[PPT] - On the Complexity of VAS Reachability Sylvain Schmitz LSV, ENS PowerPoint Presentation

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

On the Complexity

f VAS Reachability

Sylvain Schmitz LSV, ENS Paris-Saclay & CNRS, Universit´ e Paris-Saclay INFINITY 2018

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Outline

◮ VASS Reachability ◮ Decomposition Algorithm ◮ Upper Bounds ◮ Complexity

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Vector Addition Systems

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Vector Addition Systems

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Vector Addition Systems

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Springfield Power Plant

(1,1) (-1,-2)

produce electricity recycle uranium

electricity uranium waste

(0,1)

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Vector Addition Systems

Springfield Power Plant

(1,1) (-1,-2)

produce electricity recycle uranium

electricity uranium waste

(0,1)

Can we produce unbounded electricity with no left-

ver uranium waste?

3/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Vector Addition Systems

Springfield Power Plant

(1,1) (-1,-2)

produce electricity recycle uranium

electricity uranium waste

(0,1)

Can we produce unbounded electricity with no left-

ver uranium waste? Yes, (∞,0) is reachable

3/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Importance of the Problem

Reachability Problem input: a vector addition system and two configurations source and target question: source →∗ target? Discrete Resources

◮ modelling: items, money, energy, molecules, ... ◮ distributed computing: active threads in thread pool ◮ data: isomorphism types in data logics and data-centric

systems

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Importance of the Problem

Reachability Problem input: a vector addition system and two configurations source and target question: source →∗ target? Central Decision Problem [invited survey S., SIGLOG’16] Large number of problems interreducible with reachability in vector addition systems

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Importance of the Problem

Reachability Problem input: a vector addition system and two configurations source and target question: source →∗ target? Theorem (Minsky’67) Reachability is undecidable in 2-dimensional Minsky machines (vector addition systems with zero tests).

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Importance of the Problem

1962 2015

C. A. Petri: Petri nets
R. M. Karp & R. E. Miller: coverability trees

1969

R. J. Lipton: EXPSPACE lower bound

1976

J. E. Hopcroft & J.-J. Pansiot: dim. 3

not definable in Presburger arithmetic

1979

E. W. Mayr: decidability by decomposition

1981

S. R. Kosaraju: decidability by decomposition

1982

J.-L. Lambert: decidability by decomposition

1992

J. Leroux: decidability by Presburger inductive invariants

2011

this talk: Leroux & S.’15

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Demystifying Reachability in Vector Addition Systems

[Leroux & S.’15] Ideal Decomposition Theorem The Decomposition Algorithm computes the ideal decomposition of the set of runs from source to target. Upper Bound Theorem Reachability in vector addition systems is in cubic Ackermann.

5/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Demystifying Reachability in Vector Addition Systems

[Leroux & S.’15] Ideal Decomposition Theorem The Decomposition Algorithm computes the ideal decomposition of the set of runs from source to target. Upper Bound Theorem Reachability in vector addition systems is in cubic Ackermann.

5/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Demystifying Reachability in Vector Addition Systems

[Leroux & S.’15; S.’17] Ideal Decomposition Theorem The Decomposition Algorithm computes the ideal decomposition of the set of runs from source to target. Upper Bound Theorem Reachability in vector addition systems is in quadratic Ackermann.

5/20

SLIDE 15

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] (∞,∞) (1,1) (-1,-2)

(0,1) (∞,0)

c 6/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] (0,1) (0,0) (2,0) (4,0) (6,0) (0,−1) 6/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] (∞,∞) a (1,1) b (-1,-2)

(0,1) (∞,0)

c

Characteristic System

0 + 1 · a − 1 · b = c 1 + 1 · a − 2 · b = 0

Solution Path

(0,−1) 6/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] solution path

×1

(0,1) (0,0) (2,0) (4,0) (6,0) (0,−1) 6/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] solution path

×1

(0,1) (0,0) (2,0) (4,0) (6,0) (0,−1) 6/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] (∞,∞) a (1,1) b (-1,-2)

(0,1) (∞,0)

c

Homogeneous System

1 · a − 1 · b = c 1 · a − 2 · b = 0 a,b,c > 0

Unbounded Path

(2,0) 6/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] solution path

×1

unbounded path

×1

(0,1) (0,0) (2,0) (4,0) (6,0) (0,−1) 6/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] solution path

×1

unbounded path

×2

(0,1) (0,0) (2,0) (4,0) (6,0) (0,−1) 6/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] solution path

×1

unbounded path

×3

(0,1) (0,0) (2,0) (4,0) (6,0) (0,−1) 6/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92]

Pumpable Paths

unbounded path

−

pump up (0,1) (∞,∞)

−

pump down (∞,∞) (∞,0)

=

remainder

uses coverability trees [Karp & Miller’69] (J´

erˆ

me’s talk)

which relies on Dickson’s Lemma [Dickson, 1913]

6/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92]

Pumpable Paths

unbounded path

−

pump up (0,1) (∞,∞)

−

pump down (∞,∞) (∞,0)

=

remainder

uses coverability trees [Karp & Miller’69] (J´

erˆ

me’s talk)

which relies on Dickson’s Lemma [Dickson, 1913]

6/20

SLIDE 26

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] pump up

×1

(0,1) (4,0) 6/20

SLIDE 27

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] pump up

×2

(0,1) (4,0) 6/20

SLIDE 28

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] pump up

×2

solution path

×1

(0,1) (4,0) 6/20

SLIDE 29

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] pump up

×2

solution path

×1

remainder

×1

(0,1) (4,0) 6/20

SLIDE 30

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] pump up

×2

solution path

×1

remainder

×2

(0,1) (4,0) 6/20

SLIDE 31

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] pump up

×2

solution path

×1

remainder

×2

pump down

×1

(0,1) (4,0) 6/20

SLIDE 32

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] pump up

×2

solution path

×1

remainder

×2

pump down

×2

(0,1) (4,0) 6/20

SLIDE 33

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

“Simple Runs” (Θ Condition)

[Mayr’81, Kosaraju’82, Lambert’92] pump up

×3

solution path

×1

remainder

×3

pump down

×3

(0,1) (6,0) 6/20

SLIDE 34

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

can we build a “simple run”?

