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

Apr 30, 2023 •399 likes •1.58k views

Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives On the Complexity of VAS Reachability Sylvain Schmitz LSV, ENS Paris-Saclay & CNRS, Universit e Paris-Saclay INFINITY 2018 1/20 Vector Addition

  1. 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 (0,1) × 3 (6,0) 6/20

  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 � � 7/20

  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 � � 7/20

  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 � � 7/20

  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 � � 7/20

  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 � � 7/20

  7. 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 � � 7/20

  8. 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 � � 7/20

  9. 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] 8/20

  10. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Termination “Finally the checker has to verify that the process comes to an end. Here again he should be assisted by the programmer giving a further definite assertion to be verified. This may take the form of a quantity which is asserted to decrease continually and vanish when the machine stops. To the pure mathematician it is natural to give an ordinal number.” [Turing’49] 8/20

  11. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Termination of the Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92] ω ω 2 Ranking Function ∨ α 0 ∨ α 1 ∨ α 2 ∨ . . . 9/20

  12. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Termination of the Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92] ω ω 2 Ranking Function ∨ α 0 ∨ α 1 ∨ α 2 ∨ . . . 9/20

  13. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Termination of the Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92] ω ω 2 Ranking Function ∨ α 0 ∨ α 1 ∨ α 2 ∨ . . . 9/20

  14. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Termination of the Decomposition Algorithm [Mayr’81, Kosaraju’82, Lambert’92] ω ω 2 Ranking Function ∨ α 0 ∨ α 1 ∨ α 2 ∨ . . . 9/20

  15. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Demystifying Reachability in Vector Addition Systems [Leroux & S.’15; S.’17] Ideal Decomposition Theorem The Decomposition Algorithm computes the ideal decomposition of the set of runs from source to target. Upper Bound Theorem Reachability in vector addition systems is in quadratic Ackermann. 10/20

  16. 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 11/20 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 � ∗ � � � �

  17. 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 11/20 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 � ∗ � � � �

  18. 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 11/20 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 � ∗ � � � �

  19. 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 11/20 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 � ∗ � � � �

  20. 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 12/20

  21. 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 ) 12/20

  22. 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 ) 12/20

  23. 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 ) 12/20

  24. 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 ) . 12/20

  25. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Controlled Bad Sequences Proposition Over a wqo ( X , � ) , assuming { x ∈ X | � x � � n } to be finite ∀ n , ( g , n 0 ) -controlled bad sequences have a maximal length, noted L g , X ( n 0 ) . Objective Provide upper bounds for L g , X ( n 0 ) . 12/20

  26. 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 13/20

  27. 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 13/20

  28. 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 13/20

  29. 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 13/20

  30. 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 13/20

  31. 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 13/20

  32. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] ◮ Cantor Normal Form for ordinals α < ε 0 : α = ω α 1 · c 1 + ··· + ω α k · c k α > α 1 > ··· > α k 0 < c 1 ,..., c k < ω ◮ Norm of ordinals α < ε 0 : “maximal constant” def � α � = max 1 � i � k ( max ( || α i || , c i )) e.g. � ω ω 2 � = 2, � ω ω · 5 + ω 2 · 3 � = 5 14/20

  33. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] ◮ Cantor Normal Form for ordinals α < ε 0 : α = ω α 1 · c 1 + ··· + ω α k · c k α > α 1 > ··· > α k 0 < c 1 ,..., c k < ω ◮ Norm of ordinals α < ε 0 : “maximal constant” def � α � = max 1 � i � k ( max ( || α i || , c i )) e.g. � ω ω 2 � = 2, � ω ω · 5 + ω 2 · 3 � = 5 14/20

  34. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] Recall the descent equation: L g , α ( n 0 ) = max 1 + L g , α 0 ( g ( n 0 )) α 0 ∈ α , � α 0 � � n 0 Proposition ( variant of [Buchholtz, Cicho´ n & Weiermann’94]) Let 0 < α < ε 0 and � α � � n 0 . Then L g ,0 ( n 0 ) = 0 L g , α ( n 0 ) = 1 + L g , P n 0 ( α ) ( g ( n 0 )) P x ( α ) denotes the predecessor at x of α > 0: “maximal ordinal β < α s.t. � β � � x ” 14/20

