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

Dec 27, 2022 •247 likes •1.53k views

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

  1. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92] no can we build a “simple run”? yes � � , , , � decompose � � 6/24

  2. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92] no can we build a “simple run”? yes � � , , , � � decompose � � 6/24

  3. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92] no can we build a “simple run”? yes � � , , , � decompose � � 6/24

  4. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92] no can we build a “simple run”? yes � � , , , � decompose � � 6/24

  5. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92] no can we build a “simple run”? yes � � , , , � decompose � � 6/24

  6. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92] no can we build a “simple run”? yes � � , , , � decompose � � 6/24

  7. 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

  8. 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

  9. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Termination of the Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92, Leroux & S.’19] ( ω d in dim. d ) ω ω Ranking Function ∨ α 0 ∨ α 1 ∨ α 2 ∨ . . . 8/24

  10. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Termination of the Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92, Leroux & S.’19] ( ω d in dim. d ) ω ω Ranking Function ∨ α 0 ∨ α 1 ∨ α 2 ∨ . . . 8/24

  11. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Termination of the Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92, Leroux & S.’19] ( ω d in dim. d ) ω ω Ranking Function ∨ α 0 ∨ α 1 ∨ α 2 ∨ . . . 8/24

  12. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Termination of the Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92, Leroux & S.’19] ( ω d in dim. d ) ω ω Ranking Function ∨ α 0 ∨ α 1 ∨ α 2 ∨ . . . 8/24

  13. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Upper Bounds How to bound the running time of algorithms with ordinal-based termination proofs? Information and Computation 160 , 109 � 127 (2000) doi:10.1006 � inco.1999.2843, available online at http: �� www.idealibrary.com on /tcs /loca te r.com vie nce 256 (2001) 63–92 www.e lse re ! r Scie rywhe A l g l Com pute o r ve i t h ore tica s e m i The nsition syste m c A n a W e l y l l s i s d tra . Pdt Wilson, Q u o f e n ∗ a s P r ll-structure ri cation, ENS de Cachan & CNRS UMR 8643, 61 av e b e l i - o o g h n o r d e r a m h . S c r e s We , P d D w i n k e l Department of Computer Systems , Uppsala University , P . O . Box 325 , 751 05 Uppsala , Sweden o t h A . F i m a x, France Parosh Aziz Abdulla i n s 1 94235 Cachan Ce de ci cation and Ve Lab. Spe E-mail: parosh � docs.uu.se s f o r s e t m e t s y i s Institute of Mathematics and Computer Science , University of Latvia , Riga , Latvia t e - s t a t h a t n i - n i t a t s e Ka s o f e e n s S T S � rlis C 8 era c l a s b e w t h e W e n e r a l e r i n g o f t a l � ns a g - o r d t m e n t a s s i c ) a r e q u a s i e t e r a o f c l S T S s w e l - l e n s i v m p l e s v e d . E-mail: karlis � cclu.lv s ( W o f a n e x t e x a r s e e r r a c t s t e m e n c e d e a m a n y i g h s t A b s t o n s y e x s i t p r o v i l o w A l l r Department of Computer Systems , Uppsala University , P . O . Box 325 , 751 05 Uppsala , Sweden a n s t i i n t h e w e n s a l B . V . e d r t e l y o r t i c l e , n i t i o e n c e u c t u r t l s r h i s a d e d e - r S c i Bengt Jonsson e l l - s t r r e s u . I n t p r o v e s v i e W b l i i t y t i o n s u r i m 0 1 E l e c i d a t r a n s i s . O � c 2 0 i c h d h t e r e u s l t S s . w h w t i h n e w W S T a t i b e l v e r a l s o f E-mail: bengt � docs.uu.se c o m p w s e a n c e e r i n g d s h o s n i s t i - o r d a a n e e n a - q u a s i d e b e s W e l l m s t o a t i o n ; s y s t e e i r - c s ; V and y s t m e i t e s n I - n Department of Information Management , National Taiwan University , Taipei , Taiwan o d r s : K e y w Yih-Kuen Tsay h e o - o t h t o r b E-mail: tsay � im.ntu.edu.tw e l d f e s i n v e - v a n c i o n m s a c i t e a d o d u c t s y s t e v e r y s s i v m t h e I n t r a t e s a m p r e f r o Over the past few years increasing research effort has been directed 1 . t e - t s m s i c e i i o n s n n i s y s t e y s i n p i l c a t [ 1 6 ] n o f i a n d c i a l l c a p t e m s towards the automatic verification of infinite-state systems. This paper is a t o i m s e s p e a l i s t i e s y s e r i c o g r a e n s t , a l r e - n i t t e m s concerned with identifying general mathematical structures which 1 . V o f p r o p m e v e r h f o r r s y s 1 . i o n e v e l e i n s r o a c e d o f - c a t c a l d a s i b l a p p e o l p serve as sufficient conditions for achieving decidability. v e r i a r c t i d f e k i n g d e v r m a l n d p r o v e c h e c l b e s F o c h a g y p o d e l - d w e l y e a r decidability results for a class of systems s e a r h n o l o u l m c o u l c e n t 9/24 a l r e t e c c e s f s o g y n r e e which consist of a finite control r e t i c a t i o n y s u c h n o l o t e d i i s t v i e r i - c i g h l n t e c d e v o f p o m a l v h e h c a t o i b e e n a l t h The results assume f o r l d . T v e r i - h a s g w e w o r n g t h a t i s i n s t r i a l w o r k i w o r k s u r p r which d u a f a � ∗ � � � �

