Reachability in Two-Dimensional Vector Addition Systems ...info.usherbrooke.ca/mblondin/talks/lics15.pdfReachabilityinTwo-DimensionalVectorAddition SystemswithStatesisPSPACE-complete

Reachability in Two-Dimensional Vector AdditionSystems with States is PSPACE-complete

Michael Blondin1,2 Alain Finkel1 Stefan Göller1

Christoph Haase1 Pierre McKenzie1,2

1LSV, ENS Cachan & CNRS, France

2DIRO, Université de Montréal, Canada

July 6, 2015

Vector addition systemsReachability problem

New resultsDefinitionRuns

Vector addition system with states (VASS)

d-VASS:

d ≥ 1 (dimension)Q finite set (states)T ⊆ Q ×ZdZdZd × Q finite (transitions)

p q(0, 1)

(0,−2)

(1, 1)

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d-VASS:d ≥ 1 (dimension)

Q finite set (states)T ⊆ Q ×ZdZdZd × Q finite (transitions)

p q(0, 1)

(0,−2)

(1, 1)

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d-VASS:d ≥ 1 (dimension)Q finite set (states)

T ⊆ Q ×ZdZdZd × Q finite (transitions)

p q(0, 1)

(0,−2)

(1, 1)

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d-VASS:d ≥ 1 (dimension)Q finite set (states)T ⊆ Q ×ZdZdZd × Q finite (transitions)

p q(0, 1)

(0,−2)

(1, 1)

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Runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) t1−→ p(0, 1) t1−→ p(0, 2) t2−→ q(0, 0) t3−→ p(1, 1)

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Runs

p q(0, 0)

t1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0)

t1−→ p(0, 1) t1−→ p(0, 2) t2−→ q(0, 0) t3−→ p(1, 1)

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Runs

p q(0, 0)

t1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0)

t1−→ p(0, 1) t1−→ p(0, 2) t2−→ q(0, 0) t3−→ p(1, 1)

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Runs

p q(0, 1)

t1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) t1−→ p(0, 1)

t1−→ p(0, 2) t2−→ q(0, 0) t3−→ p(1, 1)

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Runs

p q(0, 1)

t1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) t1−→ p(0, 1)

t1−→ p(0, 2) t2−→ q(0, 0) t3−→ p(1, 1)

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Runs

p q(0, 1)

t1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) t1−→ p(0, 1)

t1−→ p(0, 2) t2−→ q(0, 0) t3−→ p(1, 1)

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Runs

p q(0, 2)

t1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) t1−→ p(0, 1) t1−→ p(0, 2)

t2−→ q(0, 0) t3−→ p(1, 1)

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Runs

p q(0, 2)

t1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) t1−→ p(0, 1) t1−→ p(0, 2)

t2−→ q(0, 0) t3−→ p(1, 1)

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Runs

p q(0, 0)

t1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) t1−→ p(0, 1) t1−→ p(0, 2) t2−→ q(0, 0)

t3−→ p(1, 1)

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Runs

p q(0, 0)

t1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) t1−→ p(0, 1) t1−→ p(0, 2) t2−→ q(0, 0)

t3−→ p(1, 1)

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Runs

p q(1, 1)

t1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) t1−→ p(0, 1) t1−→ p(0, 2) t2−→ q(0, 0) t3−→ p(1, 1)

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Runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) t1t1t2t3−−−−→ p(1, 1)

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Runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

p(0, 0) ∗−−−−→ p(1, 1)

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New resultsDefinitionKnown results

Reachability problem

Input: d-VASS V

and p(u), q(v) ∈ Q × Nd

Question: p(u) ∗−→ q(v)?

