[PPT] - On the Complexity of VAS Reachability Sylvain Schmitz based on PowerPoint Presentation

SLIDE 1

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

On the Complexity of VAS Reachability

Sylvain Schmitz

based on joint works with D. Figueira, S. Figueira, J. Leroux, and Ph. Schnoebelen

LSV, ENS Paris-Saclay & CNRS, Universit´ e Paris-Saclay CAALM 2019

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SLIDE 2

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Outline

well-quasi-orders (wqo):

◮ proving algorithm termination

a toolbox for wqo complexity

◮ upper bounds ◮ lower bounds ◮ complexity classes

this talk: focus on one problem

◮ reachability in vector addition systems

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SLIDE 3

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Outline

well-quasi-orders (wqo):

◮ proving algorithm termination

a toolbox for wqo complexity

◮ upper bounds ◮ lower bounds ◮ complexity classes

this talk: focus on one problem

◮ reachability in vector addition systems

2/24

SLIDE 4

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Outline

well-quasi-orders (wqo):

◮ proving algorithm termination

a toolbox for wqo complexity

◮ upper bounds ◮ lower bounds ◮ complexity classes

this talk: focus on one problem

◮ reachability in vector addition systems

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SLIDE 5

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Vector Addition Systems

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SLIDE 6

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)

SLIDE 8

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?

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SLIDE 9

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

3/24

SLIDE 10

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?

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Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Importance of the Problem

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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SLIDE 12

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Importance of the Problem

Central Decision Problem [S.’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

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 2019

R. J. Lipton: EXPSPACE lower bound

1976

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

Leroux & S.: cubic Ackermann upper bound (Fω3)

2015

S.: quadratic Ackermann upper bound (Fω2)

2017

W. Czerwinski, S. Lasota, R. Lazi´

c, J. Leroux,

F. Mazowiecki: TOWER lower bound

2018

J. Leroux & S.: Ackermann upper bound (Fω)

4/24

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 5/24

SLIDE 16

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) 5/24

SLIDE 17

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) 5/24

SLIDE 18

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) 5/24

SLIDE 19

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) 5/24

SLIDE 20

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) 5/24

SLIDE 21

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) 5/24

SLIDE 22

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) 5/24

SLIDE 23

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) 5/24

SLIDE 24

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

classically: uses coverability trees [Karp & Miller’69] in [Leroux & S.’19] Rackoff-style witnesses

5/24

SLIDE 25

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

classically: uses coverability trees [Karp & Miller’69] in [Leroux & S.’19] Rackoff-style witnesses

5/24

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) 5/24

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) 5/24

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) 5/24

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) 5/24

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) 5/24

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) 5/24

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) 5/24

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) 5/24

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

6/24

SLIDE 35

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

6/24

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

6/24

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

6/24

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

6/24

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

6/24

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

6/24

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]

7/24

SLIDE 42

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]

7/24

SLIDE 43

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Termination of the Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92, Leroux & S.’19]

Ranking Function ωω (ωd in dim. d) α0 ∨ ∨ α1 ∨ α2 ∨ . . .

8/24

SLIDE 44

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Termination of the Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92, Leroux & S.’19]

Ranking Function ωω (ωd in dim. d) α0 ∨ ∨ α1 ∨ α2 ∨ . . .

8/24

SLIDE 45

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Termination of the Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92, Leroux & S.’19]

Ranking Function ωω (ωd in dim. d) α0 ∨ ∨ α1 ∨ α2 ∨ . . .

8/24

SLIDE 46

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Termination of the Decomposition Algorithm

[Mayr’81, Kosaraju’82, Lambert’92, Leroux & S.’19]

Ranking Function ωω (ωd in dim. d) α0 ∨ ∨ α1 ∨ α2 ∨ . . .

8/24

SLIDE 47

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∗

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9/24

SLIDE 48

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

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∗

9/24

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? 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

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f

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m

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n

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r

d s : I n

n

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

c

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n

; W e l l

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f

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e

l d f

r

b

t

h t h e

r

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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

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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

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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

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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

∗

9/24

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

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

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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

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v

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n

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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

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p

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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

∗

9/24

SLIDE 51

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

A One-Player Game

◮ over Q0 × Q0 ◮ given initially (x0,y0) ◮ Eloise plays (xj,yj) s.t.

∀0 i < j, xi > xj or yi > yj

(x1,y1) (x2,y2) (x0,y0)

◮ Can Eloise win, i.e. play indefinitely? ◮ If not, how long can she last?

10/24

SLIDE 52

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

A One-Player Game

◮ over Q0 × Q0 ◮ given initially (x0,y0) ◮ Eloise plays (xj,yj) s.t.

