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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem The Parametric Complexity of Lossy Counter Machines Sylvain Schmitz ICALP , July 12, 2019, Patras 1/15 Lossy Counter Machines Controlled Sequences

  1. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Backward Coverability (Arnold & Latteux’78) q 1 c 1 −− c 1 ++ Example: coverability of q 2 ( 1,1 ) in q 3 q 2 c 2 ++ = { q ( v ) | ∃ v ′ � ( 1,1 ) . q ( v ) → � 4 def ? q 2 ( v ′ ) } c 2 = 0 U 4 ℓ U 0 U 1 q 1 U 2 ↑ q 2 ( 1,1 ) ℓ U 3 ↑ q 1 ( 0,1 ) ℓ ↑ q 3 ( 0,0 ) U 4 ℓ ↑ q 1 ( 1,0 ) q 2 ℓ ↑ q 2 ( 1,0 ) q 3 6/15

  2. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Backward Coverability (Arnold & Latteux’78) q 1 c 1 −− c 1 ++ Example: coverability of q 2 ( 1,1 ) in q 3 q 2 c 2 ++ = { q ( v ) | ∃ v ′ � ( 1,1 ) . q ( v ) → � 4 def ? q 2 ( v ′ ) } c 2 = 0 U 4 ℓ U 0 q 3 ( 1,0 ) U 1 q 1 U 2 ↑ q 2 ( 1,1 ) ℓ U 3 ↑ q 1 ( 0,1 ) ℓ ↑ q 3 ( 0,0 ) U 4 ℓ ℓ ↑ q 1 ( 1,0 ) q 2 ℓ ↑ q 2 ( 1,0 ) q 3 6/15

  3. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Backward Coverability (Arnold & Latteux’78) q 1 c 1 −− c 1 ++ Example: coverability of q 2 ( 1,1 ) in q 3 q 2 c 2 ++ = { q ( v ) | ∃ v ′ � ( 1,1 ) . q ( v ) → � 5 def ? q 2 ( v ′ ) } c 2 = 0 U 5 ℓ U 0 q 3 ( 1,0 ) U 1 q 1 U 2 ↑ q 2 ( 1,1 ) ℓ U 3 ↑ q 1 ( 0,1 ) ℓ ↑ q 3 ( 0,0 ) U 4 ℓ ℓ U 5 ↑ q 1 ( 1,0 ) q 2 ℓ ↑ q 2 ( 1,0 ) ℓ ↑ q 1 ( 0,0 ) q 3 6/15

  4. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Backward Coverability (Arnold & Latteux’78) q 1 c 1 −− c 1 ++ Example: coverability of q 2 ( 1,1 ) in q 3 q 2 c 2 ++ = { q ( v ) | ∃ v ′ � ( 1,1 ) . q ( v ) → � 6 def ? q 2 ( v ′ ) } c 2 = 0 U 6 ℓ U 0 q 3 ( 1,0 ) U 1 q 1 U 2 ↑ q 2 ( 1,1 ) ℓ U 3 ↑ q 1 ( 0,1 ) ℓ ↑ q 3 ( 0,0 ) U 4 ℓ ℓ U 5 ↑ q 1 ( 1,0 ) q 2 ℓ U 6 ↑ q 2 ( 1,0 ) ℓ ↑ q 1 ( 0,0 ) ℓ ↑ q 2 ( 0,0 ) q 3 6/15

  5. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Backward Coverability (Arnold & Latteux’78) q 1 c 1 −− c 1 ++ Example: coverability of q 2 ( 1,1 ) in q 3 q 2 c 2 ++ def = { q ( v ) | ∃ v ′ � ( 1,1 ) . q ( v ) → ∗ ? ℓ q 2 ( v ′ ) } c 2 = 0 U 6 U 0 q 3 ( 1,0 ) U 1 q 1 U 2 ↑ q 2 ( 1,1 ) ℓ U 3 ↑ q 1 ( 0,1 ) ℓ ↑ q 3 ( 0,0 ) U 4 ℓ ℓ U 5 ↑ q 1 ( 1,0 ) q 2 ℓ U 6 ↑ q 2 ( 1,0 ) ℓ ↑ q 1 ( 0,0 ) ℓ ℓ ↑ q 2 ( 0,0 ) q 3 6/15