0

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Input: d-VASS V and p(u), q(v) ∈ Q × Nd


0

p(u)

q(v)

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Input: d-VASS V and p(u), q(v) ∈ Q × Nd


0

p(u)

q(v)

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Known results timeline

1976

EXPSPACE-hard (Lipton)

1979

Decidable (Hopcroft & Pansiot)

{v : p(u) ∗−→ q(v)} semilinear

1981

Decidable (Mayr)

1982

Decidable (Kosaraju)

1986

2-EXPTIME (Howell, Huynh, Rosier & Yen)

1992

Decidable (Lambert)

2004

{(u, v) : p(u) ∗−→ q(v)} semilinear(Leroux & Sutre)

20092011

Decidable (Leroux)

20122013

PSPACE-hard (Fearnley & Jurdziński)

2015

Fω3 (Leroux & Schmitz) PSPACE-complete (our contribution)in two talks!

VASS 2-VASS

4 / 12




1976EXPSPACE-hard (Lipton)

1979

Decidable (Hopcroft & Pansiot)


1981

Decidable (Mayr)

1982


1986


1992

Decidable (Lambert)

2004


20092011

Decidable (Leroux)

20122013


2015


VASS 2-VASS

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1979 Decidable (Hopcroft & Pansiot)


1981

Decidable (Mayr)

1982


1986


1992

Decidable (Lambert)

2004


20092011

Decidable (Leroux)

20122013


2015


VASS 2-VASS

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1981Decidable (Mayr)

1982


1986


1992

Decidable (Lambert)

2004


20092011

Decidable (Leroux)

20122013


2015


VASS 2-VASS

4 / 12








1982Decidable (Kosaraju)

1986


1992

Decidable (Lambert)

2004


20092011

Decidable (Leroux)

20122013


2015


VASS 2-VASS

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1986 2-EXPTIME (Howell, Huynh, Rosier & Yen)

1992

Decidable (Lambert)

2004


20092011

Decidable (Leroux)

20122013


2015


VASS 2-VASS

4 / 12










1992Decidable (Lambert)

2004


20092011

Decidable (Leroux)

20122013


2015


VASS 2-VASS

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2004 {(u, v) : p(u) ∗−→ q(v)} semilinear(Leroux & Sutre)

20092011

Decidable (Leroux)

20122013


2015


VASS 2-VASS

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20092011Decidable (Leroux)20122013


2015


VASS 2-VASS

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20092011Decidable (Leroux)20122013 PSPACE-hard (Fearnley & Jurdziński)

2015


VASS 2-VASS

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2015Fω3 (Leroux & Schmitz)

PSPACE-complete (our contribution)

in two talks!

VASS 2-VASS

4 / 12













2015Fω3 (Leroux & Schmitz) PSPACE-complete (our contribution)in two talks!

VASS 2-VASS

4 / 12


New resultsMain theoremProof sketch

Our main theoremThere exists c ∈ N s.t. for every 2-VASS V

p(u) ∗−→ q(v) =⇒ p(u) π−→ q(v) where |π| ≤ c |V |

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Corollary

Reachability for 2-VASS ∈ PSPACE

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Corollary: proof

Exp. length runs =⇒ exp. intermediate counter values

=⇒ poly. size intermediate counter values=⇒ guess run on the fly

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Corollary: proof

Exp. length runs =⇒ exp. intermediate counter values=⇒ poly. size intermediate counter values

=⇒ guess run on the fly

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Corollary: proof

Exp. length runs =⇒ exp. intermediate counter values=⇒ poly. size intermediate counter values=⇒ guess run on the fly

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How to prove this theorem?