  14. 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 , 109 � 127 (2000) doi:10.1006 � inco.1999.2843, available online at http: �� www.idealibrary.com on /tcs /loca te r.com vie nce 256 (2001) 63–92 www.e lse re ! r Scie rywhe A l g l Com pute o r ve i t h ore tica s e m i The nsition syste m c A n a W e l y l l s i s d tra . Pdt Wilson, Q u o f e n ∗ a s P r ll-structure ri cation, ENS de Cachan & CNRS UMR 8643, 61 av e b e l i - o o g h n o r d e r a m h . S c r e s We , P d D w i n k e l Department of Computer Systems , Uppsala University , P . O . Box 325 , 751 05 Uppsala , Sweden o t h A . F i m a x, France Parosh Aziz Abdulla i n s 1 94235 Cachan Ce de ci cation and Ve Lab. Spe E-mail: parosh � docs.uu.se s f o r s e t m e t s y i s Institute of Mathematics and Computer Science , University of Latvia , Riga , Latvia t e - s t a t h a t n i - n i t a t s e Ka s o f e e n s S T S � rlis C 8 era c l a s b e w t h e W e n e r a l e r i n g o f t a l � ns a g - o r d t m e n t a s s i c ) a r e q u a s i e t e r a o f c l S T S s w e l - l e n s i v m p l e s v e d . E-mail: karlis � cclu.lv s ( W o f a n e x t e x a r s e e r r a c t s t e m e n c e d e a m a n y i g h s t A b s t o n s y e x s i t p r o v i l o w A l l r Department of Computer Systems , Uppsala University , P . O . Box 325 , 751 05 Uppsala , Sweden a n s t i i n t h e w e n s a l B . V . e d r t e l y o r t i c l e , n i t i o e n c e u c t u r t l s r h i s a d e d e - r S c i Bengt Jonsson e l l - s t r r e s u . I n t p r o v e s v i e W b l i i t y t i o n s u r i m 0 1 E l e c i d a t r a n s i s . O � c 2 0 i c h d h t e r e u s l t S s . w h w t i h n e w W S T a t i b e l v e r a l s o f E-mail: bengt � docs.uu.se c o m p w s e a n c e e r i n g d s h o s n i s t i - o r d a a n e e n a - q u a s i d e b e s W e l l m s t o a t i o n ; s y s t e e i r - c s ; V and y s t m e i t e s n I - n Department of Information Management , National Taiwan University , Taipei , Taiwan o d r s : K e y w Yih-Kuen Tsay h e o - o t h t o r b E-mail: tsay � im.ntu.edu.tw e l d f e s i n v e - v a n c i o n m s a c i t e a d o d u c t s y s t e v e r y s s i v m t h e I n t r a t e s a m p r e f r o Over the past few years increasing research effort has been directed 1 . t e - t s m s i c e i i o n s n n i s y s t e y s i n p i l c a t [ 1 6 ] n o f i a n d c i a l l c a p t e m s towards the automatic verification of infinite-state systems. This paper is a t o i m s e s p e a l i s t i e s y s e r i c o g r a e n s t , a l r e - n i t t e m s concerned with identifying general mathematical structures which 1 . V o f p r o p m e v e r h f o r r s y s 1 . i o n e v e l e i n s r o a c e d o f - c a t c a l d a s i b l a p p e o l p serve as sufficient conditions for achieving decidability. v e r i a r c t i d f e k i n g d e v r m a l n d p r o v e c h e c l b e s F o c h a g y p o d e l - d w e l y e a r decidability results for a class of systems s e a r h n o l o u l m c o u l c e n t 9/24 a l r e t e c c e s f s o g y n r e e which consist of a finite control r e t i c a t i o n y s u c h n o l o t e d i i s t v i e r i - c i g h l n t e c d e v o f p o m a l v h e h c a t o i b e e n a l t h The results assume f o r l d . T v e r i - h a s g w e w o r n g t h a t i s i n s t r i a l w o r k i w o r k s u r p r which d u a f a � ∗ � � � �

