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Affine Extensions of Integer Vector Addition Systems with States Michael Blondin 1 , Christoph Haase 2 and Filip Mazowiecki 3 1 Technische Universit at M unchen 2 University of Oxford 3 Universit e de Bordeaux Infinity 2018 Affine

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SLIDE 1

Affine Extensions of Integer Vector Addition Systems with States

Michael Blondin1, Christoph Haase2 and Filip Mazowiecki3

1Technische Universit¨

at M¨ unchen

2University of Oxford 3Universit´

e de Bordeaux

Infinity 2018

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SLIDE 2

Affine Extensions of Integer Vector Addition Systems with States

Michael Blondin1, Christoph Haase2 and Filip Mazowiecki3

1Technische Universit¨

at M¨ unchen

2University of Oxford 3Universit´

e de Bordeaux

Infinity 2018

2018/07/08 23:31:04 (14)

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SLIDE 3

Vector Addition Systems with States (VASS) Automata with counters

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 1 / 14

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SLIDE 4

Vector Addition Systems with States (VASS) Automata with counters VASS example p q (−1, 2) (2, −1) (0, 0) (0, 0)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 1 / 14

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SLIDE 5

Vector Addition Systems with States (VASS) Automata with counters VASS example p q (−1, 2) (2, −1) (0, 0) (0, 0) example run: p(1, 0) → p(0, 2) → q(0, 2) → q(2, 1) → q(4, 0) → p(4, 0)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 1 / 14

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SLIDE 6

Vector Addition Systems with States (VASS) Automata with counters VASS example p q (−1, 2) (2, −1) (0, 0) (0, 0) example run: p(1, 0) → p(0, 2) → q(0, 2) → q(2, 1) → q(4, 0) → p(4, 0) Important restriction: no negative values

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 1 / 14

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SLIDE 7

Affine VASS Interaction between counters

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 2 / 14

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SLIDE 8

Affine VASS Interaction between counters Transitions updates before: p(v) → q(v + w) Transitions updates now: p(v) → q(Av + w)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 2 / 14

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SLIDE 9

Affine VASS Interaction between counters Transitions updates before: p(v) → q(v + w) Transitions updates now: p(v) → q(Av + w) Affine VASS example p q

  • 1

1

  • ,
  • 1
  • 1

1

  • ,
  • 1
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 2 / 14

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SLIDE 10

Affine VASS Interaction between counters Transitions updates before: p(v) → q(v + w) Transitions updates now: p(v) → q(Av + w) Affine VASS example p q

  • 1

1

  • ,
  • 1
  • 1

1

  • ,
  • 1
  • p(x, y) → q(x, x + 1)

(copy)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 2 / 14

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SLIDE 11

Affine VASS Interaction between counters Transitions updates before: p(v) → q(v + w) Transitions updates now: p(v) → q(Av + w) Affine VASS example p q

  • 1

1

  • ,
  • 1
  • 1

1

  • ,
  • 1
  • p(x, y) → q(x, x + 1)

(copy) q(x, y) → p(x + y, 1) (transfer)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 2 / 14

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Affine VASS subclasses What matrices are allowed?

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 3 / 14

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Affine VASS subclasses What matrices are allowed? We consider mostly matrices over {0, 1}

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 3 / 14

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SLIDE 14

Affine VASS subclasses What matrices are allowed? We consider mostly matrices over {0, 1}

  • A has exactly one 1 in each column

(transfer VASS)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 3 / 14

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SLIDE 15

Affine VASS subclasses What matrices are allowed? We consider mostly matrices over {0, 1}

  • A has exactly one 1 in each column

(transfer VASS)

  • A has exactly one 1 in each row

(copy VASS)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 3 / 14

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SLIDE 16

Affine VASS subclasses What matrices are allowed? We consider mostly matrices over {0, 1}

  • A has exactly one 1 in each column

(transfer VASS)

  • A has exactly one 1 in each row

(copy VASS)

  • A does not contain any 1 outside of its diagonal

(reset VASS)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 3 / 14

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SLIDE 17

Affine VASS subclasses What matrices are allowed? We consider mostly matrices over {0, 1}

  • A has exactly one 1 in each column

(transfer VASS)

  • A has exactly one 1 in each row

(copy VASS)

  • A does not contain any 1 outside of its diagonal

(reset VASS)

  • A has exactly one 1 in each row and each column

(permutation VASS)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 3 / 14

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SLIDE 18

Decision problems For affine VASS

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 4 / 14

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SLIDE 19

Decision problems For affine VASS Reachability problem: Given: an affine VASS V and p(u), q(v) Decide: whether p(u) ∗ − → q(v)?