  6. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Backward Coverability (Arnold & Latteux’78) q 1 c 1 −− c 1 ++ Example: coverability of q 2 ( 1,1 ) in q 3 q 2 c 2 ++ def = { q ( v ) | ∃ v ′ � ( 1,1 ) . q ( v ) → ∗ ? ℓ q 2 ( v ′ ) } c 2 = 0 U 6 U 0 U 1 q 1 U 2 ↑ q 2 ( 1,1 ) ℓ U 3 ↑ q 1 ( 0,1 ) ℓ ↑ q 3 ( 0,0 ) U 4 ℓ U 5 ↑ q 1 ( 1,0 ) q 2 ℓ U 6 ↑ q 2 ( 1,0 ) ℓ ↑ q 1 ( 0,0 ) ℓ ↑ q 2 ( 0,0 ) q 3 The sequence q 2 ( 1,1 ) , q 1 ( 0,1 ) , q 3 ( 0,0 ) , q 1 ( 1,0 ) , q 2 ( 1,0 ) , q 1 ( 0,0 ) , q 2 ( 0,0 ) is bad 6/15

  7. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite 7/15

  8. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) 7/15

  9. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) ( 0,2 ) 7/15

  10. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) ( 0,2 ) , ( 2,1 ) 7/15

  11. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) 7/15

  12. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 6,0 ) 7/15

  13. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 6,0 ) , ( 5,0 ) 7/15

  14. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 6,0 ) , ( 5,0 ) , ( 4,0 ) 7/15

  15. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 6,0 ) , ( 5,0 ) , ( 4,0 ) , ( 3,0 ) 7/15

  16. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 6,0 ) , ( 5,0 ) , ( 4,0 ) , ( 3,0 ) , ( 2,0 ) 7/15

  17. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 6,0 ) , ( 5,0 ) , ( 4,0 ) , ( 3,0 ) , ( 2,0 ) , ( 1,0 ) 7/15

  18. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Bad Sequences Over a qo ( X , � ) ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ ( X , � ) wqo if all bad sequences are finite... ... but can be of arbitrary length Example (in N 2 ) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 6,0 ) , ( 5,0 ) , ( 4,0 ) , ( 3,0 ) , ( 2,0 ) , ( 1,0 ) , ( 0,0 ) 7/15

  19. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] 8/15

  20. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) g 0 ( 2 ) = 2 [Cicho´ n & Tahhan Bittar’98] Example (in N 2 with n 0 = 2 and g ( n ) = n + 1) ( 0,2 ) 8/15

  21. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if g 1 ( 2 ) = 3 ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Example (in N 2 with n 0 = 2 and g ( n ) = n + 1) ( 0,2 ) , ( 2,1 ) 8/15

  22. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled g 2 ( 2 ) = 4 by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Example (in N 2 with n 0 = 2 and g ( n ) = n + 1) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) 8/15

  23. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j g 3 ( 2 ) = 5 ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Example (in N 2 with n 0 = 2 and g ( n ) = n + 1) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 5,0 ) 8/15

  24. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j g 4 ( 2 ) = 6 ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Example (in N 2 with n 0 = 2 and g ( n ) = n + 1) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 5,0 ) , ( 4,0 ) 8/15

  25. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j g 5 ( 2 ) = 7 ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Example (in N 2 with n 0 = 2 and g ( n ) = n + 1) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 5,0 ) , ( 4,0 ) , ( 3,0 ) 8/15

  26. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � g 6 ( 2 ) = 8 ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Example (in N 2 with n 0 = 2 and g ( n ) = n + 1) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 5,0 ) , ( 4,0 ) , ( 3,0 ) , ( 2,0 ) 8/15

  27. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � g 7 ( 2 ) = 9 ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Example (in N 2 with n 0 = 2 and g ( n ) = n + 1) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 5,0 ) , ( 4,0 ) , ( 3,0 ) , ( 2,0 ) , ( 1,0 ) 8/15

  28. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences g 8 ( 2 ) = 10 Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Example (in N 2 with n 0 = 2 and g ( n ) = n + 1) ( 0,2 ) , ( 2,1 ) , ( 0,1 ) , ( 5,0 ) , ( 4,0 ) , ( 3,0 ) , ( 2,0 ) , ( 1,0 ) , ( 0,0 ) 8/15

  29. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Proposition In a wqo ( X , � ) , if ∀ n { x ∈ X | � x � � n } is finite, then amortised ( g , n 0 ) -controlled bad sequences have a maximal length, denoted L a g , X ( n 0 ) . 8/15

  30. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] Proposition In a wqo ( X , � ) , if ∀ n { x ∈ X | � x � � n } is finite, then amortised ( g , n 0 ) -controlled bad sequences have a maximal length, denoted L a g , X ( n 0 ) . Theorem ( S. & Schnoebelen’12) def def For LCM Reachability, g ( x ) = x + 1 and n 0 = � v f � fit, and L a g , Q × N d ( n 0 ) ≈ F d + 1 ( n ) . 8/15