  15. 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 , 109 � 127 (2000) doi:10.1006 � inco.1999.2843, available online at http: �� www.idealibrary.com on /tcs /loca te r.com vie nce 256 (2001) 63–92 www.e lse re ! r Scie rywhe A l g l Com pute o r ve i t h ore tica s e m i The nsition syste m c A n a W e l y l l s i s d tra . Pdt Wilson, Q u o f e n ∗ a s P r ll-structure ri cation, ENS de Cachan & CNRS UMR 8643, 61 av e b e l i - o o g h n o r d e r a m h . S c r e s We , P d D w i n k e l Department of Computer Systems , Uppsala University , P . O . Box 325 , 751 05 Uppsala , Sweden o t h A . F i m a x, France Parosh Aziz Abdulla i n s 1 94235 Cachan Ce de ci cation and Ve Lab. Spe E-mail: parosh � docs.uu.se s f o r s e t m e t s y i s Institute of Mathematics and Computer Science , University of Latvia , Riga , Latvia t e - s t a t h a t i n - n i a t t s e Ka s o f e e n s S T S � rlis C 8 era c l a s b e t w h e W e n e r a l e r i n g o f t a l � ns a g - o r d t m e n t a s s i c ) a r e q u a s i e t e r a o f c l S T S s w e l - l e n s i v m p l e s v e d . E-mail: karlis � cclu.lv s ( W o f a n e x t e x a r s e e r r a c t s t e m e n c e d e a m a n y i g h s t A b s t o n s y e x s i t p r o v i l o w A l l r Department of Computer Systems , Uppsala University , P . O . Box 325 , 751 05 Uppsala , Sweden a n s t i i n t h e w e n s a l B . V . e d r t e l y o r t i c l e , n i t i o e n c e u c t u r t l s r h i s a d e d e - r S c i Bengt Jonsson e l l - s t r r e s u . I n t p r o v e s v i e W b l i i t y t i o n s u r i m 0 1 E l e c i d a t r a n s i s . O � c 2 0 i c h d h t e r e u s l t S s . w h w t i h n e w W S T a t i b e l v e r a l s o f E-mail: bengt � docs.uu.se c o m p w s e a n c e e r i n g d s h o s n i s t i - o r d a a n e e n a - q u a s i d e b e s W e l l m s t o a t i o n ; s y s t e e i r - c s ; V and y s t m e i t e s n I - n Department of Information Management , National Taiwan University , Taipei , Taiwan o d r s : K e y w Yih-Kuen Tsay h e o - o t h t o r b E-mail: tsay � im.ntu.edu.tw e l d f e s i n v e - v a n c i o n m s a c i t e a d o d u c t s y s t e v e r y s s i v m t h e I n t r a t e s a m p r e f r o Over the past few years increasing research effort has been directed 1 . t e - t s m s i c e i i o n s n n i s y s t e y s i n p i l c a t [ 1 6 ] n o f i a n d c i a l l c a p t e m s towards the automatic verification of infinite-state systems. This paper is a t o i m s e s p e a l i s t i e s y s e r i c o g r a e n s t , a l r e - n i t t e m s concerned with identifying general mathematical structures which 1 . V o f p r o p m e v e r h f o r r s y s 1 . i o n e v e l e i n s r o a c e d o f - c a t c a l d a s i b l a p p e o l p serve as sufficient conditions for achieving decidability. v e r i a r c t i d f e k i n g d e v r m a l n d p r o v e c h e c l b e s F o c h a g y p o d e - l d w e l y e a r decidability results for a class of systems s e a r h n o l o u l m c o u l c e n t 9/24 a l r e t e c c e s s f o g y n e r e which consist of a finite control r e t i c a t o i n y s u c h n o l o e t d i s i t i v e r i - c i g h l n t e c d e v o f p o m a l v h e h c a t i o b e e n a l t h The results assume f o r d l . T v e r i - h a s g w e w o r n g t h a t i s i n s t r i a l w o r k i w o r k s u r p r which d u a f a � ∗ � � � �

  16. 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 , 109 � 127 (2000) doi:10.1006 � inco.1999.2843, available online at http: �� www.idealibrary.com on /tcs /loca te r.com vie nce 256 (2001) 63–92 www.e lse re ! r Scie rywhe A l g l Com pute o r ve i t h ore tica s e m i The nsition syste m c A n a W e l y l l s i s d tra . Pdt Wilson, Q u o f e n ∗ a s P r ll-structure ri cation, ENS de Cachan & CNRS UMR 8643, 61 av e b e l i - o o g h n o r d e r a m h . S c r e s We , P d D w i n k e l Department of Computer Systems , Uppsala University , P . O . Box 325 , 751 05 Uppsala , Sweden o t h A . F i m a x, France Parosh Aziz Abdulla i n s 1 94235 Cachan Ce de ci cation and Ve Lab. Spe E-mail: parosh � docs.uu.se s f o r s e t m e t s y i s Institute of Mathematics and Computer Science , University of Latvia , Riga , Latvia t e - s t a t h a t i n - n i a t t s e Ka s o f e e n s S T S � rlis C 8 era c l a s b e t w h e W e n e r a l e r i n g o f t a l � ns a g - o r d t m e n t a s s i c ) a r e q u a s i e t e r a o f c l S T S s w e l - l e n s i v m p l e s v e d . E-mail: karlis � cclu.lv s ( W o f a n e x t e x a r s e e r r a c t s t e m e n c e d e a m a n y i g h s t A b s t o n s y e x s i t p r o v i l o w A l l r Department of Computer Systems , Uppsala University , P . O . Box 325 , 751 05 Uppsala , Sweden a n s t i i n t h e w e n s a l B . V . e d r t e l y o r t i c l e , n i t i o e n c e u c t u r t l s r h i s a d e d e - r S c i Bengt Jonsson e l l - s t r r e s u . I n t p r o v e s v i e W b l i i t y t i o n s u r i m 0 1 E l e c i d a t r a n s i s . O � c 2 0 i c h d h t e r e u s l t S s . w h w t i h n e w W S T a t i b e l v e r a l s o f E-mail: bengt � docs.uu.se c o m p w s e a n c e e r i n g d s h o s n i s t i - o r d a a n e e n a - q u a s i d e b e s W e l l m s t o a t i o n ; s y s t e e i r - c s ; V and y s t m e i t e s n I - n Department of Information Management , National Taiwan University , Taipei , Taiwan o d r s : K e y w Yih-Kuen Tsay h e o - o t h t o r b E-mail: tsay � im.ntu.edu.tw e l d f e s i n v e - v a n c i o n m s a c i t e a d o d u c t s y s t e v e r y s s i v m t h e I n t r a t e s a m p r e f r o Over the past few years increasing research effort has been directed 1 . t e - t s m s i c e i i o n s n n i s y s t e y s i n p i l c a t [ 1 6 ] n o f i a n d c i a l l c a p t e m s towards the automatic verification of infinite-state systems. This paper is a t o i m s e s p e a l i s t i e s y s e r i c o g r a e n s t , a l r e - n i t t e m s concerned with identifying general mathematical structures which 1 . V o f p r o p m e v e r h f o r r s y s 1 . i o n e v e l e i n s r o a c e d o f - c a t c a l d a s i b l a p p e o l p serve as sufficient conditions for achieving decidability. v e r i a r c t i d f e k i n g d e v r m a l n d p r o v e c h e c l b e s F o c h a g y p o d e - l d w e l y e a r decidability results for a class of systems s e a r h n o l o u l m c o u l c e n t 9/24 a l r e t e c c e s s f o g y n e r e which consist of a finite control r e t i c a t o i n y s u c h n o l o e t d i s i t i v e r i - c i g h l n t e c d e v o f p o m a l v h e h c a t i o b e e n a l t h The results assume f o r d l . T v e r i - h a s g w e w o r n g t h a t i s i n s t r i a l w o r k i w o r k s u r p r which d u a f a � ∗ � � � �

  17. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives A One-Player Game ◮ over Q � 0 × Q � 0 ( x 0, y 0 ) ◮ given initially ( x 0 , y 0 ) ( x 2, y 2 ) ◮ Eloise plays ( x j , y j ) s.t. ( x 1, y 1 ) ∀ 0 � i < j , x i > x j or y i > y j ◮ Can Eloise win, i.e. play indefinitely? ◮ If not, how long can she last? 10/24

  18. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives A One-Player Game ◮ over Q � 0 × Q � 0 ( x 0, y 0 ) ◮ given initially ( x 0 , y 0 ) ( x 2, y 2 ) ◮ Eloise plays ( x j , y j ) s.t. ( x 1, y 1 ) ∀ 0 � i < j , x i > x j or y i > y j ◮ Can Eloise win, i.e. play indefinitely? ◮ If not, how long can she last? 10/24