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 4 / 14

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SLIDE 20

Decision problems For affine VASS Reachability problem: Given: an affine VASS V and p(u), q(v) Decide: whether p(u) ∗ − → q(v)? Coverability problem: Given: an affine VASS V and p(u), q(v) Decide: whether exists v′ s.t. p(u) ∗ − → q(v′) and v′ ≥ v?

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 4 / 14

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Decision problems For affine VASS Reachability problem: Given: an affine VASS V and p(u), q(v) Decide: whether p(u) ∗ − → q(v)? Coverability problem: Given: an affine VASS V and p(u), q(v) Decide: whether exists v′ s.t. p(u) ∗ − → q(v′) and v′ ≥ v?

  • Usually affine VASS → some specific class
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 4 / 14

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Decision problems For affine VASS Reachability problem: Given: an affine VASS V and p(u), q(v) Decide: whether p(u) ∗ − → q(v)? Coverability problem: Given: an affine VASS V and p(u), q(v) Decide: whether exists v′ s.t. p(u) ∗ − → q(v′) and v′ ≥ v?

  • Usually affine VASS → some specific class
  • In this talk mostly affine Z-VASS

(counters can be negative)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 4 / 14

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State of art Over N VASS:

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 5 / 14

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State of art Over N VASS:

  • Coverability: EXPSPACE-complete
  • Reachability: decidable, EXPSPACE-hard
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 5 / 14

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State of art Over N VASS:

  • Coverability: EXPSPACE-complete
  • Reachability: decidable, EXPSPACE-hard

transfer/reset VASS:

  • Coverability: decidable, Ackermann-complete

[Schnoebelen, 2002],[Figueira, Figueira, Schmitz and Schnoebelen, 2011]

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 5 / 14

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State of art Over N VASS:

  • Coverability: EXPSPACE-complete
  • Reachability: decidable, EXPSPACE-hard

transfer/reset VASS:

  • Coverability: decidable, Ackermann-complete

[Schnoebelen, 2002],[Figueira, Figueira, Schmitz and Schnoebelen, 2011]

  • Reachability: undecidable [Araki and Kasami, 1976]
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 5 / 14

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SLIDE 27

State of art Over N VASS:

  • Coverability: EXPSPACE-complete
  • Reachability: decidable, EXPSPACE-hard

transfer/reset VASS:

  • Coverability: decidable, Ackermann-complete

[Schnoebelen, 2002],[Figueira, Figueira, Schmitz and Schnoebelen, 2011]

  • Reachability: undecidable [Araki and Kasami, 1976]

Over Z

  • Reachability and Coverability are inter-reducible
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 5 / 14

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SLIDE 28

State of art Over N VASS:

  • Coverability: EXPSPACE-complete
  • Reachability: decidable, EXPSPACE-hard

transfer/reset VASS:

  • Coverability: decidable, Ackermann-complete

[Schnoebelen, 2002],[Figueira, Figueira, Schmitz and Schnoebelen, 2011]

  • Reachability: undecidable [Araki and Kasami, 1976]

Over Z

  • Reachability and Coverability are inter-reducible
  • VASS and reset VASS NP-complete [Haase and Halfon, 2014]
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 5 / 14

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SLIDE 29

Z-VASS instead of VASS This simplifies the problem (?)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 6 / 14

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SLIDE 30

Z-VASS instead of VASS This simplifies the problem (?) Coverability for affine VASS with matrices over N

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 6 / 14

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SLIDE 31

Z-VASS instead of VASS This simplifies the problem (?) Coverability for affine VASS with matrices over N