  31. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] ◮ x 0 , x 1 ,... is strongly controlled by g : N → N and n 0 ∈ N if � x 0 � � n 0 and ∀ i . � x i + 1 � � g ( � x i � ) 8/15

  32. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] ◮ x 0 , x 1 ,... is strongly controlled by g : N → N and n 0 ∈ N if � x 0 � � n 0 and ∀ i . � x i + 1 � � g ( � x i � ) Corollary In a wqo ( X , � ) , if ∀ n { x ∈ X | � x � � n } is finite, then strongly ( g , n 0 ) -controlled bad sequences have a maximal length, denoted L s g , X ( n 0 ) . 8/15

  33. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) [Cicho´ n & Tahhan Bittar’98] ◮ x 0 , x 1 ,... is strongly controlled by g : N → N and n 0 ∈ N if � x 0 � � n 0 and ∀ i . � x i + 1 � � g ( � x i � ) Corollary In a wqo ( X , � ) , if ∀ n { x ∈ X | � x � � n } is finite, then strongly ( g , n 0 ) -controlled bad sequences have a maximal length, denoted L s g , X ( n 0 ) and a maximal norm, denoted N s g , X ( n 0 ) . 8/15

  34. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Bad Sequences Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is bad if ∀ i < j . x i � x j ◮ x 0 , x 1 ,... is strongly controlled by g : N → N and n 0 ∈ N if � x 0 � � n 0 and ∀ i . � x i + 1 � � g ( � x i � ) Corollary In a wqo ( X , � ) , if ∀ n { x ∈ X | � x � � n } is finite, then ( g , n 0 ) -controlled bad sequences have a maximal length, denoted L s g , X ( n 0 ) and a maximal norm, denoted N s g , X ( n 0 ) . Theorem def def For LCM Reachability, g ( x ) = x + 1 and n 0 = � v f � fit, and L s g , Q × N d ( n 0 ) ≈ F d ( n ) . 8/15

  35. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Antichains Over a qo ( X , � ) ◮ x 0 , x 1 ,... is an antichain if ∀ i < j . x i ⊥ x j (i.e. xi � xj and xj � xi ) 9/15

  36. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Antichains Over a qo ( X , � ) ◮ x 0 , x 1 ,... is an antichain if ∀ i < j . x i ⊥ x j (i.e. xi � xj and xj � xi ) ◮ ( X , � ) wqo i ff it is well-founded and all antichains are finite 9/15

  37. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Antichains Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is an antichain if ∀ i < j . x i ⊥ x j (i.e. xi � xj and xj � xi ) ◮ ( X , � ) wqo i ff it is well-founded and all antichains are finite ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) 9/15

  38. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Controlled Antichains Over a qo ( X , � ) with norm � · � ◮ x 0 , x 1 ,... is an antichain if ∀ i < j . x i ⊥ x j (i.e. xi � xj and xj � xi ) ◮ ( X , � ) wqo i ff it is well-founded and all antichains are finite ◮ x 0 , x 1 ,... is amortised controlled by g : N → N and n 0 ∈ N if ∀ i . � x i � � g i ( n 0 ) Corollary In a wqo ( X , � ) , if ∀ n { x ∈ X | � x � � n } is finite, then amortised ( g , n 0 ) -controlled antichains have a maximal length, denoted W a g , X ( n 0 ) . 9/15

  39. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) 10/15

  40. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) ( 3,4 ) 10/15

  41. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) ( 3,4 ) ( 5,2 ) 10/15

  42. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) ( 3,4 ) ( 5,2 ) ( 4,3 ) 10/15

  43. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) ( 3,4 ) > ( 5,2 ) ( 4,2 ) ( 4,3 ) 10/15

  44. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) ( 3,4 ) > ( 5,2 ) ( 4,2 ) ( 4,3 ) ( 5,1 ) 10/15

  45. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > ( 3,4 ) ( 2,3 ) > ( 5,2 ) ( 4,2 ) ( 4,3 ) ( 5,1 ) 10/15

  46. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > ( 3,4 ) ( 2,3 ) > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 4,3 ) ( 5,1 ) 10/15

  47. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > ( 3,4 ) ( 2,3 ) > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 4,3 ) ( 5,1 ) ( 5,0 ) 10/15

  48. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > ( 3,4 ) ( 2,3 ) > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 4,3 ) ( 5,1 ) ( 5,0 ) ( 1,4 ) 10/15

  49. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > ( 3,4 ) ( 2,3 ) > > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 3,1 ) ( 4,3 ) ( 5,1 ) ( 5,0 ) ( 1,4 ) 10/15