  19. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives A One-Player Game ◮ over Q � 0 × Q � 0 ( x 0, y 0 ) ◮ given initially ( x 0 , y 0 ) ( x 2, y 2 ) ◮ Eloise plays ( x j , y j ) s.t. ( x 1, y 1 ) ∀ 0 � i < j , x i > x j or y i > y j ◮ Can Eloise win, i.e. play indefinitely? ◮ If not, how long can she last? 10/24

  20. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives If ( x 0 , y 0 ) � ( 0,0 ) , then choosing ( x j , y j ) = ( x 0 2 j , y 0 2 j ) wins. 11/24

  21. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives A One-Player Game ◮ over N × N ( x 0, y 0 ) ◮ given initially ( x 0 , y 0 ) ( x 2, y 2 ) ◮ Eloise plays ( x j , y j ) s.t. ( x 1, y 1 ) ∀ 0 � i < j , x i > x j or y i > y j ◮ Can Eloise win, i.e. play indefinitely? ◮ If not, how long can she last? 12/24

  22. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Assume there exists an infinite sequence ( x j , y j ) j of moves over N 2 . 13/24

  23. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Assume there exists an infinite sequence ( x j , y j ) j of moves over N 2 . Consider the pairs of indices i < j : color ( i , j ) purple if x i > x j but y i � y j , red if x i > x j and y i > y j , orange if y i > y j but x i � x j . ... ( 3,4 ) ( 5,2 ) ( 2,3 ) 13/24

  24. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Assume there exists an infinite sequence ( x j , y j ) j of moves over N 2 . Consider the pairs of indices i < j : color ( i , j ) purple if x i > x j but y i � y j , red if x i > x j and y i > y j , orange if y i > y j but x i � x j . ... ( 3,4 ) ( 5,2 ) ( 2,3 ) By the infinite Ramsey Theorem, there exists an infinite monochromatic subset of indices. 13/24

  25. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Assume there exists an infinite sequence ( x j , y j ) j of moves over N 2 . Consider the pairs of indices i < j : color ( i , j ) purple if x i > x j but y i � y j , red if x i > x j and y i > y j , orange if y i > y j but x i � x j . ... ( 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

  26. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives A One-Player Game ◮ over N × N ( x 0, y 0 ) ◮ given initially ( x 0 , y 0 ) ( x 2, y 2 ) ◮ Eloise plays ( x j , y j ) s.t. ( x 1, y 1 ) ∀ 0 � i < j , x i > x j or y i > y j ◮ Can Eloise win, i.e. play indefinitely? ◮ If not, how long can she last? 14/24

  27. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo i ff all bad sequences are finite ◮ 15/24

  28. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo i ff all bad sequences are finite ◮ Example (over N 2 ) 15/24

  29. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo i ff all bad sequences are finite ◮ but can be of arbitrary length Example (over N 2 ) 16/24

  30. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo i ff all bad sequences are finite ◮ controlled by g : N → N monotone and inflationary and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] 16/24

  31. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo i ff all bad sequences are finite g 0 ( 2 ) = 2 ◮ controlled by g : N → N monotone and inflationary and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Example (over N 2 with n 0 = 2 and g ( n ) = n + 1 ) 16/24

  32. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo i ff all bad sequences are finite ◮ controlled by g : N → N monotone and inflationary and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Proposition Over ( X , � ) , assuming ∀ n { x ∈ X | � x � � n } finite, ( g , n 0 ) -controlled bad sequences have a maximal length, noted L g , X ( n 0 ) . 16/24

  33. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Descent Equation ( g , n 0 ) -controlled bad sequence x 0 , x 1 , x 2 , x 3 ,... over a wqo ( X , � ) : norms � xi � g 2 ( g ( n 0 )) = g 3 ( n 0 ) x 3 x 2 g 1 ( g ( n 0 )) = g 2 ( n 0 ) g 0 ( g ( n 0 )) = g 1 ( n 0 ) � � � � g 0 ( n 0 ) x 0 x 1 � � indices i 17/24

  34. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Descent Equation ( g , n 0 ) -controlled bad sequence x 0 , x 1 , x 2 , x 3 ,... over a wqo ( X , � ) : norms � xi � g 2 ( g ( n 0 )) = g 3 ( n 0 ) x 3 x 2 g 1 ( g ( n 0 )) = g 2 ( n 0 ) over the su ffi x x 1 , x 2 , x 3 ,..., ∀ i > 0, g 0 ( g ( n 0 )) = g 1 ( n 0 ) � � x 0 � x i � � g 0 ( n 0 ) x 0 x 1 � � indices i 17/24