  • decidable by [Figueira, Figueira, Schmitz and Schnoebelen, 2011]
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 6 / 14

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SLIDE 32

Z-VASS instead of VASS This simplifies the problem (?) Coverability for affine VASS with matrices over N

  • decidable by [Figueira, Figueira, Schmitz and Schnoebelen, 2011]

Reachability for affine Z-VASS with matrices over N

  • undecidable already in dimension 2 [Reichert, 2015]
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 6 / 14

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SLIDE 33

Z-VASS instead of VASS This simplifies the problem (?) Coverability for affine VASS with matrices over N

  • decidable by [Figueira, Figueira, Schmitz and Schnoebelen, 2011]

Reachability for affine Z-VASS with matrices over N

  • undecidable already in dimension 2 [Reichert, 2015]

Proof: reduce from PCP over {0, 1} to a 2-VASS

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 6 / 14

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SLIDE 34

Z-VASS instead of VASS This simplifies the problem (?) Coverability for affine VASS with matrices over N

  • decidable by [Figueira, Figueira, Schmitz and Schnoebelen, 2011]

Reachability for affine Z-VASS with matrices over N

  • undecidable already in dimension 2 [Reichert, 2015]

Proof: reduce from PCP over {0, 1} to a 2-VASS we build two words: w, v; counters store bin(1w) and bin(1v)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 6 / 14

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SLIDE 35

Z-VASS instead of VASS This simplifies the problem (?) Coverability for affine VASS with matrices over N

  • decidable by [Figueira, Figueira, Schmitz and Schnoebelen, 2011]

Reachability for affine Z-VASS with matrices over N

  • undecidable already in dimension 2 [Reichert, 2015]

Proof: reduce from PCP over {0, 1} to a 2-VASS we build two words: w, v; counters store bin(1w) and bin(1v) matrices used to shift (double the values in counters)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 6 / 14

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SLIDE 36

Z-VASS instead of VASS This simplifies the problem (?) Coverability for affine VASS with matrices over N

  • decidable by [Figueira, Figueira, Schmitz and Schnoebelen, 2011]

Reachability for affine Z-VASS with matrices over N

  • undecidable already in dimension 2 [Reichert, 2015]

Proof: reduce from PCP over {0, 1} to a 2-VASS we build two words: w, v; counters store bin(1w) and bin(1v) matrices used to shift (double the values in counters) But for Z-VASS reachability and coverability are inter-reducible So undecidability for coverability of affine Z-VASS (in dimension 4)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 6 / 14

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SLIDE 37

Motivation

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 7 / 14

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SLIDE 38

Motivation

  • Why consider relaxed variants (like Z-VASS)?
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 7 / 14

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SLIDE 39

Motivation

  • Why consider relaxed variants (like Z-VASS)?

A configuration not reachable for relaxed semantics = ⇒ not reachable

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 7 / 14

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SLIDE 40

Motivation

  • Why consider relaxed variants (like Z-VASS)?

A configuration not reachable for relaxed semantics = ⇒ not reachable Used to prune the search space

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 7 / 14

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SLIDE 41

Motivation

  • Why consider relaxed variants (like Z-VASS)?

A configuration not reachable for relaxed semantics = ⇒ not reachable Used to prune the search space

  • Transfer VASS used in reasoning about
  • broadcast protocols
  • multi-threaded non-recursive C programs
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 7 / 14

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SLIDE 42

Motivation

  • Why consider relaxed variants (like Z-VASS)?

A configuration not reachable for relaxed semantics = ⇒ not reachable Used to prune the search space

  • Transfer VASS used in reasoning about
  • broadcast protocols
  • multi-threaded non-recursive C programs
  • Recent implementations of Coverability for VASS
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 7 / 14

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SLIDE 43

Motivation

  • Why consider relaxed variants (like Z-VASS)?

A configuration not reachable for relaxed semantics = ⇒ not reachable Used to prune the search space

  • Transfer VASS used in reasoning about
  • broadcast protocols
  • multi-threaded non-recursive C programs
  • Recent implementations of Coverability for VASS

Used Z-VASS or continuous VASS

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 7 / 14

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SLIDE 44

Motivation

  • Why consider relaxed variants (like Z-VASS)?