∀0 i < j, xi > xj or yi > yj

(x1,y1) (x2,y2) (x0,y0)

◮ Can Eloise win, i.e. play indefinitely? ◮ If not, how long can she last?

10/24

SLIDE 53

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

A One-Player Game

◮ over Q0 × Q0 ◮ given initially (x0,y0) ◮ Eloise plays (xj,yj) s.t.

∀0 i < j, xi > xj or yi > yj

(x1,y1) (x2,y2) (x0,y0)

◮ Can Eloise win, i.e. play indefinitely? ◮ If not, how long can she last?

10/24

SLIDE 54

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

If (x0,y0) (0,0), then choosing (xj,yj) = (x0

2j , y0 2j ) wins.

11/24

SLIDE 55

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

A One-Player Game

◮ over N × N ◮ given initially (x0,y0) ◮ Eloise plays (xj,yj) s.t.

∀0 i < j, xi > xj or yi > yj

(x1,y1) (x2,y2) (x0,y0)

◮ Can Eloise win, i.e. play indefinitely? ◮ If not, how long can she last?

12/24

SLIDE 56

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Assume there exists an infinite sequence (xj,yj)j of moves over N2.

13/24

SLIDE 57

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Assume there exists an infinite sequence (xj,yj)j of moves over N2. Consider the pairs of indices i < j: color (i,j) purple if xi > xj but yi yj, red if xi > xj and yi > yj,

range if yi > yj but xi xj.

(3,4) (5,2) (2,3) ...

13/24

SLIDE 58

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Assume there exists an infinite sequence (xj,yj)j of moves over N2. Consider the pairs of indices i < j: color (i,j) purple if xi > xj but yi yj, red if xi > xj and yi > yj,

range if yi > yj but xi xj.

(3,4) (5,2) (2,3) ... By the infinite Ramsey Theorem, there exists an infinite monochromatic subset of indices.

13/24

SLIDE 59

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Assume there exists an infinite sequence (xj,yj)j of moves over N2. Consider the pairs of indices i < j: color (i,j) purple if xi > xj but yi yj, red if xi > xj and yi > yj,

range if yi > yj but xi xj.

(3,4) (5,2) (2,3) ... By the infinite Ramsey Theorem, there exists an infinite monochromatic subset of indices. In all cases, it implies the existence of an infinite decreasing sequence in N, a contradiction.

13/24

SLIDE 60

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

A One-Player Game

◮ over N × N ◮ given initially (x0,y0) ◮ Eloise plays (xj,yj) s.t.

∀0 i < j, xi > xj or yi > yj

(x1,y1) (x2,y2) (x0,y0)

◮ Can Eloise win, i.e. play indefinitely? ◮ If not, how long can she last?

14/24

SLIDE 61

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

◮

15/24

SLIDE 62

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Bad Sequences

15/24

Over a qo (X,)

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

are finite

◮

Example (over N2)

SLIDE 63

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Bad Sequences

16/24

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 64

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] 16/24

SLIDE 65

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Controlled Bad Sequences

16/24

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 66

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).

16/24

SLIDE 67

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

17/24

SLIDE 68

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

17/24

SLIDE 69

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}

17/24

SLIDE 70

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))

17/24

SLIDE 71

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))

17/24

SLIDE 72

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))

17/24

SLIDE 73

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

Lg,α(n0) = max

α0∈α,α0n0

1 + Lg,α0(g(n0))

18/24

SLIDE 74

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

Lg,α(n0) = max

α0∈α,α0n0

1 + Lg,α0(g(n0))

18/24

SLIDE 75

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

For a suitable norm function, there is a “maximising”

rdinal Pn0(α):

Lg,0(n0) = 0 Lg,α(n0) = 1 + Lg,Pn0(α)(g(n0)) These functions form the Cich´

n hierarchy.

18/24

SLIDE 76

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Case of Ordinals

[S.’14]

For a suitable norm function, there is a “maximising”

rdinal Pn0(α):

Lg,0(n0) = 0 Lg,α(n0) = 1 + Lg,Pn0(α)(g(n0)) These functions form the Cich´

n hierarchy.

18/24

SLIDE 77

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: Lg,0(x)

def

= 0 Lg,α(x)

def

= 1 + Lg,Px(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

19/24

SLIDE 78

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Relating Norm and Length

[Cicho´ n & Tahhan Bittar’98] length: Cicho´ n function Lg,α(n0) norm: Hardy function gα(n0) norms xi indices i g0(n0) g1(n0) g2(n0) g3(n0) x0 x1 x2 x3

gα(x) = gLg,α(x)(x) gα(x) Lg,α(x) + x

19/24

SLIDE 79

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Relating Norm and Length

[Cicho´ n & Tahhan Bittar’98] length: Cicho´ n function Lg,α(n0) norm: Hardy function gα(n0) norms xi indices i g0(n0) g1(n0) g2(n0) g3(n0) x0 x1 x2 x3

gα(x) = gLg,α(x)(x) gα(x) Lg,α(x) + x

19/24

SLIDE 80

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Length of Decomposition Branches

α0 ∨ α1 ∨ α2 ∨ . . .

