w

− − →} defined by the PDA.

prefix-closed!

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Another Perspective

Definition (Context-free Controlled VAS)

a VAS A ⊆ Zd together with a context-free language L ⊆ A∗. There is a step s − − → t between s, t ∈ Nd if a1a2 . . . ak ∈ L and s

a1

− − →

a2

− − → · · ·

ak

− − → t.

Theorem

For Cf-Controlled VAS, Coverability (and Reachability) logspace reduces to Boundedness.

Observation

Relevant for the PVAS boundedness problem is the trace language {w ∈ A∗ | (p0, ⊥)

w

− − →} defined by the PDA.

Main Theorem

Boundedness of 1-dim VAS controlled by a prefix-closed language is in ExpTime.

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Another Perspective

Definition (Context-free Controlled VAS)

a VAS A ⊆ Zd together with a context-free language L ⊆ A∗. There is a step s

X

− − → t between s, t ∈ Nd if X

∗

= = ⇒ a1a2 . . . ak and s

a1

− − →

a2

− − → · · ·

ak

− − → t.

Theorem

For Cf-Controlled VAS, Coverability (and Reachability) logspace reduces to Boundedness.

Observation

Relevant for the PVAS boundedness problem is the trace language {w ∈ A∗ | (p0, ⊥)

w

− − →} defined by the PDA.

Main Theorem

Boundedness of 1-dim VAS controlled by a prefix-closed language is in ExpTime.

given as GfG

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}.

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X −1 Y Z

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X −1 Y Z 1 Y

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X −1 Y Z 1 Y 1

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X −1 Y Z 1 Y 1 −1

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5

−1 Y Z 1 Y 1 −1

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5

−1

5

Y Z 1 Y 1 −1

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5

−1

5 4

Y Z 1 Y 1 −1

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5

−1

5 4

Y

4

Z 1 Y 1 −1

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5

−1

5 4

Y

4

Z 1

4

Y 1 −1

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5

−1

5 4

Y

4

Z 1

4 5

Y 1 −1

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5

−1

5 4

Y

4

Z 1

4 5

Y

5

1 −1

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5

−1

5 4

Y

4

Z 1

4 5

Y

5

1

5

−1

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5 5

−1

5 4

Y

4 6

Z

6 5

1

4 5

Y

5 6

1

5 6

−1

6 5

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5 5

−1

5 4

Y

4 6

Z

6 5

1

4 5

Y

5 6

1

5 6

−1

6 5

X

a b means a X

− − → b′ ≥ b; X

−∞ b means ∃a ∈ N. a X

− − → b′ ≥ b.

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5 5

−1

5 4

Y

4 6

Z

6 5

1

4 5

Y

5 6

1

5 6

−1

6 5

T ⊑ T ′ if

• 1. |T| < |T ′| or
• 2. |T| = |T ′| and (labels > −∞) on T is smaller than on T ′

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Flow Trees

A derivation tree with consistent in/out labels in Z ∪ {−∞}. X

5 5

−1

5 4

Y

4 6

Z

6 5

1

4 5

Y

5 6

1

5 6

−1

6 5

T ⊑ T ′ if

• 1. |T| < |T ′| or
• 2. |T| = |T ′| and (labels > −∞) on T is smaller than on T ′

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Certificates

Definition

A certificate is a flow tree with a node X

b b′ and a descendant

X

c c′ such that

• 1. b < c or
• 2. b = c and c′ < b′.

S

a

X

b b’

X

c c’

> 0

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Certificates

Definition

A certificate is a flow tree with a node X

b b′ and a descendant

X

c c′ such that

• 1. b < c or
• 2. b = c and c′ < b′.

S

a

X

b b’

X

c c’

= 0 > 0

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Certificates (cont.)

Theorem

{a′ | a

S

− − → a′} is infinite iff there is a certificate with root S

(≤ a)

.

Unboundedness = ⇒ Certificate:

◮ a S

− − → b for sufficiently large b

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Certificates (cont.)

Theorem

{a′ | a

S

− − → a′} is infinite iff there is a certificate with root S

(≤ a)

.

Unboundedness = ⇒ Certificate:

◮ a S

− − → b for sufficiently large b

◮ a minimal flow tree must have long branch

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Certificates (cont.)

Theorem

{a′ | a

S

− − → a′} is infinite iff there is a certificate with root S

(≤ a)

.

Unboundedness = ⇒ Certificate:

◮ a S

− − → b for sufficiently large b

◮ a minimal flow tree must have long branch ◮ wqo (≤, =) on (input ×V ) implies matching nodes with b ≤ c

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Certificates (cont.)

Theorem

{a′ | a

S

− − → a′} is infinite iff there is a certificate with root S

(≤ a)

.

Unboundedness = ⇒ Certificate:

◮ a S

− − → b for sufficiently large b

◮ a minimal flow tree must have long branch ◮ wqo (≤, =) on (input ×V ) implies matching nodes with b ≤ c ◮ minimality excludes b = c ∧ c′ > b′

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Certificates (cont.)

Theorem

{a′ | a

S

− − → a′} is infinite iff there is a certificate with root S

(≤ a)

.

Unboundedness = ⇒ Certificate:

◮ a S

− − → b for sufficiently large b

◮ a minimal flow tree must have long branch ◮ wqo (≤, =) on (input ×V ) implies matching nodes with b ≤ c ◮ minimality excludes b = c ∧ c′ > b′

Unboundedness ⇐ = Certificate:

◮ yield is uvwxy ∈ L with v ≥ 0 and v + x > 0

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Certificates (cont.)

Theorem

{a′ | a

S

− − → a′} is infinite iff there is a certificate with root S

(≤ a)

.