A configuration not reachable for relaxed semantics = ⇒ not reachable Used to prune the search space

  • Transfer VASS used in reasoning about
  • broadcast protocols
  • multi-threaded non-recursive C programs
  • Recent implementations of Coverability for VASS

Used Z-VASS or continuous VASS Quite successful [Esparza et al., 2014], [Blondin et al., 2016] [Geffroy, Leroux and Sutre, 2016]

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 7 / 14

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SLIDE 45

Matrix monoid of the affine VASS V MV – the matrix monoid generated by matrices in V

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 8 / 14

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SLIDE 46

Matrix monoid of the affine VASS V MV – the matrix monoid generated by matrices in V d-dimension

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 8 / 14

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SLIDE 47

Matrix monoid of the affine VASS V MV – the matrix monoid generated by matrices in V d-dimension

  • A has exactly one 1 in each column,

|MV| ≤ dd (transfer VASS)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 8 / 14

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SLIDE 48

Matrix monoid of the affine VASS V MV – the matrix monoid generated by matrices in V d-dimension

  • A has exactly one 1 in each column,

|MV| ≤ dd (transfer VASS)

  • A has exactly one 1 in each row,

|MV| ≤ dd (copy VASS)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 8 / 14

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SLIDE 49

Matrix monoid of the affine VASS V MV – the matrix monoid generated by matrices in V d-dimension

  • A has exactly one 1 in each column,

|MV| ≤ dd (transfer VASS)

  • A has exactly one 1 in each row,

|MV| ≤ dd (copy VASS)

  • A does not contain any 1 outside of its diagonal,

|MV| ≤ 2d (reset VASS)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 8 / 14

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SLIDE 50

Matrix monoid of the affine VASS V MV – the matrix monoid generated by matrices in V d-dimension

  • A has exactly one 1 in each column,

|MV| ≤ dd (transfer VASS)

  • A has exactly one 1 in each row,

|MV| ≤ dd (copy VASS)

  • A does not contain any 1 outside of its diagonal,

|MV| ≤ 2d (reset VASS)

  • A has exactly one 1 in each row and each column,

|MV| ≤ d! (permutation VASS)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 8 / 14

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SLIDE 51

Our results Consider affine Z-VASS V

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 9 / 14

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SLIDE 52

Our results Consider affine Z-VASS V

  • if MV is finite

we reduce reachability in V to reachability in VASS V′

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 9 / 14

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SLIDE 53

Our results Consider affine Z-VASS V

  • if MV is finite

we reduce reachability in V to reachability in VASS V′ where |V′| = |V| · MV

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 9 / 14

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SLIDE 54

Our results Consider affine Z-VASS V

  • if MV is finite

we reduce reachability in V to reachability in VASS V′ where |V′| = |V| · MV if MV is of exponential size reachability in PSPACE (all previous cases)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 9 / 14

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SLIDE 55

Our results Consider affine Z-VASS V

  • if MV is finite

we reduce reachability in V to reachability in VASS V′ where |V′| = |V| · MV if MV is of exponential size reachability in PSPACE (all previous cases) and PSPACE lower bound for permutation+reset

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 9 / 14

slide-56
SLIDE 56

Our results Consider affine Z-VASS V

  • if MV is finite

we reduce reachability in V to reachability in VASS V′ where |V′| = |V| · MV if MV is of exponential size reachability in PSPACE (all previous cases) and PSPACE lower bound for permutation+reset

  • if MV is infinite

undecidability for transfer+copy

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 9 / 14

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SLIDE 57

Our results Consider affine Z-VASS V

  • if MV is finite

we reduce reachability in V to reachability in VASS V′ where |V′| = |V| · MV if MV is of exponential size reachability in PSPACE (all previous cases) and PSPACE lower bound for permutation+reset

  • if MV is infinite

undecidability for transfer+copy (even in dimension 3)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 9 / 14

slide-58
SLIDE 58

Results overview Finite monoids

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 10 / 14

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SLIDE 59

Results overview Finite monoids Z-VASS (NP)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 10 / 14