  35. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Descent Equation ( g , n 0 ) -controlled bad sequence x 0 , x 1 , x 2 , x 3 ,... over a wqo ( X , � ) : norms � xi � g 2 ( g ( n 0 )) = g 3 ( n 0 ) x 3 x 2 g 1 ( g ( n 0 )) = g 2 ( n 0 ) over the su ffi x x 1 , x 2 , x 3 ,..., ∀ i > 0, g 0 ( g ( n 0 )) = g 1 ( n 0 ) � � � � def x i ∈ X \ ↑ x 0 = { x ∈ X | x 0 � x } g 0 ( n 0 ) x 0 x 1 � � indices i 17/24

  36. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Descent Equation ( g , n 0 ) -controlled bad sequence x 0 , x 1 , x 2 , x 3 ,... over a wqo ( X , � ) : norms � xi � g 2 ( g ( n 0 )) = g 3 ( n 0 ) x 3 x 2 g 1 ( g ( n 0 )) = g 2 ( n 0 ) over the su ffi x x 1 , x 2 , x 3 ,..., ∀ i > 0, g 0 ( g ( n 0 )) = g 1 ( n 0 ) � � � � def x i ∈ X \ ↑ x 0 = { x ∈ X | x 0 � x } g 0 ( n 0 ) x 0 � x i � � g i − 1 ( g ( n 0 )) x 1 � � indices i 17/24

  37. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Descent Equation ( g , n 0 ) -controlled bad sequence x 0 , x 1 , x 2 , x 3 ,... over a wqo ( X , � ) : norms � xi � g 2 ( g ( n 0 )) = g 3 ( n 0 ) x 3 x 2 g 1 ( g ( n 0 )) = g 2 ( n 0 ) over the su ffi x x 1 , x 2 , x 3 ,..., ∀ i > 0, g 0 ( g ( n 0 )) = g 1 ( n 0 ) � � � � def x i ∈ X \ ↑ x 0 = { x ∈ X | x 0 � x } g 0 ( n 0 ) x 0 � x i � � g i − 1 ( g ( n 0 )) x 1 � � indices i L g , X ( n 0 ) = max 1 + L g , X \ ↑ x 0 ( g ( n 0 )) x 0 ∈ X , � x 0 � � n 0 17/24

  38. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Descent Equation ( g , n 0 ) -controlled bad sequence α 0 , α 1 , α 2 , α 3 ,... over an ordinal α : norms � αi � g 2 ( g ( n 0 )) = g 3 ( n 0 ) α 3 α 2 g 1 ( g ( n 0 )) = g 2 ( n 0 ) over the su ffi x α 1 , α 2 , α 3 ,..., ∀ i > 0, g 0 ( g ( n 0 )) = g 1 ( n 0 ) � � � � def α i ∈ α 0 = { β ∈ α | β � � α 0 } g 0 ( n 0 ) α 0 � α i � � g i − 1 ( g ( n 0 )) α 1 � � indices i L g , α ( n 0 ) = max 1 + L g , α 0 ( g ( n 0 )) α 0 ∈ α , � α 0 � � n 0 17/24

  39. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] max 1 + L g , α 0 ( g ( n 0 )) L g , α ( n 0 ) = α 0 ∈ α , � α 0 � � n 0 18/24

  40. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] max 1 + L g , α 0 ( g ( n 0 )) L g , α ( n 0 ) = α 0 ∈ α , � α 0 � � n 0 18/24

  41. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] For a suitable norm function, there is a “maximising” ordinal P n 0 ( α ) : L g ,0 ( n 0 ) = 0 L g , α ( n 0 ) = 1 + L g , P n 0 ( α ) ( g ( n 0 )) These functions form the Cich´ on hierarchy. 18/24

  42. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] For a suitable norm function, there is a “maximising” ordinal P n 0 ( α ) : L g ,0 ( n 0 ) = 0 L g , α ( n 0 ) = 1 + L g , P n 0 ( α ) ( g ( n 0 )) These functions form the Cich´ on hierarchy. 18/24

  43. 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: def def L g ,0 ( x ) = 0 Lg , α ( x ) = 1 + L g , P x ( α ) ( g ( x )) for α > 0 Definition ( Hardy Hierarchy) For g : N → N , define ( g α : N → N ) α by g 0 ( x ) = g P x ( α ) ( g ( x )) for α > 0 def g α ( x ) def = x 19/24