,

, ,

yes

no decompose

7/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

can we build a “simple run”?

,

, ,

yes

no decompose

7/20

SLIDE 36

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

can we build a “simple run”?

,

, ,

yes

no decompose

7/20

SLIDE 37

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

can we build a “simple run”?

,

, ,

yes

no decompose

7/20

SLIDE 38

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

can we build a “simple run”?

,

, ,

yes

no decompose

7/20

SLIDE 39

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

can we build a “simple run”?

,

, ,

yes

no decompose

7/20

SLIDE 40

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

can we build a “simple run”?

,

, ,

yes

no decompose

7/20

SLIDE 41

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Termination

“Finally the checker has to verify that the process comes to an end. Here again he should be assisted by the programmer giving a further definite assertion to be verified. This may take the form of a quantity which is asserted to decrease continually and vanish when the machine stops.” [Turing’49]

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Termination

“Finally the checker has to verify that the process comes to an end. Here again he should be assisted by the programmer giving a further definite assertion to be verified. This may take the form of a quantity which is asserted to decrease continually and vanish when the machine stops. To the pure mathematician it is natural to give an ordinal number.” [Turing’49]

8/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Termination of the Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

Ranking Function ωω2 α0 ∨ ∨ α1 ∨ α2 ∨ . . .

9/20

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Termination of the Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

Ranking Function ωω2 α0 ∨ ∨ α1 ∨ α2 ∨ . . .

9/20

SLIDE 45

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Termination of the Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

Ranking Function ωω2 α0 ∨ ∨ α1 ∨ α2 ∨ . . .

9/20

SLIDE 46

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Termination of the Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92]

Ranking Function ωω2 α0 ∨ ∨ α1 ∨ α2 ∨ . . .

9/20

SLIDE 47

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Demystifying Reachability in Vector Addition Systems

[Leroux & S.’15; S.’17] Ideal Decomposition Theorem The Decomposition Algorithm computes the ideal decomposition of the set of runs from source to target. Upper Bound Theorem Reachability in vector addition systems is in quadratic Ackermann.

10/20

SLIDE 48

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Upper Bounds

How to bound the running time of algorithms with

rdinal-based termination proofs?

Information and Computation 160, 109127 (2000)

A l g

r

i t h m i c A n a l y s i s

f

P r

g

r a m s w i t h W e l l Q u a s i

r

d e r e d D

m

a i n s

1 Parosh Aziz Abdulla

Department of Computer Systems, Uppsala University, P.O. Box 325, 751 05 Uppsala, Sweden E-mail: paroshdocs.uu.se Ka rlis C 8 era ns Institute of Mathematics and Computer Science, University of Latvia, Riga, Latvia E-mail: karliscclu.lv Bengt Jonsson Department of Computer Systems, Uppsala University, P.O. Box 325, 751 05 Uppsala, Sweden E-mail: bengtdocs.uu.se and Yih-Kuen Tsay Department of Information Management, National Taiwan University, Taipei, Taiwan E-mail: tsayim.ntu.edu.tw Over the past few years increasing research effort has been directed towards the automatic verification of infinite-state systems. This paper is concerned with identifying general mathematical structures which serve as sufficient conditions for achieving decidability. decidability results for a class of systems which consist of a finite control The results assume which doi:10.1006inco.1999.2843, available online at http:www.idealibrary.com on

The
re

tica l Com pute r Scie nce 256 (2001) 63–92 www.e lse vie r.com /loca te /tcs

We ll-structure d tra nsition syste m s e ve rywhe re !

A . F i n k e l , P h . S c h n

e

b e l e n∗

Lab. Spe

ci cation and Ve ri cation, ENS de Cachan & CNRS UMR 8643, 61 av . Pdt Wilson, 94235 Cachan Ce de x, France A b s t r a c t W e l l

s

t r u c t u r e d t r a n s i t i

n

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f

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r

w h i c h d e c i d a b i l i t y r e s u l t s r e l y

n

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f

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d e r i n g b e t w e e n s t a t e s t h a t i s c

m

p a t i b l e w i t h t h e t r a n s i t i

n

s . I n t h i s a r t i c l e , w e p r

v

i d e a n e x t e n s i v e t r e a t m e n t

f

t h e W S T S i d e a a n d s h

w

s e v e r a l n e w r e s u l t s . O u r i m p r

v

e d d e

n

i t i

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w

m a n y e x a m p l e s

f

c l a s s i c a l s y s t e m s t

b

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f

W S T S s . c

2

1 E l s e v i e r S c i e n c e B . V . A l l r i g h t s r e s e r v e d . K e y w

r

d s : I n

n

i t e s y s t e m s ; V e r i

c

a t i

n

; W e l l

q

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d

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n

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n
f

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t a t e s y s t e m s F

r

m a l v e r i

c

a t i

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f

p r

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r a m s a n d s y s t e m s i s a v e r y a c t i v e

e

l d f

r

b

t

h t h e

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e t i c a l r e s e a r c h a n d p r a c t i c a l d e v e l

p

m e n t s , e s p e c i a l l y s i n c e i m p r e s s i v e a d v a n c e s i n f

r

m a l v e r i

c

a t i

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t e c h n

l
g

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s f r

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r

l d . T h e h i g h l y s u c c e s s f u l m

d

e l

c

h e c k i n g a p p r

a

c h f

r
n

i t e s y s t e m s [ 1 6 ] a w

r

k i n g v e r i

c

a t i

n

t e c h n

l
g

y c

u

l d w e l l b e d e v e l

p

e d f

r

s y s t e m s f w

r

k t h a t h a s b e e n d e v

t

e d i n r e c e n t y e a r s a s u r p r i s i n g w e a l t h

f

p

s

i t i v e

∗

11/20

SLIDE 49

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Upper Bounds

How to bound the running time of algorithms with wqo-based termination proofs?