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Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2 (t3t1∗t2)

∗

(t3t1∗t2) · · · (t3t1∗t2)

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Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2

(t3t1∗t2)

∗

(t3t1∗t2) · · · (t3t1∗t2)

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Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2 (t3t1∗t2)

∗(t3t1∗t2) · · · (t3t1∗t2)

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Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2 (t3t1∗t2)

∗

(t3t1∗t2)

· · · (t3t1∗t2)

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Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2 (t3t1∗t2)

∗

(t3t1∗t2) · · · (t3t1∗t2)

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Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2 (t3t1∗t2)

∗

(t3t1∗t2) · · · (t3t1∗t2)

6 / 12



Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2 (t3t1∗t2)

∗

(t3t1∗t2) · · · (t3t1∗t2)

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Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2 (t3

t1∗

t2)

∗

(t3

t1∗

t2) · · · (t3

t1∗

t2)

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Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2 (t3

t1∗

t2)

∗

(t3

t1∗

t2) · · · (t3

t1∗

t2)

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Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2 (t3

t1∗

t2)∗

(t3

t1∗

t2) · · · (t3

t1∗

t2)

6 / 12



Bounding the runs

p qt1: (0, 1)

t2: (0,−2)

t3: (1, 1)

Runs from p to q:

t1∗t2 (t3

t1∗

t2)∗

(t3

t1∗

t2) · · · (t3

t1∗

t2)

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2-VASS can always be flattened (Leroux & Sutre ’04)

∃S =⋃finite

α0β1∗α1 · · ·βk

∗αk︸︷︷︸linear path scheme

such that

p(u) ∗−→ q(v) =⇒ p(u) π∈S−−→ q(v)

2-VASS have small linear path schemes (our contribution)

|αi |, |βi | ≤k ∈

∗-exponents ≤

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∃S =⋃finite

α0β1∗α1 · · ·βk


such that

p(u) ∗−→ q(v) =⇒ p(u) π∈S−−→ q(v)


|αi |, |βi | ≤k ∈

∗-exponents ≤

7 / 12




∃S =⋃finite

α0β1∗α1 · · ·βk


such that

p(u) ∗−→ q(v) =⇒ p(u) π∈S−−→ q(v)


|αi |, |βi | ≤k ∈

∗-exponents ≤

7 / 12




∃S =⋃finite

α0β1∗α1 · · ·βk


such that

p(u) ∗−→ q(v) =⇒ p(u) π∈S−−→ q(v)


|αi |, |βi | ≤ (|Q|+ ‖T‖)O(1)

k ∈ O(|Q|2)∗-exponents ≤ (|Q|+ ‖T‖+ ‖u‖+ ‖v‖)O(1)

7 / 12




∃S =⋃finite

α0β1∗α1 · · ·βk


such that

p(u) ∗−→ q(v) =⇒ p(u) π∈S−−→ q(v)


|αi |, |βi | ≤ (|Q|+ ‖T‖)O(1)

k ∈ O(|Q|2)

∗-exponents ≤ (|Q|+ ‖T‖+ ‖u‖+ ‖v‖)O(1)

7 / 12




∃S =⋃finite

α0β1∗α1 · · ·βk


such that

p(u) ∗−→ q(v) =⇒ p(u) π∈S−−→ q(v)


|αi |, |βi | ≤ (|Q|+ ‖T‖)O(1)

k ∈ O(|Q|2)∗-exponents ≤ (|Q|+ ‖T‖+ ‖u‖+ ‖v‖)O(1)

7 / 12




∃S =⋃finite

α0β1∗α1 · · ·βk


such that

p(u) ∗−→ q(v) =⇒ p(u) π∈S−−→ q(v)


|αi |, |βi | ≤ exponentialk ∈ polynomial

∗-exponents ≤ exponential

7 / 12




∃S =⋃finite

α0β1∗α1 · · ·βk


such that

p(u) ∗−→ q(v) =⇒ p(u) π∈S−−→ q(v)


|αi |, |βi | ≤k ∈

∗-exponents ≤

How to obtain such linear path schemes?