  35. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] Recall the descent equation: L g , α ( n 0 ) = max 1 + L g , α 0 ( g ( n 0 )) α 0 ∈ α , � α 0 � � n 0 Proposition ( variant of [Buchholtz, Cicho´ n & Weiermann’94]) Let 0 < α < ε 0 and � α � � n 0 . Then L g ,0 ( n 0 ) = 0 L g , α ( n 0 ) = 1 + L g , P n 0 ( α ) ( g ( n 0 )) P x ( α ) denotes the predecessor at x of α > 0: “maximal ordinal β < α s.t. � β � � x ” 14/20

  36. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] Proposition ( variant of [Buchholtz, Cicho´ n & Weiermann’94]) Let 0 < α < ε 0 and � α � � n 0 . Then L g ,0 ( n 0 ) = 0 L g , α ( n 0 ) = 1 + L g , P n 0 ( α ) ( g ( n 0 )) P x ( α ) denotes the predecessor at x of α > 0: “maximal ordinal β < α s.t. � β � � x ” Example P 3 ( ω 2 ) = ω · 3 + 3 P 3 ( ω ω 2 ) = ω ω · 3 + 3 · 3 + ω ω · 3 + 2 · 3 + ω ω · 3 + 1 · 3 + ω ω · 3 · 3 + ω ω · 2 + 3 · 3 + ω ω · 2 + 2 · 3 + ω ω · 2 + 1 · 3 + ω ω · 2 · 3 + ω ω + 3 · 3 + ω ω + 2 · 3 + ω ω + 1 · 3 + ω ω · 3 + ω 3 · 3 + ω 2 · 3 + ω · 3 + 3 14/20

  37. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] Proposition ( variant of [Buchholtz, Cicho´ n & Weiermann’94]) Let 0 < α < ε 0 and � α � � n 0 . Then L g ,0 ( n 0 ) = 0 L g , α ( n 0 ) = 1 + L g , P n 0 ( α ) ( g ( n 0 )) P x ( α ) denotes the predecessor at x of α > 0: “maximal ordinal β < α s.t. � β � � x ” Example P 3 ( ω 2 ) = ω · 3 + 3 P 3 ( ω ω 2 ) = ω ω · 3 + 3 · 3 + ω ω · 3 + 2 · 3 + ω ω · 3 + 1 · 3 + ω ω · 3 · 3 + ω ω · 2 + 3 · 3 + ω ω · 2 + 2 · 3 + ω ω · 2 + 1 · 3 + ω ω · 2 · 3 + ω ω + 3 · 3 + ω ω + 2 · 3 + ω ω + 1 · 3 + ω ω · 3 + ω 3 · 3 + ω 2 · 3 + ω · 3 + 3 14/20

  38. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] Proposition ( variant of [Buchholtz, Cicho´ n & Weiermann’94]) Let 0 < α < ε 0 and � α � � n 0 . Then L g ,0 ( n 0 ) = 0 L g , α ( n 0 ) = 1 + L g , P n 0 ( α ) ( g ( n 0 )) This function was already known in the literature! Definition ( Cicho´ n Hierarchy [Cicho´ n & Tahhan Bittar’98] ) For g : N → N , define ( g α : N → N ) α by def def g 0 ( x ) = 0 g α ( x ) = 1 + g P x ( α ) ( g ( x )) for α > 0 14/20

  39. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Case of Ordinals [S.’14] Length Function Theorem (for Ordinals) Let α < ε 0 and n 0 � � α � . Then the longest ( g , n 0 ) -controlled descending sequence over α is of length L g , α ( n 0 ) = g α ( n 0 ) 14/20

  40. 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 g 0 ( x ) = 0 g α ( x ) = 1 + 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 15/20

  41. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Relating Norm and Length [Cicho´ n & Tahhan Bittar’98] def def g 0 ( x ) = 0 g α ( x ) = 1 + g P x ( α ) ( g ( x )) for α > 0 g 0 ( x ) def g α ( x ) = g P x ( α ) ( g ( x )) def = x for α > 0 norms � xi � g 3 ( n 0 ) x 3 norm: Hardy function gα ( n 0 ) x 2 g 2 ( n 0 ) g α ( x ) = g g α ( x ) ( x ) g 1 ( n 0 ) g α ( x ) � g α ( x ) + x g 0 ( n 0 ) x 0 x 1 indices i length: Cicho´ n function gα ( n 0 ) 15/20