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SLIDE 60

Results overview Finite monoids Z-VASS (NP) Reset Z-VASS (NP) Permutation Z-VASS (NP/PSPACE)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 10 / 14

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SLIDE 61

Results overview Finite monoids Z-VASS (NP) Reset Z-VASS (NP) Permutation Z-VASS (NP/PSPACE) Reset+permutation Z-VASS (PSPACE) Transfer Z-VASS (PSPACE) Copy Z-VASS (PSPACE)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 10 / 14

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SLIDE 62

Results overview Finite monoids Z-VASS (NP) Reset Z-VASS (NP) Permutation Z-VASS (NP/PSPACE) Reset+permutation Z-VASS (PSPACE) Transfer Z-VASS (PSPACE) Copy Z-VASS (PSPACE) Infinite monoids

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 10 / 14

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SLIDE 63

Results overview Finite monoids Z-VASS (NP) Reset Z-VASS (NP) Permutation Z-VASS (NP/PSPACE) Reset+permutation Z-VASS (PSPACE) Transfer Z-VASS (PSPACE) Copy Z-VASS (PSPACE) Infinite monoids Transfer+copy Z-VASS (UNDEC) Affine Z-VASS (UNDEC)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 10 / 14

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SLIDE 64

States and semilinearity for affine Z-VASS Some remarks

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 11 / 14

slide-65
SLIDE 65

States and semilinearity for affine Z-VASS Some remarks

  • States make a difference for Z-VASS

(the +3 dimensions reduction doesn’t work)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 11 / 14

slide-66
SLIDE 66

States and semilinearity for affine Z-VASS Some remarks

  • States make a difference for Z-VASS

(the +3 dimensions reduction doesn’t work)

  • The reachability relation is semilinear

for affine Z-VASS with finite MV

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 11 / 14

slide-67
SLIDE 67

States and semilinearity for affine Z-VASS Some remarks

  • States make a difference for Z-VASS

(the +3 dimensions reduction doesn’t work)

  • The reachability relation is semilinear

for affine Z-VASS with finite MV

  • For |Q| = 1 and MV with one generator: semilinearity iff MV finite

[Boigelot, 1998], [Finkel and Leroux, 2002]

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 11 / 14

slide-68
SLIDE 68

States and semilinearity for affine Z-VASS Some remarks

  • States make a difference for Z-VASS

(the +3 dimensions reduction doesn’t work)

  • The reachability relation is semilinear

for affine Z-VASS with finite MV

  • For |Q| = 1 and MV with one generator: semilinearity iff MV finite

[Boigelot, 1998], [Finkel and Leroux, 2002] Not in our case: p p q

  • 1

1 1 1

  • , 0

I,

  • 1
  • I,
  • 1
  • I,
  • −1
  • I,
  • −1
  • 1

1 1 1

  • , 0
  • , 0
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 11 / 14

slide-69
SLIDE 69

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-70
SLIDE 70

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • Just encode MV into the states

so Q′ = Q × MV

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-71
SLIDE 71

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • Just encode MV into the states

so Q′ = Q × MV the matrix tells you how to update the value

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-72
SLIDE 72

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • Just encode MV into the states

so Q′ = Q × MV the matrix tells you how to update the value Example: transfer VASS Two counters (x, y) with occasional transfer x → y

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-73
SLIDE 73

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • Just encode MV into the states

so Q′ = Q × MV the matrix tells you how to update the value Example: transfer VASS Two counters (x, y) with occasional transfer x → y (0, 0)

+(5,1)

− − − − → (5, 1)

+(−3,1)

− − − − − → (2, 2)

x→y

− − − → (0, 4)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-74
SLIDE 74

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • Just encode MV into the states

so Q′ = Q × MV the matrix tells you how to update the value Example: transfer VASS Two counters (x, y) with occasional transfer x → y (0, 0)

+(5,1)

− − − − → (5, 1)

+(−3,1)

− − − − − → (2, 2)

x→y

− − − → (0, 4) (0, 0)

+(0,6)