7 / 12



Obtaining linear path schemes for 3 types of runs

0

q(u)

q(v)c

0

p(u)

q(v)

c

0

p(u)

q(v)

Type 1

8 / 12




0

q(u)

q(v)c

0

p(u)

q(v)

c

0

p(u)

q(v)

Type 1

Remove zigzags

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0

q(u)

q(v)c

0

p(u)

q(v)

c

0

p(u)

q(v)

Type 1

Remove zigzags

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Type 1: removing zig-zags

q(u)

q(v)

q(u)

q(v)

q(u)

q(v)

Arbitrary run

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q(u)

q(v)

q(u)

q(v)

q(u)

q(v)

Arbitrary run⋃finite

α0β∗1α1· · ·β∗kαk

9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)



9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)



9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)



9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)



9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)



9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)



9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)


α0β∗1α1 · · ·β∗kαk

⋃finite

γ0θ∗1γ1θ

∗2γ2

9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)


α0β∗1α1 · · ·β∗kαk

⋃finite

γ0θ∗1γ1θ

∗2γ2

9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)


α0β∗1α1 · · ·β∗kαk

⋃finite

γ0θ∗1γ1θ

∗2γ2

9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)


α0β∗1α1 · · ·β∗kαk

⋃finite

γ0θ∗1γ1θ

∗2γ2

9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)


α0β∗1α1 · · ·β∗kαk

⋃finite

γ0θ∗1γ1θ

∗2γ2

9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)


α0β∗1α1 · · ·β∗kαk

⋃finite

γ0θ∗1γ1θ

∗2γ2

9 / 12




q(u)

q(v)

q(u)

q(v)

q(u)

q(v)


α0β∗1α1 · · ·β∗kαk

⋃finite

γ0θ∗1γ1θ

∗2γ2

9 / 12




q(u)

q(v)

q(u)

q(v)

q(x)

q(y)


α0β∗1α1 · · ·β∗kαk

⋃finite

γ0θ∗1γ1θ

∗2γ2

9 / 12




0

q(u)

q(v)c

0

p(u)

q(v)

c

0

p(u)

q(v)

Type 1 Type 2

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0

q(u)

q(v)c

0

p(u)

q(v)

c

0

p(u)

q(v)

Type 1 Type 2

Composition of type 1

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Type 2: decomposition

0

p(u)rr

s sq(v)

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0

p(u)rr

s sq(v)

Small run

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0

p(u)rr

s sq(v)

Type 1

10 / 12




0

p(u)rr

s sq(v)

Small run

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0

p(u)rr

s sq(v)

Type 1

10 / 12




0

p(u)rr

s sq(v)

Small run

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0

q(u)

q(v)c

0

p(u)

q(v)

c

0

p(u)

q(v)

Type 1 Type 2 Type 3

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0

q(u)

q(v)c

0

p(u)

q(v)

c

0

p(u)

q(v)

Type 1 Type 2 Type 3

' 1-VASS

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Every run decomposes into ≤ |Q|+ 1 runs of type 1, 2 & 3

0

p(u)

q(v)

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0

p(u)

q(v)

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0

‖T‖p(u)

q(v)

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0

p(u)

q(v)

q

p

qrp s

p

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0

p(u)

q(v)

q

p

qrp s

p

Type 2

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0

p(u)

q(v)

q

p

qrp s

p

Type 1

11 / 12




0

p(u)

q(v)

q

p

qrp s

p

Type 2

11 / 12




0

p(u)

q(v)

q

p

qrp s

p

Type 1

11 / 12




0

p(u)

q(v)

q

p

qrp s

p

Type 3

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New results

Open questions

2-VASS, unary encoding: NL-hard and ∈ NP. NL-complete?

3-VASS: PSPACE-hard and ∈ Fω3 . Better bounds?

12 / 12


New results

Open questions

2-VASS, unary encoding: NL-hard and ∈ NP. NL-complete?

3-VASS: PSPACE-hard and ∈ Fω3 . Better bounds?

12 / 12


New results

Thank you!

ありがとうございます!(arigato gozaimasu)

Documents

Reachability in Two-Dimensional Vector Addition Systems ...info.usherbrooke.ca/mblondin/talks/lics15.pdfReachabilityinTwo-DimensionalVectorAddition SystemswithStatesisPSPACE-complete