  42. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Relating Norm and Length [Cicho´ n & Tahhan Bittar’98] def def g 0 ( x ) = 0 g α ( x ) = 1 + g P x ( α ) ( g ( x )) for α > 0 g 0 ( x ) def g α ( x ) = g P x ( α ) ( g ( x )) def = x for α > 0 norms � xi � g 3 ( n 0 ) x 3 norm: Hardy function gα ( n 0 ) x 2 g 2 ( n 0 ) g α ( x ) = g g α ( x ) ( x ) g 1 ( n 0 ) g α ( x ) � g α ( x ) + x g 0 ( n 0 ) x 0 x 1 indices i length: Cicho´ n function gα ( n 0 ) 15/20

  43. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Length of Decomposition Branches α 0 ∨ α 1 ∨ α 2 ∨ . . . 16/20

  44. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Length of Decomposition Branches α 0 ∨ α 1 ∨ α 2 ∨ . . . Corollary Let n 0 � 2 and g : N → N be such that the sequence of ordinal ranks computed by the decomposition algorithm is ( g , n 0 ) -controlled. The algorithm runs in SPACE ( g ω ω 2 ( n 0 )) . 16/20

  45. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Length of Decomposition Branches α 0 ∨ α 1 ∨ α 2 ∨ . . . Corollary Let n 0 � 2 and g : N → N be such that the sequence of ordinal ranks computed by the decomposition algorithm is ( g , n 0 ) -controlled. The algorithm runs in SPACE ( g ω ω 2 ( n 0 )) . 16/20

  46. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives The Length of Decomposition Branches α 0 ∨ α 1 ∨ α 2 ∨ . . . Consequence of ( Figueira, Figueira, S. & Schnoebelen’11) = H ω ω ( e ( x )) for H ( x ) def def The control g ( x ) = x + 1 and an elementary function e , and n 0 the size of the reachability instance fit. Thus the decomposition algorithm runs in SPACE (( H ω ω ◦ e ) ω ω 2 ( n ) . 16/20

  47. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result “ SPACE (( H ω ω ◦ e ) ω ω 2 ( n ) ” is unreadable! 1. give names ◮ H ω ω is the Ackermann function ◮ H ω ω 2 is the “quadratic Ackermann” function 2. define coarse-grained complexity classes DTIME ( H ω α ( f ( n )) def � FDTIME ( H γ ( n )) def � F <α = = F α γ<ω α f ∈ F <α Consequence of (S.’16, Thm. 4.4) VAS Reachability is in F ω 2 . 17/20

  48. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result “ SPACE (( H ω ω ◦ e ) ω ω 2 ( n ) ” is unreadable! 1. give names ◮ H ω ω is the Ackermann function ◮ H ω ω 2 is the “quadratic Ackermann” function 2. define coarse-grained complexity classes DTIME ( H ω α ( f ( n )) def � FDTIME ( H γ ( n )) def � F <α = = F α γ<ω α f ∈ F <α Consequence of (S.’16, Thm. 4.4) VAS Reachability is in F ω 2 . 17/20

  49. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result “ SPACE (( H ω ω ◦ e ) ω ω 2 ( n ) ” is unreadable! 1. give names ◮ H ω ω is the Ackermann function ◮ H ω ω 2 is the “quadratic Ackermann” function 2. define coarse-grained complexity classes DTIME ( H ω α ( f ( n )) def � FDTIME ( H γ ( n )) def � F <α = = F α γ<ω α f ∈ F <α Consequence of (S.’16, Thm. 4.4) VAS Reachability is in F ω 2 . 17/20

  50. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Restating the Result “ SPACE (( H ω ω ◦ e ) ω ω 2 ( n ) ” is unreadable! 1. give names ◮ H ω ω is the Ackermann function ◮ H ω ω 2 is the “quadratic Ackermann” function 2. define coarse-grained complexity classes DTIME ( H ω α ( f ( n )) def � FDTIME ( H γ ( n )) def � F <α = = F α γ<ω α f ∈ F <α Consequence of (S.’16, Thm. 4.4) VAS Reachability is in F ω 2 . 17/20

  51. 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 ExpSpace F ω 2 18/20

  52. 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 ExpSpace F ω 2 � def DTime ( tower ( e ( n ))) F 3 = e elementary 18/20

  53. 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 ExpSpace F ω 2 Examples of Tower-Complete Problems: ◮ satisfiability of first-order logic on words [Meyer’75] ◮ β -equivalence of simply typed λ terms [Statman’79] ◮ model-checking higher-order recursion schemes [Ong’06] 18/20