20/24

SLIDE 81

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Length of Decomposition Branches

α0 ∨ α1 ∨ α2 ∨ . . . Consequence of (Leroux & S.’19) An elementary control g and n the size of the reachability instance fit. Thus the decomposition algorithm runs in

SPACE(gωω(n)), and SPACE(gωd(n))) in fixed dimension d.

20/24

SLIDE 82

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

The Length of Decomposition Branches

α0 ∨ α1 ∨ α2 ∨ . . . Consequence of (Leroux & S.’19) An elementary control g and n the size of the reachability instance fit. Thus the decomposition algorithm runs in

SPACE(gωω(n)), and SPACE(gωd(n))) in fixed dimension d.

20/24

SLIDE 83

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

“SPACE

gωd(n)
” is unreadable!

21/24

SLIDE 84

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

Hardy hierarchy with base function H(x)

def

= x + 1:

H0(x) = x Hk(x) =

k times

H ◦ ··· ◦ H(x)

= x + k Hω(x) = Hx+1(x) =

x+1 times

H ◦ ··· ◦ H(x)

= 2x + 1 Hω2(x) = Hω·(x+1) =

x+1 times

Hω ◦ ··· ◦ Hω(x)

≈ 2x Hω3(x) = Hω2·(x+1) =

x+1 times

Hω2 ◦ ··· ◦ Hω2(x)

≈ tower(x) . . . Hωω(x) = Hωx+1(x) ≈ ack(x)

21/24

SLIDE 85

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

Hardy hierarchy with base function H(x)

def

= x + 1:

H0(x) = x Hk(x) =

k times

H ◦ ··· ◦ H(x)

= x + k Hω(x) = Hx+1(x) =

x+1 times

H ◦ ··· ◦ H(x)

= 2x + 1 Hω2(x) = Hω·(x+1) =

x+1 times

Hω ◦ ··· ◦ Hω(x)

≈ 2x Hω3(x) = Hω2·(x+1) =

x+1 times

Hω2 ◦ ··· ◦ Hω2(x)

≈ tower(x) . . . Hωω(x) = Hωx+1(x) ≈ ack(x)

21/24

SLIDE 86

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

Hardy hierarchy with base function H(x)

def

= x + 1:

H0(x) = x Hk(x) =

k times

H ◦ ··· ◦ H(x)

= x + k Hω(x) = Hx+1(x) =

x+1 times

H ◦ ··· ◦ H(x)

= 2x + 1 Hω2(x) = Hω·(x+1) =

x+1 times

Hω ◦ ··· ◦ Hω(x)

≈ 2x Hω3(x) = Hω2·(x+1) =

x+1 times

Hω2 ◦ ··· ◦ Hω2(x)

≈ tower(x) . . . Hωω(x) = Hωx+1(x) ≈ ack(x)

21/24

SLIDE 87

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

Hardy hierarchy with base function H(x)

def

= x + 1:

H0(x) = x Hk(x) =

k times

H ◦ ··· ◦ H(x)

= x + k Hω(x) = Hx+1(x) =

x+1 times

H ◦ ··· ◦ H(x)

= 2x + 1 Hω2(x) = Hω·(x+1) =

x+1 times

Hω ◦ ··· ◦ Hω(x)

≈ 2x Hω3(x) = Hω2·(x+1) =

x+1 times

Hω2 ◦ ··· ◦ Hω2(x)

≈ tower(x) . . . Hωω(x) = Hωx+1(x) ≈ ack(x)

21/24

SLIDE 88

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

Hardy hierarchy with base function H(x)

def

= x + 1:

H0(x) = x Hk(x) =

k times

H ◦ ··· ◦ H(x)

= x + k Hω(x) = Hx+1(x) =

x+1 times

H ◦ ··· ◦ H(x)

= 2x + 1 Hω2(x) = Hω·(x+1) =

x+1 times

Hω ◦ ··· ◦ Hω(x)

≈ 2x Hω3(x) = Hω2·(x+1) =

x+1 times

Hω2 ◦ ··· ◦ Hω2(x)

≈ tower(x) . . . Hωω(x) = Hωx+1(x) ≈ ack(x)

21/24

SLIDE 89

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

Hardy hierarchy with base function H(x)

def

= x + 1:

H0(x) = x Hk(x) =

k times

H ◦ ··· ◦ H(x)

= x + k Hω(x) = Hx+1(x) =

x+1 times

H ◦ ··· ◦ H(x)

= 2x + 1 Hω2(x) = Hω·(x+1) =

x+1 times

Hω ◦ ··· ◦ Hω(x)

≈ 2x Hω3(x) = Hω2·(x+1) =

x+1 times

Hω2 ◦ ··· ◦ Hω2(x)

≈ tower(x) . . . Hωω(x) = Hωx+1(x) ≈ ack(x)

21/24

SLIDE 90

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

Define coarse-grained classes: F<α

def

=

β<ωα

FDTIME(Hβ(n))

Fα

def

=

f∈F<α

DTIME(Hωα(f(n)))

Consequence of (S.’16, Thm. 4.4) VAS Reachability is in Fω, and in Fd+3 in fixed dimension d.

21/24

SLIDE 91

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Restating the Result

Define coarse-grained classes: F<α

def

=

β<ωα

FDTIME(Hβ(n))

Fα

def

=

f∈F<α

DTIME(Hωα(f(n)))

Consequence of (S.’16, Thm. 4.4) VAS Reachability is in Fω, and in Fd+3 in fixed dimension d.

21/24

SLIDE 92

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Fast-Growing Complexity

22/24

SLIDE 93

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Fast-Growing Complexity

F3

def

=

e elementary

DTime(tower(e(n)))

22/24

SLIDE 94

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

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]

22/24

SLIDE 95

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Fast-Growing Complexity

Fω

def

=

p primitive recursive

DTime(ack(p(n)))

22/24

SLIDE 96

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

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]

22/24

SLIDE 97

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Fast-Growing Complexity

22/24

SLIDE 98

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Fast-Growing Complexity

22/24

SLIDE 99

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 Fω

23/24

SLIDE 100

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Perspectives

1. complexity gap for VAS reachability

◮ TOWER-hard [Czerwinski et al.’18]

better lower bounds?

◮ decomposition algorithm: requires Fω (Ackermannian) time,

because downward language inclusion is Fω-hard [Zetzsche’16]

2. reachability in VAS extensions

◮ decidable in VAS with hierarchical zero tests [Reinhardt’08] ◮ what about ◮ branching VAS ◮ unordered data Petri nets ◮ pushdown VAS 24/24

SLIDE 101

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Perspectives

1. complexity gap for VAS reachability

◮ TOWER-hard [Czerwinski et al.’18]

better lower bounds?

◮ decomposition algorithm: requires Fω (Ackermannian) time,

because downward language inclusion is Fω-hard [Zetzsche’16]

2. reachability in VAS extensions

◮ decidable in VAS with hierarchical zero tests [Reinhardt’08] ◮ what about ◮ branching VAS ◮ unordered data Petri nets ◮ pushdown VAS 24/24

SLIDE 102

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives

Perspectives

1. complexity gap for VAS reachability

◮ TOWER-hard [Czerwinski et al.’18]

better lower bounds?

◮ decomposition algorithm: requires Fω (Ackermannian) time,

because downward language inclusion is Fω-hard [Zetzsche’16]

2. reachability in VAS extensions

◮ decidable in VAS with hierarchical zero tests [Reinhardt’08] ◮ what about ◮ branching VAS ◮ unordered data Petri nets ◮ pushdown VAS 24/24

SLIDE 103

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.

25/24

SLIDE 104

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})

26/24

SLIDE 105

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})

26/24

SLIDE 106

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)

26/24

SLIDE 107

Decomposition Theorem

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

I0

I1 I2 I3 I4 ↓Runs

27/24

SLIDE 108

Decomposition Theorem

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

I0

I1 I2 I3 I4 ↓Runs

27/24

SLIDE 109

Decomposition Theorem

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

I0

I1 I2 I3 I4 ↓Runs

27/24

SLIDE 110

Decomposition Theorem

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

I0

I1 I2 I3 I4 ↓Runs

27/24

SLIDE 111

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

28/24

SLIDE 112

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

28/24

SLIDE 113

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

28/24

SLIDE 114

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

28/24

SLIDE 115

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

28/24

SLIDE 116

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

28/24

SLIDE 117

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

28/24

SLIDE 118

Branching VAS Reachability

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

’15]

◮ application domains:

29/24

SLIDE 119

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 29/24

SLIDE 120

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 29/24

SLIDE 121

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 29/24

SLIDE 122

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 29/24

SLIDE 123

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 29/24

SLIDE 124

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 29/24

SLIDE 125

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 29/24

SLIDE 126

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 29/24

SLIDE 127

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] 29/24