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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

  1. 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

  2. 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 � 2018/07/08 23:31:04 (14) �

  3. Vector Addition Systems with States (VASS) Automata with counters 1 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  4. Vector Addition Systems with States (VASS) Automata with counters VASS example ( − 1 , 2) (2 , − 1) (0 , 0 ) p q (0 , 0) 1 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  5. Vector Addition Systems with States (VASS) Automata with counters VASS example ( − 1 , 2) (2 , − 1) (0 , 0 ) p q (0 , 0) example run: p (1 , 0) → p (0 , 2) → q (0 , 2) → q (2 , 1) → q (4 , 0) → p (4 , 0) 1 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  6. Vector Addition Systems with States (VASS) Automata with counters VASS example ( − 1 , 2) (2 , − 1) (0 , 0 ) p q (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 1 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  7. Affine VASS Interaction between counters 2 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  8. Affine VASS Interaction between counters Transitions updates before: p ( v ) → q ( v + w ) Transitions updates now: p ( v ) → q ( A v + w ) 2 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  9. Affine VASS Interaction between counters Transitions updates before: p ( v ) → q ( v + w ) Transitions updates now: p ( v ) → q ( A v + w ) � � � � 1 0 0 , 1 0 1 p q Affine VASS example � � � � 1 1 0 , 0 0 1 2 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  10. Affine VASS Interaction between counters Transitions updates before: p ( v ) → q ( v + w ) Transitions updates now: p ( v ) → q ( A v + w ) � � � � 1 0 0 , 1 0 1 p q Affine VASS example � � � � 1 1 0 p ( x, y ) → q ( x, x + 1) , 0 0 1 (copy) 2 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  11. Affine VASS Interaction between counters Transitions updates before: p ( v ) → q ( v + w ) Transitions updates now: p ( v ) → q ( A v + w ) � � � � 1 0 0 , 1 0 1 p q Affine VASS example � � � � 1 1 0 p ( x, y ) → q ( x, x + 1) , 0 0 1 (copy) q ( x, y ) → p ( x + y, 1) (transfer) 2 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  12. Affine VASS subclasses What matrices are allowed? 3 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  13. Affine VASS subclasses What matrices are allowed? We consider mostly matrices over { 0 , 1 } 3 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  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) 3 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  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) 3 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  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) 3 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  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) 3 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  18. Decision problems For affine VASS 4 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  19. Decision problems For affine VASS Reachability problem: Given : an affine VASS V and p ( u ) , q ( v ) Decide : whether p ( u ) ∗ − → q ( v ) ? 4 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  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 ? − 4 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  21. 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 4 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  22. 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) 4 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  23. State of art Over N VASS: 5 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  24. State of art Over N VASS: • Coverability: EXPSPACE-complete • Reachability: decidable, EXPSPACE-hard 5 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  25. 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] 5 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  26. 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] 5 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  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 5 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  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] 5 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  29. Z -VASS instead of VASS This simplifies the problem (?) 6 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  30. Z -VASS instead of VASS This simplifies the problem (?) Coverability for affine VASS with matrices over N 6 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  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] 6 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

  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] 6 / 14 M. Blondin, C. Haase and F. Mazowiecki Affine Extensions of Integer VASS

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