  54. 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 ExpSpace F ω 2 � def DTime ( ackermann ( p ( n ))) F ω = p primitive recursive 18/20

  55. 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 ExpSpace F ω 2 Examples of Ackermann-Complete Problems: ◮ reachability in lossy Minsky machines [Urquhart’98, Schnoebelen’02] ◮ satisfiability of safety Metric Temporal Logic [Lazi´ c et al.’16] ◮ satisfiability of Vertical XPath [Figueira and Segoufin’17] 18/20

  56. 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 ExpSpace F ω 2 18/20

  57. 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 ExpSpace F ω 2 18/20

  58. 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 2 19/20

  59. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Perspectives 1. complexity gap for VAS reachability ◮ ExpSpace -hard [Lipton’76] better lower bounds? (Wojciech’s talk) ◮ decomposition algorithm: at least F ω (Ackermannian) time [Zetzsche’16] 2. reachability in VAS extensions ◮ decidable in VAS with hierarchical zero tests [Reinhardt’08] ◮ what about (J´ erˆ ome’s talk) ◮ branching VAS ◮ unordered data Petri nets ◮ pushdown VAS 20/20

  60. Vector Addition Systems Decomposition Algorithm Upper Bounds Complexity Perspectives Perspectives 1. complexity gap for VAS reachability ◮ ExpSpace -hard [Lipton’76] better lower bounds? (Wojciech’s talk) ◮ decomposition algorithm: at least F ω (Ackermannian) time [Zetzsche’16] 2. reachability in VAS extensions ◮ decidable in VAS with hierarchical zero tests [Reinhardt’08] ◮ what about (J´ erˆ ome’s talk) ◮ branching VAS ◮ unordered data Petri nets ◮ pushdown VAS 20/20

  61. Demystifying Reachability in Vector Addition Systems [Leroux & S.’15] Ideal Decomposition Theorem The Decomposition Algorithm computes the ideal decomposition of the set of runs from source to target. Upper Bound Theorem Reachability in vector addition systems is in cubic Ackermann. 21/20

  62. Ideals of Well-Quasi-Orders ( X , � ) ◮ Canonical decompositions [Bonnet’75] if D ⊆ X is ↓ -closed, then D = I 1 ∪ ··· ∪ I n for (maximal) ideals I 1 ,..., I n Example (over N 2 ) D = ( { 0,...,2 } × N ) ∪ ( { 0,...,5 } × { 0,...,7 } ) ∪ ( N × { 0,...,4 } ) 22/20

  63. Ideals of Well-Quasi-Orders ( X , � ) ◮ Canonical decompositions [Bonnet’75] if D ⊆ X is ↓ -closed, then D = I 1 ∪ ··· ∪ I n for (maximal) ideals I 1 ,..., I n Example (over N 2 ) D = ( { 0,...,2 } × N ) ∪ ( { 0,...,5 } × { 0,...,7 } ) ∪ ( N × { 0,...,4 } ) 22/20

  64. Ideals of Well-Quasi-Orders ( X , � ) ◮ Canonical decompositions [Bonnet’75] if D ⊆ X is ↓ -closed, then D = I 1 ∪ ··· ∪ I n for (maximal) ideals I 1 ,..., I n ◮ E ff ective representations [Goubault-Larrecq et al.’17] Example (over N 2 ) D = � ( 2, ∞ ) � ∪ � ( 5,7 ) � ∪ � ( ∞ ,4 ) � 22/20

  65. Decomposition Theorem Well-Quasi-Order on Runs combination of Dickson’s and Higman’s lemmata Syntax Semantics I 0 I 2 I 3 I 1 I 4 � � ↓ Runs � � 23/20

  66. Decomposition Theorem Well-Quasi-Order on Runs combination of Dickson’s and Higman’s lemmata Syntax Semantics I 0 I 2 I 3 I 1 I 4 � � ↓ Runs � � 23/20

  67. Decomposition Theorem Well-Quasi-Order on Runs combination of Dickson’s and Higman’s lemmata Syntax Semantics I 0 I 2 I 3 I 1 I 4 � � ↓ Runs � � 23/20

  68. Decomposition Theorem Well-Quasi-Order on Runs combination of Dickson’s and Higman’s lemmata Syntax Semantics I 0 I 2 I 3 I 1 I 4 � � ↓ Runs � � 23/20

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