Information and Computation 160, 109127 (2000)

A l g

r

i t h m i c A n a l y s i s

f

P r

g

r a m s w i t h W e l l Q u a s i

r

d e r e d D

m

a i n s

1 Parosh Aziz Abdulla

Department of Computer Systems, Uppsala University, P.O. Box 325, 751 05 Uppsala, Sweden E-mail: paroshdocs.uu.se Ka rlis C 8 era ns Institute of Mathematics and Computer Science, University of Latvia, Riga, Latvia E-mail: karliscclu.lv Bengt Jonsson Department of Computer Systems, Uppsala University, P.O. Box 325, 751 05 Uppsala, Sweden E-mail: bengtdocs.uu.se and Yih-Kuen Tsay Department of Information Management, National Taiwan University, Taipei, Taiwan E-mail: tsayim.ntu.edu.tw Over the past few years increasing research effort has been directed towards the automatic verification of infinite-state systems. This paper is concerned with identifying general mathematical structures which serve as sufficient conditions for achieving decidability. decidability results for a class of systems which consist of a finite control The results assume which doi:10.1006inco.1999.2843, available online at http:www.idealibrary.com on

The
re

tica l Com pute r Scie nce 256 (2001) 63–92 www.e lse vie r.com /loca te /tcs

We ll-structure d tra nsition syste m s e ve rywhe re !

A . F i n k e l , P h . S c h n

e

b e l e n∗

Lab. Spe

ci cation and Ve ri cation, ENS de Cachan & CNRS UMR 8643, 61 av . Pdt Wilson, 94235 Cachan Ce de x, France A b s t r a c t W e l l

s

t r u c t u r e d t r a n s i t i

n

s y s t e m s ( W S T S s ) a r e a g e n e r a l c l a s s

f

i n

n

i t e

s

t a t e s y s t e m s f

r

w h i c h d e c i d a b i l i t y r e s u l t s r e l y

n

t h e e x i s t e n c e

f

a w e l l

q

u a s i

r

d e r i n g b e t w e e n s t a t e s t h a t i s c

m

p a t i b l e w i t h t h e t r a n s i t i

n

s . I n t h i s a r t i c l e , w e p r

v

i d e a n e x t e n s i v e t r e a t m e n t

f

t h e W S T S i d e a a n d s h

w

s e v e r a l n e w r e s u l t s . O u r i m p r

v

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n

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s a l l

w

m a n y e x a m p l e s

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b

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W S T S s . c

2

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r

d s : I n

n

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c

a t i

n

; W e l l

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r a m s a n d s y s t e m s i s a v e r y a c t i v e

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r

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c

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p

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f

p

s

i t i v e

∗

11/20

SLIDE 50

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Upper Bounds

How to bound the running time of algorithms with wqo-based termination proofs? wqos ubiquitous in infinite-state verification

Information and Computation 160, 109127 (2000)

A l g

r

i t h m i c A n a l y s i s

f

P r

g

r a m s w i t h W e l l Q u a s i

r

d e r e d D

m

a i n s

1 Parosh Aziz Abdulla

Department of Computer Systems, Uppsala University, P.O. Box 325, 751 05 Uppsala, Sweden E-mail: paroshdocs.uu.se Ka rlis C 8 era ns Institute of Mathematics and Computer Science, University of Latvia, Riga, Latvia E-mail: karliscclu.lv Bengt Jonsson Department of Computer Systems, Uppsala University, P.O. Box 325, 751 05 Uppsala, Sweden E-mail: bengtdocs.uu.se and Yih-Kuen Tsay Department of Information Management, National Taiwan University, Taipei, Taiwan E-mail: tsayim.ntu.edu.tw Over the past few years increasing research effort has been directed towards the automatic verification of infinite-state systems. This paper is concerned with identifying general mathematical structures which serve as sufficient conditions for achieving decidability. decidability results for a class of systems which consist of a finite control The results assume which doi:10.1006inco.1999.2843, available online at http:www.idealibrary.com on

The
re

tica l Com pute r Scie nce 256 (2001) 63–92 www.e lse vie r.com /loca te /tcs

We ll-structure d tra nsition syste m s e ve rywhe re !

A . F i n k e l , P h . S c h n

e

b e l e n∗

Lab. Spe

ci cation and Ve ri cation, ENS de Cachan & CNRS UMR 8643, 61 av . Pdt Wilson, 94235 Cachan Ce de x, France A b s t r a c t W e l l

s

t r u c t u r e d t r a n s i t i

n

s y s t e m s ( W S T S s ) a r e a g e n e r a l c l a s s

f

i n

n

i t e

s

t a t e s y s t e m s f

r

w h i c h d e c i d a b i l i t y r e s u l t s r e l y

n

t h e e x i s t e n c e

f

a w e l l

q

u a s i

r

d e r i n g b e t w e e n s t a t e s t h a t i s c

m

p a t i b l e w i t h t h e t r a n s i t i

n

s . I n t h i s a r t i c l e , w e p r

v

i d e a n e x t e n s i v e t r e a t m e n t

f

t h e W S T S i d e a a n d s h

w

s e v e r a l n e w r e s u l t s . O u r i m p r

v

e d d e

n

i t i

n

s a l l

w

m a n y e x a m p l e s

f

c l a s s i c a l s y s t e m s t

b

e s e e n a s i n s t a n c e s

f

W S T S s . c

2

1 E l s e v i e r S c i e n c e B . V . A l l r i g h t s r e s e r v e d . K e y w

r

d s : I n

n

i t e s y s t e m s ; V e r i

c

a t i

n

; W e l l

q

u a s i

r

d e r i n g 1 . I n t r

d

u c t i

n

1 . 1 . V e r i c a t i

n
f

i n n i t e

s

t a t e s y s t e m s F

r

m a l v e r i

c

a t i

n
f

p r

g

r a m s a n d s y s t e m s i s a v e r y a c t i v e

e

l d f

r

b

t

h t h e

r

e t i c a l r e s e a r c h a n d p r a c t i c a l d e v e l

p

m e n t s , e s p e c i a l l y s i n c e i m p r e s s i v e a d v a n c e s i n f