− − − − → (0, 6)

+(0,−2)

− − − − − → (0, 4)

update M

− − − − − − → (0, 4)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-75
SLIDE 75

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • Just encode MV into the states

so Q′ = Q × MV the matrix tells you how to update the value Example: transfer VASS Two counters (x, y) with occasional transfer x → y (0, 0)

+(5,1)

− − − − → (5, 1)

+(−3,1)

− − − − − → (2, 2)

x→y

− − − → (0, 4) (0, 0)

+(0,6)

− − − − → (0, 6)

+(0,−2)

− − − − − → (0, 4)

update M

− − − − − − → (0, 4) OK OK

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-76
SLIDE 76

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • Just encode MV into the states

so Q′ = Q × MV the matrix tells you how to update the value Example: transfer VASS Two counters (x, y) with occasional transfer x → y (0, 0)

+(5,1)

− − − − → (5, 1)

+(−6,1)

− − − − − → (−1, 2)

x→y

− − − → (0, 1) (0, 0)

+(0,6)

− − − − → (0, 6)

+(0,−5)

− − − − − → (0, 1)

update M

− − − − − − → (0, 1)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-77
SLIDE 77

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • Just encode MV into the states

so Q′ = Q × MV the matrix tells you how to update the value Example: transfer VASS Two counters (x, y) with occasional transfer x → y (0, 0)

+(5,1)

− − − − → (5, 1)

+(−6,1)

− − − − − → (−1, 2)

x→y

− − − → (0, 1) (0, 0)

+(0,6)

− − − − → (0, 6)

+(0,−5)

− − − − − → (0, 1)

update M

− − − − − − → (0, 1) NOT OK (in N) OK

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-78
SLIDE 78

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • Just encode MV into the states

so Q′ = Q × MV the matrix tells you how to update the value Example: transfer VASS Two counters (x, y) with occasional transfer x → y (0, 0)

+(5,1)

− − − − → (5, 1)

+(−6,1)

− − − − − → (−1, 2)

x→y

− − − → (0, 1) (0, 0)

+(0,6)

− − − − → (0, 6)

+(0,−5)

− − − − − → (0, 1)

update M

− − − − − − → (0, 1) NOT OK (in N) OK

  • Using flatness results [Blondin et al., 2015]

we get PSPACE if MV is exponential

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-79
SLIDE 79

From affine Z-VASS to Z-VASS Given affine Z-VASS V construct Z-VASS V′

  • Just encode MV into the states

so Q′ = Q × MV the matrix tells you how to update the value Example: transfer VASS Two counters (x, y) with occasional transfer x → y (0, 0)

+(5,1)

− − − − → (5, 1)

+(−6,1)

− − − − − → (−1, 2)

x→y

− − − → (0, 1) (0, 0)

+(0,6)

− − − − → (0, 6)

+(0,−5)

− − − − − → (0, 1)

update M

− − − − − − → (0, 1) NOT OK (in N) OK

  • Using flatness results [Blondin et al., 2015]

we get PSPACE if MV is exponential for reset VASS we get NP (already known)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 12 / 14

slide-80
SLIDE 80

Permutation + reset VASS is PSPACE-hard Its contained in transfer VASS and copy VASS

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 13 / 14

slide-81
SLIDE 81

Permutation + reset VASS is PSPACE-hard Its contained in transfer VASS and copy VASS Simulate a linear space machine with alphabet Γ and states P tape size n

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 13 / 14

slide-82
SLIDE 82

Permutation + reset VASS is PSPACE-hard Its contained in transfer VASS and copy VASS Simulate a linear space machine with alphabet Γ and states P tape size n Define affine Z-VASS dimension: d = |Γ| · n

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 13 / 14

slide-83
SLIDE 83

Permutation + reset VASS is PSPACE-hard Its contained in transfer VASS and copy VASS Simulate a linear space machine with alphabet Γ and states P tape size n Define affine Z-VASS dimension: d = |Γ| · n denote xi,a for a ∈ Γ, 1 ≤ i ≤ n “in cell i letter a”