Unboundedness = ⇒ Certificate:

◮ a S

− − → b for sufficiently large b

◮ a minimal flow tree must have long branch ◮ wqo (≤, =) on (input ×V ) implies matching nodes with b ≤ c ◮ minimality excludes b = c ∧ c′ > b′

Unboundedness ⇐ = Certificate:

◮ yield is uvwxy ∈ L with v ≥ 0 and v + x > 0 ◮ All uvnwxn are in L and executable.

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Certificates (cont.)

Theorem

{a′ | a

S

− − → a′} is infinite iff there is a certificate with root S

(≤ a)

.

Unboundedness = ⇒ Certificate:

◮ a S

− − → b for sufficiently large b

◮ a minimal flow tree must have long branch ◮ wqo (≤, =) on (input ×V ) implies matching nodes with b ≤ c ◮ minimality excludes b = c ∧ c′ > b′

Unboundedness ⇐ = Certificate:

◮ yield is uvwxy ∈ L with v ≥ 0 and v + x > 0 ◮ All uvnwxn are in L and executable. ◮ Prefix-closedness of L implies uvn and uvnwxn ∈ L.

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Certificates (cont.)

Theorem

{a′ | a

S

− − → a′} is infinite iff there is a certificate with root S

(≤ a)

.

Unboundedness = ⇒ Certificate:

◮ a S

− − → b for sufficiently large b

◮ a minimal flow tree must have long branch ◮ wqo (≤, =) on (input ×V ) implies matching nodes with b ≤ c ◮ minimality excludes b = c ∧ c′ > b′

Unboundedness ⇐ = Certificate:

◮ yield is uvwxy ∈ L with v ≥ 0 and v + x > 0 ◮ All uvnwxn are in L and executable. ◮ Prefix-closedness of L implies uvn and uvnwxn ∈ L.

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Bounding ⊏-minimal Certificates

(maybe on blackboard if time)

Theorem

Let G = (V , A, R, S) be a CfG generating a prefix-closed language

• ver A = {−1, 0, 1} and n ∈ N an initial value. Then

{m | n

S

− − → m} is infinite iff it admits a certificate with height and all input/output values bounded by n + 44(|V |+1).

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Conclusion

Discussed here

◮ Pushdown VAS; Boundedness of counter/stack/both ◮ Cf-controlled VAS; Flow Trees ◮ prefix-closed control ∼ counter-Boundedness ◮ Counter-Boundedness in 1-PVAS is in ExpTime

Open Problems

◮ Decidability of PVAS Reachability (even in dim 1) ◮ is Boundedness reducible to Reachability in Cf-C-VAS? ◮ Complexity of 1-PVAS counter-Boundedness

(NP– ExpTime)

◮ Complexity of 1-PVAS Coverability (NP– ExpSpace)

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Conclusion

Discussed here

◮ Pushdown VAS; Boundedness of counter/stack/both ◮ Cf-controlled VAS; Flow Trees ◮ prefix-closed control ∼ counter-Boundedness ◮ Counter-Boundedness in 1-PVAS is in ExpTime

Open Problems

◮ Decidability of PVAS Reachability (even in dim 1) ◮ is Boundedness reducible to Reachability in Cf-C-VAS? ◮ Complexity of 1-PVAS counter-Boundedness

(NP– ExpTime)

◮ Complexity of 1-PVAS Coverability (NP– ExpSpace)

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Additional Stuff

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Weak Computation of Ackermann Functions Am :

Am(n)

def

=

• n + 1

if m = 0 An+1

m−1(1)

if m > 0

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Weak Computation of Ackermann Functions Am :

Am(n)

def

=

• n + 1

if m = 0 An+1

m−1(1)

if m > 0 A0(n) = n + 1 A1(n) = n + 2 A2(n) = 2n + 2 A3(n) = 2n − 1 . . .

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Weak Computation of Ackermann Functions Am :

Am(n)

def

=

• n + 1

if m = 0 An+1

m−1(1)

if m > 0

s0

pop(0), +1

Weak Computation of Ackermann Functions Am :

Am(n)

def

=

• n + 1

if m = 0 An+1

m−1(1)

if m > 0

s0

pop(0), +1

s1

pop(1) push(0), +1 push(0) −1

Weak Computation of Ackermann Functions Am :

Am(n)

def

=

• n + 1

if m = 0 An+1

m−1(1)

if m > 0

s0

pop(0), +1

s1

pop(1) push(0), +1 push(0) −1

s2

pop(2) push(1),+1 push(1) −1

Weak Computation of Ackermann Functions Am :

Am(n)

def

=

• n + 1

if m = 0 An+1

m−1(1)

if m > 0

s0

pop(0), +1

s1

pop(1) push(0), +1 push(0) −1

s2

pop(2) push(1),+1 push(1) −1

sm

pop(m) push(m − 1), +1 push(m − 1) −1

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Weak Computation of Ackermann Functions Am :

Am(n)

def

=

• n + 1

if m = 0 An+1

m−1(1)

if m > 0

s0

pop(0), +1

s1

pop(1) push(0), +1 push(0) −1

s2

pop(2) push(1),+1 push(1) −1

sm

pop(m) push(m − 1), +1 push(m − 1) −1

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Weak Computation of Ackermann Functions Am :

Am(n)

def

=

• n + 1

if m = 0 An+1

m−1(1)

if m > 0

s0

pop(0), +1

s1

pop(1) push(0), +1 push(0) −1

s2

pop(2) push(1),+1 push(1) −1

sm

pop(m) push(m − 1), +1 push(m − 1) −1

(s0, m⊥, n)

∗

− − → (s0, ⊥, Am(n)) If (s0, m⊥, n)

∗

− − → (s0, ⊥, n′) then n′ ≤ Am(n)

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