r

m a l v e r i

c

a t i

n

t e c h n

l
g

y p r

v

e d f e a s i b l e i n s e v e r a l r e a l i s t i c a p p l i c a t i

n

s f r

m

t h e d u s t r i a l w

r

l d . T h e h i g h l y s u c c e s s f u l m

d

e l

c

h e c k i n g a p p r

a

c h f

r
n

i t e s y s t e m s [ 1 6 ] a w

r

k i n g v e r i

c

a t i

n

t e c h n

l
g

y c

u

l d w e l l b e d e v e l

p

e d f

r

s y s t e m s f w

r

k t h a t h a s b e e n d e v

t

e d i n r e c e n t y e a r s a s u r p r i s i n g w e a l t h

f

p

s

i t i v e

∗

11/20

SLIDE 51

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Upper Bounds

How to bound the running time of algorithms with wqo-based termination proofs? wqos ubiquitous in infinite-state verification

Information and Computation 160, 109127 (2000)

A l g

r

i t h m i c A n a l y s i s

f

P r

g

r a m s w i t h W e l l Q u a s i

r

d e r e d D

m

a i n s

1 Parosh Aziz Abdulla

Department of Computer Systems, Uppsala University, P.O. Box 325, 751 05 Uppsala, Sweden E-mail: paroshdocs.uu.se Ka rlis C 8 era ns Institute of Mathematics and Computer Science, University of Latvia, Riga, Latvia E-mail: karliscclu.lv Bengt Jonsson Department of Computer Systems, Uppsala University, P.O. Box 325, 751 05 Uppsala, Sweden E-mail: bengtdocs.uu.se and Yih-Kuen Tsay Department of Information Management, National Taiwan University, Taipei, Taiwan E-mail: tsayim.ntu.edu.tw Over the past few years increasing research effort has been directed towards the automatic verification of infinite-state systems. This paper is concerned with identifying general mathematical structures which serve as sufficient conditions for achieving decidability. decidability results for a class of systems which consist of a finite control The results assume which doi:10.1006inco.1999.2843, available online at http:www.idealibrary.com on

The
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tica l Com pute r Scie nce 256 (2001) 63–92 www.e lse vie r.com /loca te /tcs

We ll-structure d tra nsition syste m s e ve rywhe re !

A . F i n k e l , P h . S c h n

e

b e l e n∗

Lab. Spe

ci cation and Ve ri cation, ENS de Cachan & CNRS UMR 8643, 61 av . Pdt Wilson, 94235 Cachan Ce de x, France A b s t r a c t W e l l

s

t r u c t u r e d t r a n s i t i

n

s y s t e m s ( W S T S s ) a r e a g e n e r a l c l a s s

f

i n

n

i t e

s

t a t e s y s t e m s f

r

w h i c h d e c i d a b i l i t y r e s u l t s r e l y

n

t h e e x i s t e n c e

f

a w e l l

q

u a s i

r

d e r i n g b e t w e e n s t a t e s t h a t i s c

m

p a t i b l e w i t h t h e t r a n s i t i

n

s . I n t h i s a r t i c l e , w e p r

v

i d e a n e x t e n s i v e t r e a t m e n t

f

t h e W S T S i d e a a n d s h

w

s e v e r a l n e w r e s u l t s . O u r i m p r

v

e d d e

n

i t i

n

s a l l

w

m a n y e x a m p l e s

f

c l a s s i c a l s y s t e m s t

b

e s e e n a s i n s t a n c e s

f

W S T S s . c

2

1 E l s e v i e r S c i e n c e B . V . A l l r i g h t s r e s e r v e d . K e y w

r

d s : I n

n

i t e s y s t e m s ; V e r i

c

a t i

n

; W e l l

q

u a s i

r

d e r i n g 1 . I n t r

d

u c t i

n

1 . 1 . V e r i c a t i

n
f

i n n i t e

s

t a t e s y s t e m s F

r

m a l v e r i

c

a t i

n
f

p r

g

r a m s a n d s y s t e m s i s a v e r y a c t i v e

e

l d f

r

b

t

h t h e

r

e t i c a l r e s e a r c h a n d p r a c t i c a l d e v e l

p

m e n t s , e s p e c i a l l y s i n c e i m p r e s s i v e a d v a n c e s i n f

r

m a l v e r i

c

a t i

n

t e c h n

l
g

y p r

v

e d f e a s i b l e i n s e v e r a l r e a l i s t i c a p p l i c a t i

n

s f r

m

t h e d u s t r i a l w

r

l d . T h e h i g h l y s u c c e s s f u l m

d

e l

c

h e c k i n g a p p r

a

c h f

r
n

i t e s y s t e m s [ 1 6 ] a w

r

k i n g v e r i

c

a t i

n

t e c h n

l
g

y c

u

l d w e l l b e d e v e l

p

e d f

r

s y s t e m s f w

r

k t h a t h a s b e e n d e v

t

e d i n r e c e n t y e a r s a s u r p r i s i n g w e a l t h

f

p

s

i t i v e

∗

11/20

SLIDE 52

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo iff all bad sequences

are finite

◮ but can be of arbitrary length

12/20

SLIDE 53

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Bad Sequences

12/20

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo iff all bad sequences

are finite

◮ but can be of arbitrary length

Example (over N2)

SLIDE 54

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Bad Sequences

12/20

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo iff all bad sequences

are finite

◮ but can be of arbitrary length

Example (over N2)

SLIDE 55

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Controlled Bad Sequences

12/20

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo iff all bad sequences

are finite

◮ controlled by g:N → N

monotone and inflationary and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] g0(2) = 2

Example (over N2 with n0 = 2 and g(n) = n + 1)

SLIDE 56

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo iff all bad sequences

are finite

◮ controlled by g:N → N

monotone and inflationary and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98]

Proposition Over (X,), assuming ∀n {x ∈ X | x n} finite, (g,n0)-controlled bad sequences have a maximal length, noted Lg,X(n0).