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 13 / 14

slide-84
SLIDE 84

Permutation + reset VASS is PSPACE-hard Its contained in transfer VASS and copy VASS Simulate a linear space machine with alphabet Γ and states P tape size n Define affine Z-VASS dimension: d = |Γ| · n denote xi,a for a ∈ Γ, 1 ≤ i ≤ n “in cell i letter a” invariant: only 0 and 1 in counters and the sum is ≤ n

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 13 / 14

slide-85
SLIDE 85

Permutation + reset VASS is PSPACE-hard Its contained in transfer VASS and copy VASS Simulate a linear space machine with alphabet Γ and states P tape size n Define affine Z-VASS dimension: d = |Γ| · n denote xi,a for a ∈ Γ, 1 ≤ i ≤ n “in cell i letter a” invariant: only 0 and 1 in counters and the sum is ≤ n Transitions: swap content of registers (permutation VASS)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 13 / 14

slide-86
SLIDE 86

Permutation + reset VASS is PSPACE-hard Its contained in transfer VASS and copy VASS Simulate a linear space machine with alphabet Γ and states P tape size n Define affine Z-VASS dimension: d = |Γ| · n denote xi,a for a ∈ Γ, 1 ≤ i ≤ n “in cell i letter a” invariant: only 0 and 1 in counters and the sum is ≤ n Transitions: swap content of registers (permutation VASS) Checking mistakes: with resets

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 13 / 14

slide-87
SLIDE 87

Permutation + reset VASS is PSPACE-hard Its contained in transfer VASS and copy VASS Simulate a linear space machine with alphabet Γ and states P tape size n Define affine Z-VASS dimension: d = |Γ| · n denote xi,a for a ∈ Γ, 1 ≤ i ≤ n “in cell i letter a” invariant: only 0 and 1 in counters and the sum is ≤ n Transitions: swap content of registers (permutation VASS) Checking mistakes: with resets if there was a mistake the sum is < n

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 13 / 14

slide-88
SLIDE 88

Conclusion

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 14 / 14

slide-89
SLIDE 89

Conclusion

  • What is the size of MV? Exponential?

That would give a PSPACE-upper bound for reachability

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 14 / 14

slide-90
SLIDE 90

Conclusion

  • What is the size of MV? Exponential?

That would give a PSPACE-upper bound for reachability (Note: I ignored the size of elements in matrices)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 14 / 14

slide-91
SLIDE 91

Conclusion

  • What is the size of MV? Exponential?

That would give a PSPACE-upper bound for reachability (Note: I ignored the size of elements in matrices) TOWER upper bound is known [Mandel and Simon, 1977] For 1 generator or for groups exponential upper bound

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 14 / 14

slide-92
SLIDE 92

Conclusion

  • What is the size of MV? Exponential?

That would give a PSPACE-upper bound for reachability (Note: I ignored the size of elements in matrices) TOWER upper bound is known [Mandel and Simon, 1977] For 1 generator or for groups exponential upper bound

  • Is reachability undecidable for any class with infinite MV?

(or can we find a dichotomy)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 14 / 14

slide-93
SLIDE 93

Conclusion

  • What is the size of MV? Exponential?

That would give a PSPACE-upper bound for reachability (Note: I ignored the size of elements in matrices) TOWER upper bound is known [Mandel and Simon, 1977] For 1 generator or for groups exponential upper bound

  • Is reachability undecidable for any class with infinite MV?

(or can we find a dichotomy)

  • What is the complexity of permutation Z-VASS?

(between NP and PSPACE)

  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 14 / 14

slide-94
SLIDE 94

Conclusion

  • What is the size of MV? Exponential?

That would give a PSPACE-upper bound for reachability (Note: I ignored the size of elements in matrices) TOWER upper bound is known [Mandel and Simon, 1977] For 1 generator or for groups exponential upper bound

  • Is reachability undecidable for any class with infinite MV?

(or can we find a dichotomy)

  • What is the complexity of permutation Z-VASS?

(between NP and PSPACE)

  • Tools for transfer VASS?
  • M. Blondin, C. Haase and F. Mazowiecki

Affine Extensions of Integer VASS 14 / 14