  44. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Relating Norm and Length [Cicho´ n & Tahhan Bittar’98] norms � xi � g 3 ( n 0 ) x 3 norm: Hardy function gα ( n 0 ) x 2 g 2 ( n 0 ) g α ( x ) = g L g , α ( x ) ( x ) g 1 ( n 0 ) g α ( x ) � L g , α ( x ) + x g 0 ( n 0 ) x 0 x 1 indices i length: Cicho´ n function Lg , α ( n 0 ) 19/24

  45. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Relating Norm and Length [Cicho´ n & Tahhan Bittar’98] norms � xi � g 3 ( n 0 ) x 3 norm: Hardy function gα ( n 0 ) x 2 g 2 ( n 0 ) g α ( x ) = g L g , α ( x ) ( x ) g 1 ( n 0 ) g α ( x ) � L g , α ( x ) + x g 0 ( n 0 ) x 0 x 1 indices i length: Cicho´ n function Lg , α ( n 0 ) 19/24

  46. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Length of Decomposition Branches α 0 ∨ α 1 ∨ α 2 ∨ . . . 20/24

  47. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Length of Decomposition Branches α 0 ∨ α 1 ∨ α 2 ∨ . . . 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 ( g ω ω ( n )) , and SPACE ( g ω d ( n ))) in fixed dimension d . 20/24

  48. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Length of Decomposition Branches α 0 ∨ α 1 ∨ α 2 ∨ . . . 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 ( g ω ω ( n )) , and SPACE ( g ω d ( n ))) in fixed dimension d . 20/24

  49. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result g ω d ( n ) � � “ SPACE ” is unreadable! 21/24

  50. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result def Hardy hierarchy with base function H ( x ) = x + 1: H 0 ( x ) = x k times � �������� �� �������� � H k ( x ) = H ◦ ··· ◦ H ( x ) = x + k x + 1 times � �������� �� �������� � H ω ( x ) = H x + 1 ( x ) = H ◦ ··· ◦ H ( x ) = 2 x + 1 x + 1 times � �������������� �� �������������� � H ω 2 ( x ) = H ω · ( x + 1 ) = H ω ◦ ··· ◦ H ω ( x ) ≈ 2 x x + 1 times � ���������������� �� ���������������� � H ω 2 ◦ ··· ◦ H ω 2 ( x ) H ω 3 ( x ) = H ω 2 · ( x + 1 ) = ≈ tower ( x ) . . . H ω ω ( x ) = H ω x + 1 ( x ) ≈ ack ( x ) 21/24

  51. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result def Hardy hierarchy with base function H ( x ) = x + 1: H 0 ( x ) = x k times � �������� �� �������� � H k ( x ) = H ◦ ··· ◦ H ( x ) = x + k x + 1 times � �������� �� �������� � H ω ( x ) = H x + 1 ( x ) = H ◦ ··· ◦ H ( x ) = 2 x + 1 x + 1 times � �������������� �� �������������� � H ω 2 ( x ) = H ω · ( x + 1 ) = H ω ◦ ··· ◦ H ω ( x ) ≈ 2 x x + 1 times � ���������������� �� ���������������� � H ω 2 ◦ ··· ◦ H ω 2 ( x ) H ω 3 ( x ) = H ω 2 · ( x + 1 ) = ≈ tower ( x ) . . . H ω ω ( x ) = H ω x + 1 ( x ) ≈ ack ( x ) 21/24

  52. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result def Hardy hierarchy with base function H ( x ) = x + 1: H 0 ( x ) = x k times � �������� �� �������� � H k ( x ) = H ◦ ··· ◦ H ( x ) = x + k x + 1 times � �������� �� �������� � H ω ( x ) = H x + 1 ( x ) = H ◦ ··· ◦ H ( x ) = 2 x + 1 x + 1 times � �������������� �� �������������� � H ω 2 ( x ) = H ω · ( x + 1 ) = H ω ◦ ··· ◦ H ω ( x ) ≈ 2 x x + 1 times � ���������������� �� ���������������� � H ω 2 ◦ ··· ◦ H ω 2 ( x ) H ω 3 ( x ) = H ω 2 · ( x + 1 ) = ≈ tower ( x ) . . . H ω ω ( x ) = H ω x + 1 ( x ) ≈ ack ( x ) 21/24