12/20

SLIDE 57

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Controlled Bad Sequences

Proposition Over a wqo (X,), assuming {x ∈ X | x n} to be finite ∀n, (g,n0)-controlled bad sequences have a maximal length, noted Lg,X(n0). Objective Provide upper bounds for Lg,X(n0).

12/20

SLIDE 58

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Descent Equation

(g,n0)-controlled bad sequence x0,x1,x2,x3,... over a wqo (X,):

norms xi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = x0 x1 x2 x3

13/20

SLIDE 59

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Descent Equation

(g,n0)-controlled bad sequence x0,x1,x2,x3,... over a wqo (X,):

norms xi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = x0 x1 x2 x3

ver

the suffix x1,x2,x3,..., ∀i > 0, x0 xi

13/20

SLIDE 60

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Descent Equation

(g,n0)-controlled bad sequence x0,x1,x2,x3,... over a wqo (X,):

norms xi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = x0 x1 x2 x3

ver

the suffix x1,x2,x3,..., ∀i > 0, xi ∈ X\↑x0

def

= {x ∈ X | x0 x}

13/20

SLIDE 61

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Descent Equation

(g,n0)-controlled bad sequence x0,x1,x2,x3,... over a wqo (X,):

norms xi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = x0 x1 x2 x3

ver

the suffix x1,x2,x3,..., ∀i > 0, xi ∈ X\↑x0

def

= {x ∈ X | x0 x} xi gi−1(g(n0))

13/20

SLIDE 62

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Descent Equation

(g,n0)-controlled bad sequence x0,x1,x2,x3,... over a wqo (X,):

norms xi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = x0 x1 x2 x3

ver

the suffix x1,x2,x3,..., ∀i > 0, xi ∈ X\↑x0

def

= {x ∈ X | x0 x} xi gi−1(g(n0))

Lg,X(n0) = max

x0∈X,x0n0

1 + Lg,X\↑x0(g(n0))

13/20

SLIDE 63

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Descent Equation

(g,n0)-controlled bad sequence α0,α1,α2,α3,... over an

rdinal α:

norms αi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = α0 α1 α2 α3

ver

the suffix α1,α2,α3,..., ∀i > 0, αi ∈ α0

def

= {β ∈ α | β α0} αi gi−1(g(n0))

Lg,α(n0) = max

α0∈α,α0n0

1 + Lg,α0(g(n0))

13/20

SLIDE 64

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

◮ Cantor Normal Form for ordinals α < ε0:

α = ωα1 · c1 + ··· + ωαk · ck α > α1 > ··· > αk 0 < c1,...,ck < ω

◮ Norm of ordinals α < ε0: “maximal constant”

α

def

= max

1ik(max(||αi||,ci))

e.g. ωω2 = 2, ωω·5 + ω2 · 3 = 5

14/20

SLIDE 65

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

◮ Cantor Normal Form for ordinals α < ε0:

α = ωα1 · c1 + ··· + ωαk · ck α > α1 > ··· > αk 0 < c1,...,ck < ω

◮ Norm of ordinals α < ε0: “maximal constant”

α

def

= max

1ik(max(||αi||,ci))

e.g. ωω2 = 2, ωω·5 + ω2 · 3 = 5

14/20

SLIDE 66

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

Recall the descent equation: Lg,α(n0) = max

α0∈α,α0n0

1 + Lg,α0(g(n0)) Proposition (variant of [Buchholtz, Cicho´

n & Weiermann’94])

Let 0 < α < ε0 and α n0. Then Lg,0(n0) = 0 Lg,α(n0) = 1 + Lg,Pn0(α)(g(n0)) Px(α) denotes the predecessor at x of α > 0: “maximal

rdinal β < α s.t. β x”

14/20

SLIDE 67

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

Recall the descent equation: Lg,α(n0) = max

α0∈α,α0n0

1 + Lg,α0(g(n0)) Proposition (variant of [Buchholtz, Cicho´

n & Weiermann’94])

Let 0 < α < ε0 and α n0. Then Lg,0(n0) = 0 Lg,α(n0) = 1 + Lg,Pn0(α)(g(n0)) Px(α) denotes the predecessor at x of α > 0: “maximal

rdinal β < α s.t. β x”

14/20

SLIDE 68

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

Proposition (variant of [Buchholtz, Cicho´

n & Weiermann’94])

Let 0 < α < ε0 and α n0. Then Lg,0(n0) = 0 Lg,α(n0) = 1 + Lg,Pn0(α)(g(n0)) Px(α) denotes the predecessor at x of α > 0: “maximal

rdinal β < α s.t. β x”

Example

P3(ω2) = ω · 3 + 3 P3(ωω2) = ωω·3+3 · 3 + ωω·3+2 · 3 + ωω·3+1 · 3 + ωω·3 · 3 + ωω·2+3 · 3 + ωω·2+2 · 3 + ωω·2+1 · 3 + ωω·2 · 3 + ωω+3 · 3 + ωω+2 · 3 + ωω+1 · 3 + ωω · 3 + ω3 · 3 + ω2 · 3 + ω · 3 + 3

14/20

SLIDE 69

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

Proposition (variant of [Buchholtz, Cicho´

n & Weiermann’94])

Let 0 < α < ε0 and α n0. Then Lg,0(n0) = 0 Lg,α(n0) = 1 + Lg,Pn0(α)(g(n0)) Px(α) denotes the predecessor at x of α > 0: “maximal

rdinal β < α s.t. β x”

Example

P3(ω2) = ω · 3 + 3 P3(ωω2) = ωω·3+3 · 3 + ωω·3+2 · 3 + ωω·3+1 · 3 + ωω·3 · 3 + ωω·2+3 · 3 + ωω·2+2 · 3 + ωω·2+1 · 3 + ωω·2 · 3 + ωω+3 · 3 + ωω+2 · 3 + ωω+1 · 3 + ωω · 3 + ω3 · 3 + ω2 · 3 + ω · 3 + 3

14/20

SLIDE 70

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

Proposition (variant of [Buchholtz, Cicho´

n & Weiermann’94])

Let 0 < α < ε0 and α n0. Then Lg,0(n0) = 0 Lg,α(n0) = 1 + Lg,Pn0(α)(g(n0)) This function was already known in the literature! Definition (Cicho´ n Hierarchy [Cicho´

n & Tahhan Bittar’98])

For g : N → N, define (gα : N → N)α by g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x)) for α > 0

14/20

SLIDE 71

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

Length Function Theorem (for Ordinals) Let α < ε0 and n0 α. Then the longest (g,n0)-controlled descending sequence over α is of length Lg,α(n0) = gα(n0)

14/20

SLIDE 72

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Relating Norm and Length

[Cicho´ n & Tahhan Bittar’98]

Recall the definition of the Cicho´ n Hierarchy: g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x)) for α > 0 Definition (Hardy Hierarchy) For g : N → N, define (gα : N → N)α by g0(x)

def

= x gα(x)

def

= gPx(α)(g(x)) for α > 0

15/20

SLIDE 73

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Relating Norm and Length

[Cicho´ n & Tahhan Bittar’98]

g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x)) for α > 0 g0(x)

def

= x gα(x)

def

= gPx(α)(g(x)) for α > 0

length: Cicho´ n function gα(n0) norm: Hardy function gα(n0) norms xi indices i g0(n0) g1(n0) g2(n0) g3(n0) x0 x1 x2 x3

gα(x) = ggα(x)(x) gα(x) gα(x) + x

15/20

SLIDE 74

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Relating Norm and Length

[Cicho´ n & Tahhan Bittar’98]

g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x)) for α > 0 g0(x)

def

= x gα(x)

def

= gPx(α)(g(x)) for α > 0

length: Cicho´ n function gα(n0) norm: Hardy function gα(n0) norms xi indices i g0(n0) g1(n0) g2(n0) g3(n0) x0 x1 x2 x3

gα(x) = ggα(x)(x) gα(x) gα(x) + x

15/20

SLIDE 75

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Length of Decomposition Branches

α0 ∨ α1 ∨ α2 ∨ . . .

16/20

SLIDE 76

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Length of Decomposition Branches

α0 ∨ α1 ∨ α2 ∨ . . . Corollary Let n0 2 and g : N → N be such that the sequence of

rdinal ranks computed by the decomposition algorithm

is (g,n0)-controlled. The algorithm runs in

SPACE(gωω2

(n0)).

16/20

SLIDE 77

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Length of Decomposition Branches

α0 ∨ α1 ∨ α2 ∨ . . . Corollary Let n0 2 and g : N → N be such that the sequence of

rdinal ranks computed by the decomposition algorithm

is (g,n0)-controlled. The algorithm runs in

SPACE(gωω2

(n0)).

16/20

SLIDE 78

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Length of Decomposition Branches

α0 ∨ α1 ∨ α2 ∨ . . . Consequence of (Figueira, Figueira, S. & Schnoebelen’11) The control g(x)

def

= Hωω(e(x)) for H(x)

def

= x + 1 and an elementary function e, and n0 the size of the reachability instance fit. Thus the decomposition algorithm runs in

SPACE((Hωω ◦ e)ωω2

(n).

16/20

SLIDE 79

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

“SPACE((Hωω ◦ e)ωω2 (n)” is unreadable!

1. give names

◮ Hωω is the Ackermann function ◮ Hωω2

is the “quadratic Ackermann” function

2. define coarse-grained complexity classes

F<α

def

=

γ<ωα

FDTIME(Hγ(n))

Fα

def

=

f∈F<α

DTIME(Hωα(f(n))

Consequence of (S.’16, Thm. 4.4) VAS Reachability is in Fω2.

17/20

SLIDE 80

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

“SPACE((Hωω ◦ e)ωω2 (n)” is unreadable!

1. give names

◮ Hωω is the Ackermann function ◮ Hωω2

is the “quadratic Ackermann” function

2. define coarse-grained complexity classes

F<α

def

=

γ<ωα

FDTIME(Hγ(n))

Fα

def

=

f∈F<α

DTIME(Hωα(f(n))

Consequence of (S.’16, Thm. 4.4) VAS Reachability is in Fω2.

17/20

SLIDE 81

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

“SPACE((Hωω ◦ e)ωω2 (n)” is unreadable!

1. give names

◮ Hωω is the Ackermann function ◮ Hωω2

is the “quadratic Ackermann” function

2. define coarse-grained complexity classes

F<α

def

=

γ<ωα

FDTIME(Hγ(n))

Fα

def

=

f∈F<α

DTIME(Hωα(f(n))

Consequence of (S.’16, Thm. 4.4) VAS Reachability is in Fω2.

17/20

SLIDE 82

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

“SPACE((Hωω ◦ e)ωω2 (n)” is unreadable!

1. give names

◮ Hωω is the Ackermann function ◮ Hωω2

is the “quadratic Ackermann” function

2. define coarse-grained complexity classes

F<α

def

=

γ<ωα

FDTIME(Hγ(n))

Fα

def

=

f∈F<α

DTIME(Hωα(f(n))

Consequence of (S.’16, Thm. 4.4) VAS Reachability is in Fω2.