  53. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result def Hardy hierarchy with base function H ( x ) = x + 1: H 0 ( x ) = x k times � �������� �� �������� � H k ( x ) = H ◦ ··· ◦ H ( x ) = x + k x + 1 times � �������� �� �������� � H ω ( x ) = H x + 1 ( x ) = H ◦ ··· ◦ H ( x ) = 2 x + 1 x + 1 times � �������������� �� �������������� � H ω 2 ( x ) = H ω · ( x + 1 ) = H ω ◦ ··· ◦ H ω ( x ) ≈ 2 x x + 1 times � ���������������� �� ���������������� � H ω 2 ◦ ··· ◦ H ω 2 ( x ) H ω 3 ( x ) = H ω 2 · ( x + 1 ) = ≈ tower ( x ) . . . H ω ω ( x ) = H ω x + 1 ( x ) ≈ ack ( x ) 21/24

  54. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result def Hardy hierarchy with base function H ( x ) = x + 1: H 0 ( x ) = x k times � �������� �� �������� � H k ( x ) = H ◦ ··· ◦ H ( x ) = x + k x + 1 times � �������� �� �������� � H ω ( x ) = H x + 1 ( x ) = H ◦ ··· ◦ H ( x ) = 2 x + 1 x + 1 times � �������������� �� �������������� � H ω 2 ( x ) = H ω · ( x + 1 ) = H ω ◦ ··· ◦ H ω ( x ) ≈ 2 x x + 1 times � ���������������� �� ���������������� � H ω 2 ◦ ··· ◦ H ω 2 ( x ) H ω 3 ( x ) = H ω 2 · ( x + 1 ) = ≈ tower ( x ) . . . H ω ω ( x ) = H ω x + 1 ( x ) ≈ ack ( x ) 21/24

  55. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result def Hardy hierarchy with base function H ( x ) = x + 1: H 0 ( x ) = x k times � �������� �� �������� � H k ( x ) = H ◦ ··· ◦ H ( x ) = x + k x + 1 times � �������� �� �������� � H ω ( x ) = H x + 1 ( x ) = H ◦ ··· ◦ H ( x ) = 2 x + 1 x + 1 times � �������������� �� �������������� � H ω 2 ( x ) = H ω · ( x + 1 ) = H ω ◦ ··· ◦ H ω ( x ) ≈ 2 x x + 1 times � ���������������� �� ���������������� � H ω 2 ◦ ··· ◦ H ω 2 ( x ) H ω 3 ( x ) = H ω 2 · ( x + 1 ) = ≈ tower ( x ) . . . H ω ω ( x ) = H ω x + 1 ( x ) ≈ ack ( x ) 21/24

  56. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result Define coarse-grained classes: def � FDTIME ( H β ( n )) F <α = β<ω α DTIME ( H ω α ( f ( n ))) � def = F α f ∈ F <α Consequence of (S.’16, Thm. 4.4) VAS Reachability is in F ω , and in F d + 3 in fixed dimension d . 21/24

  57. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result Define coarse-grained classes: def � FDTIME ( H β ( n )) F <α = β<ω α DTIME ( H ω α ( f ( n ))) � def = F α f ∈ F <α Consequence of (S.’16, Thm. 4.4) VAS Reachability is in F ω , and in F d + 3 in fixed dimension d . 21/24

  58. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Complexity Classes Beyond Elementary [S.’16] Multiply Recursive Fast-Growing Complexity F ω Primitive Recursive = Ackermann F 3 = Tower Elementary 22/24

  59. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Complexity Classes Beyond Elementary [S.’16] Multiply Recursive Fast-Growing Complexity F ω Primitive Recursive = Ackermann F 3 = Tower Elementary � def DTime ( tower ( e ( n ))) F 3 = e elementary 22/24

  60. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Complexity Classes Beyond Elementary [S.’16] Multiply Recursive Fast-Growing Complexity F ω Primitive Recursive = Ackermann F 3 = Tower Elementary 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

  61. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Complexity Classes Beyond Elementary [S.’16] Multiply Recursive Fast-Growing Complexity F ω Primitive Recursive = Ackermann F 3 = Tower Elementary � def DTime ( ack ( p ( n ))) F ω = p primitive recursive 22/24

  62. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Complexity Classes Beyond Elementary [S.’16] Multiply Recursive Fast-Growing Complexity F ω Primitive Recursive = Ackermann F 3 = Tower Elementary 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

  63. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Complexity Classes Beyond Elementary [S.’16] Multiply Recursive Fast-Growing Complexity F ω Primitive Recursive = Ackermann F 3 = Tower Elementary 22/24

  64. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Complexity Classes Beyond Elementary [S.’16] Multiply Recursive Fast-Growing Complexity F ω Primitive Recursive = Ackermann F 3 = Tower Elementary 22/24

  65. 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: ( F α ) α this talk: focus on one problem ◮ reachability in vector addition systems in F ω 23/24

  66. 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 F ω (Ackermannian) time, because downward language inclusion is F ω -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

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