17/20

SLIDE 83

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

ExpSpace Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann Fω2

Fast-Growing Complexity

18/20

SLIDE 84

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

ExpSpace Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann Fω2

Fast-Growing Complexity

F3

def

=

e elementary

DTime(tower(e(n)))

18/20

SLIDE 85

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

ExpSpace Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann Fω2

Fast-Growing Complexity Examples of Tower-Complete Problems: ◮ satisfiability of first-order logic on words [Meyer’75] ◮ β-equivalence of simply typed λ terms [Statman’79] ◮ model-checking higher-order recursion schemes [Ong’06]

18/20

SLIDE 86

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

ExpSpace Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann Fω2

Fast-Growing Complexity

Fω

def

=

p primitive recursive

DTime(ackermann(p(n)))

18/20

SLIDE 87

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

ExpSpace Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann Fω2

Fast-Growing Complexity Examples of Ackermann-Complete Problems: ◮ reachability in lossy Minsky machines [Urquhart’98, Schnoebelen’02] ◮ satisfiability of safety Metric Temporal Logic [Lazi´

c et al.’16]

◮ satisfiability of Vertical XPath [Figueira and Segoufin’17]

18/20

SLIDE 88

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

ExpSpace Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann Fω2

Fast-Growing Complexity

18/20

SLIDE 89

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

ExpSpace Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann Fω2

Fast-Growing Complexity

18/20

SLIDE 90

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Summary

well-quasi-orders (wqo):

◮ proving algorithm termination

a toolbox for wqo-based complexity

◮ upper bounds: length function theorems

(for ordinals, Dickson’s Lemma, Higman’s Lemma, and combinations)

◮ lower bounds ◮ complexity classes: (Fα)α

this talk: focus on one problem

◮ reachability in vector addition systems

in Fω2

19/20

SLIDE 91

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Perspectives

1. complexity gap for VAS reachability

◮ ExpSpace-hard [Lipton’76]

better lower bounds? (Wojciech’s talk)

◮ decomposition algorithm: at least Fω (Ackermannian) time

[Zetzsche’16]

2. reachability in VAS extensions

◮ decidable in VAS with hierarchical zero tests [Reinhardt’08] ◮ what about (J´

erˆ

me’s talk)

◮ branching VAS ◮ unordered data Petri nets ◮ pushdown VAS 20/20

SLIDE 92

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Perspectives

1. complexity gap for VAS reachability

◮ ExpSpace-hard [Lipton’76]

better lower bounds? (Wojciech’s talk)

◮ decomposition algorithm: at least Fω (Ackermannian) time

[Zetzsche’16]

2. reachability in VAS extensions

◮ decidable in VAS with hierarchical zero tests [Reinhardt’08] ◮ what about (J´

erˆ

me’s talk)

◮ branching VAS ◮ unordered data Petri nets ◮ pushdown VAS 20/20

SLIDE 93

Demystifying Reachability in Vector Addition Systems

[Leroux & S.’15] Ideal Decomposition Theorem The Decomposition Algorithm computes the ideal decomposition of the set of runs from source to target. Upper Bound Theorem Reachability in vector addition systems is in cubic Ackermann.

21/20

SLIDE 94

Ideals of Well-Quasi-Orders (X,)

◮ Canonical decompositions

[Bonnet’75]

if D ⊆ X is ↓-closed, then D = I1 ∪ ··· ∪ In for (maximal) ideals I1,...,In Example (over N2)

D = ({0,...,2} × N) ∪ ({0,...,5} × {0,...,7}) ∪ (N × {0,...,4})

22/20

SLIDE 95

Ideals of Well-Quasi-Orders (X,)

◮ Canonical decompositions

[Bonnet’75]

if D ⊆ X is ↓-closed, then D = I1 ∪ ··· ∪ In for (maximal) ideals I1,...,In Example (over N2)

D = ({0,...,2} × N) ∪ ({0,...,5} × {0,...,7}) ∪ (N × {0,...,4})

22/20

SLIDE 96

Ideals of Well-Quasi-Orders (X,)

◮ Canonical decompositions

[Bonnet’75]

if D ⊆ X is ↓-closed, then D = I1 ∪ ··· ∪ In for (maximal) ideals I1,...,In

◮ Effective representations

[Goubault-Larrecq et al.’17]

Example (over N2)

D = (2,∞) ∪ (5,7) ∪ (∞,4)

22/20

SLIDE 97

Decomposition Theorem

Well-Quasi-Order on Runs combination of Dickson’s and Higman’s lemmata Syntax Semantics

I0

I1 I2 I3 I4 ↓Runs

23/20

SLIDE 98

Decomposition Theorem

Well-Quasi-Order on Runs combination of Dickson’s and Higman’s lemmata Syntax Semantics

I0

I1 I2 I3 I4 ↓Runs

23/20

SLIDE 99

Decomposition Theorem

Well-Quasi-Order on Runs combination of Dickson’s and Higman’s lemmata Syntax Semantics

I0

I1 I2 I3 I4 ↓Runs

23/20

SLIDE 100

Decomposition Theorem

Well-Quasi-Order on Runs combination of Dickson’s and Higman’s lemmata Syntax Semantics

I0

I1 I2 I3 I4 ↓Runs

23/20

SLIDE 101

Adherence Membership

◮ I is adherent to Runs if

I ⊆ ↓(I ∩ Runs)

◮ semantic equivalent to

Θ condition

◮ undecidable for arbitrary ideals ◮ decidable for the ideals arising in

the decomposition algorithm Runs ↓Runs I I adherent I not adherent

24/20

SLIDE 102

Adherence Membership

◮ I is adherent to Runs if

I ⊆ ↓(I ∩ Runs)

◮ semantic equivalent to

Θ condition

◮ undecidable for arbitrary ideals ◮ decidable for the ideals arising in

the decomposition algorithm Runs ↓Runs I I adherent I not adherent

24/20

SLIDE 103

Adherence Membership

◮ I is adherent to Runs if

I ⊆ ↓(I ∩ Runs)

◮ semantic equivalent to

Θ condition

◮ undecidable for arbitrary ideals ◮ decidable for the ideals arising in

the decomposition algorithm Runs ↓Runs I I adherent I not adherent

24/20

SLIDE 104

Adherence Membership

◮ I is adherent to Runs if

I ⊆ ↓(I ∩ Runs)

◮ semantic equivalent to

Θ condition

◮ undecidable for arbitrary ideals ◮ decidable for the ideals arising in

the decomposition algorithm Runs ↓Runs I I adherent I not adherent

24/20

SLIDE 105

Adherence Membership

◮ I is adherent to Runs if

I ⊆ ↓(I ∩ Runs)

◮ semantic equivalent to

Θ condition

◮ undecidable for arbitrary ideals ◮ decidable for the ideals arising in

the decomposition algorithm Runs ↓Runs I I adherent I not adherent

24/20

SLIDE 106

Adherence Membership

◮ I is adherent to Runs if

I ⊆ ↓(I ∩ Runs)

◮ semantic equivalent to

Θ condition

◮ undecidable for arbitrary ideals ◮ decidable for the ideals arising in

the decomposition algorithm Runs ↓Runs I I adherent I not adherent

24/20

SLIDE 107

Adherence Membership

◮ I is adherent to Runs if

I ⊆ ↓(I ∩ Runs)

◮ semantic equivalent to

Θ condition

◮ undecidable for arbitrary ideals ◮ decidable for the ideals arising in

the decomposition algorithm Runs ↓Runs I I adherent I not adherent

24/20

SLIDE 108

Branching VAS Reachability

◮ important open problem [Boja´ nczyk’14] ◮ incorrect decidability proof in [Bimb´

’15]

◮ application domains:

25/20

SLIDE 109

Branching VAS Reachability

◮ important open problem [Boja´ nczyk’14] ◮ incorrect decidability proof in [Bimb´

’15]

◮ application domains:

Complexity Theory Distributed Computing Computational Biology Proof Theory Database Theory Programming Languages Security Computational Linguistics 25/20

SLIDE 110

Branching VAS Reachability

◮ important open problem [Boja´ nczyk’14] ◮ incorrect decidability proof in [Bimb´

’15]

◮ application domains:

Complexity Theory Tower-hard [Lazi´ c & S., ToCL’15] Distributed Computing Computational Biology Proof Theory Database Theory Programming Languages Security Computational Linguistics 25/20

SLIDE 111

Branching VAS Reachability

◮ important open problem [Boja´ nczyk’14] ◮ incorrect decidability proof in [Bimb´

’15]

◮ application domains:

Complexity Theory Tower-hard [Lazi´ c & S., ToCL’15] Distributed Computing recursive parallel programs [Bouajjani & Emmi’13] Computational Biology Proof Theory Database Theory Programming Languages Security Computational Linguistics 25/20

SLIDE 112

Branching VAS Reachability

◮ important open problem [Boja´ nczyk’14] ◮ incorrect decidability proof in [Bimb´

’15]

◮ application domains:

Complexity Theory Tower-hard [Lazi´ c & S., ToCL’15] Distributed Computing recursive parallel programs [Bouajjani & Emmi’13] Computational Biology Proof Theory linear and relevance logics [de Groote et al.’04 Lazi´ c & S., ToCL’15 S., JSL’16] Database Theory Programming Languages Security Computational Linguistics 25/20

SLIDE 113

Branching VAS Reachability

◮ important open problem [Boja´ nczyk’14] ◮ incorrect decidability proof in [Bimb´

’15]

◮ application domains:

Complexity Theory Tower-hard [Lazi´ c & S., ToCL’15] Distributed Computing recursive parallel programs [Bouajjani & Emmi’13] Computational Biology population protocols [Bertrand et al.’17] Proof Theory linear and relevance logics [de Groote et al.’04 Lazi´ c & S., ToCL’15 S., JSL’16] Database Theory Programming Languages Security Computational Linguistics 25/20

SLIDE 114

Branching VAS Reachability

◮ important open problem [Boja´ nczyk’14] ◮ incorrect decidability proof in [Bimb´

’15]

◮ application domains:

Complexity Theory Tower-hard [Lazi´ c & S., ToCL’15] Distributed Computing recursive parallel programs [Bouajjani & Emmi’13] Computational Biology population protocols [Bertrand et al.’17] Proof Theory linear and relevance logics [de Groote et al.’04 Lazi´ c & S., ToCL’15 S., JSL’16] Database Theory Programming Languages

bservational equivalence

[Cotton-Barratt et al.’17] Security Computational Linguistics 25/20

SLIDE 115

Branching VAS Reachability

◮ important open problem [Boja´ nczyk’14] ◮ incorrect decidability proof in [Bimb´

’15]

◮ application domains:

Complexity Theory Tower-hard [Lazi´ c & S., ToCL’15] Distributed Computing recursive parallel programs [Bouajjani & Emmi’13] Computational Biology population protocols [Bertrand et al.’17] Proof Theory linear and relevance logics [de Groote et al.’04 Lazi´ c & S., ToCL’15 S., JSL’16] Database Theory data logics [Boja´ nczyk et al.’09, Abriola et al.’17] Programming Languages

bservational equivalence

[Cotton-Barratt et al.’17] Security Computational Linguistics 25/20

SLIDE 116

Branching VAS Reachability

◮ important open problem [Boja´ nczyk’14] ◮ incorrect decidability proof in [Bimb´

’15]

◮ application domains:

Complexity Theory Tower-hard [Lazi´ c & S., ToCL’15] Distributed Computing recursive parallel programs [Bouajjani & Emmi’13] Computational Biology population protocols [Bertrand et al.’17] Proof Theory linear and relevance logics [de Groote et al.’04 Lazi´ c & S., ToCL’15 S., JSL’16] Database Theory data logics [Boja´ nczyk et al.’09, Abriola et al.’17] Programming Languages

bservational equivalence

[Cotton-Barratt et al.’17] Security security protocols [Verma & Goubault-Larrecq’05] Computational Linguistics 25/20

SLIDE 117

Branching VAS Reachability

◮ important open problem [Boja´ nczyk’14] ◮ incorrect decidability proof in [Bimb´

’15]

◮ application domains:

Complexity Theory Tower-hard [Lazi´ c & S., ToCL’15] Distributed Computing recursive parallel programs [Bouajjani & Emmi’13] Computational Biology population protocols [Bertrand et al.’17] Proof Theory linear and relevance logics [de Groote et al.’04 Lazi´ c & S., ToCL’15 S., JSL’16] Database Theory data logics [Boja´ nczyk et al.’09, Abriola et al.’17] Programming Languages

bservational equivalence

[Cotton-Barratt et al.’17] Security security protocols [Verma & Goubault-Larrecq’05] Computational Linguistics dominance grammars [Rambow’94; S., ACL’10] minimalist syntax [Salvati’10] 25/20