  50. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > ( 3,4 ) ( 2,3 ) > > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 3,1 ) ( 4,3 ) ( 5,1 ) ( 5,0 ) ( 0,4 ) ( 1,4 ) 10/15

  51. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > ( 3,4 ) ( 2,3 ) > > > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 3,1 ) ( 3,0 ) ( 4,3 ) ( 5,1 ) ( 5,0 ) ( 0,4 ) ( 1,4 ) 10/15

  52. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > > ( 3,4 ) ( 2,3 ) ( 1,1 ) > > > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 3,1 ) ( 3,0 ) ( 4,3 ) ( 5,1 ) ( 5,0 ) ( 0,4 ) ( 1,4 ) 10/15

  53. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > > ( 3,4 ) ( 2,3 ) ( 1,1 ) > > > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 3,1 ) ( 3,0 ) ( 4,3 ) ( 5,1 ) ( 5,0 ) ( 0,4 ) ( 1,4 ) Property Every branch is a strongly ( x �→ x + 1,4 ) -controlled antichain over N 2 . 10/15

  54. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > > ( 3,4 ) ( 2,3 ) ( 1,1 ) > > > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 3,1 ) ( 3,0 ) ( 4,3 ) ( 5,1 ) ( 5,0 ) ( 0,4 ) ( 1,4 ) Here, height is 3, hence ◮ maximal norm at most g 3 ( n 0 ) = n 0 + 3 = 7, ◮ length of the bad sequence at most ( 7 + 1 ) 2 = 64 10/15

  55. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > > ( 3,4 ) ( 2,3 ) ( 1,1 ) > > > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 3,1 ) ( 3,0 ) ( 4,3 ) ( 5,1 ) ( 5,0 ) ( 0,4 ) ( 1,4 ) Here, height is 3, hence ◮ maximal norm at most g 3 ( n 0 ) = n 0 + 3 = 7, ◮ length of the bad sequence at most ( 7 + 1 ) 2 = 64 10/15

  56. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Antichain Factorisation Example (strongly ( x �→ x + 1,4 ) -controlled bad sequence over N 2 ) ( 3,4 ) , ( 5,2 ) , ( 4,3 ) , ( 4,2 ) , ( 5,1 ) , ( 2,3 ) , ( 4,1 ) , ( 5,0 ) , ( 1,4 ) , ( 3,1 ) , ( 0,4 ) , ( 3,0 ) , ( 1,1 ) > > ( 3,4 ) ( 2,3 ) ( 1,1 ) > > > ( 5,2 ) ( 4,2 ) ( 4,1 ) ( 3,1 ) ( 3,0 ) ( 4,3 ) ( 5,1 ) ( 5,0 ) ( 0,4 ) ( 1,4 ) Here, height is 3, hence ◮ maximal norm at most g 3 ( n 0 ) = n 0 + 3 = 7, ◮ length of the bad sequence at most ( 7 + 1 ) 2 = 64 10/15

  57. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Factorisation Lemma Lemma Let ( X , � ) be a wqo with norm � . � : X → N monotone. Then g , X ( n 0 ) � g W s g , X ( n 0 ) ( n 0 ) . N s 11/15

  58. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Factorisation Lemma Lemma Let ( X , � ) be a wqo with norm � . � : X → N monotone. Then g , X ( n 0 ) � g W s g , X ( n 0 ) ( n 0 ) . N s 11/15

  59. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Width Function Theorem Informal statement W a g , Q × N d ( n 0 ) ≈ F d ( n ) in the case of LCMs. Proof ingredients 1. descent equation 2. normed reflections 3. subrecursive hierarchies 12/15

  60. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Width Function Theorem Informal statement W a g , Q × N d ( n 0 ) ≈ F d ( n ) in the case of LCMs. Proof ingredients 1. descent equation 2. normed reflections 3. subrecursive hierarchies 12/15

  61. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Width Function Theorem Informal statement W a g , Q × N d ( n 0 ) ≈ F d ( n ) in the case of LCMs. Proof ingredients 1. descent equation 2. normed reflections 3. subrecursive hierarchies 12/15

  62. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 1: Descent Equation amortised ( g , n 0 ) -controlled antichain 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 13/15

  63. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 1: Descent Equation amortised ( g , n 0 ) -controlled antichain 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 13/15

  64. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 1: Descent Equation amortised ( g , n 0 ) -controlled antichain 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 13/15

  65. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 1: Descent Equation amortised ( g , n 0 ) -controlled antichain 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 13/15

  66. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 1: Descent Equation amortised ( g , n 0 ) -controlled antichain 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 W a 1 + W a g , X ( n 0 ) = max g , X ⊥ x 0 ( g ( n 0 )) x 0 ∈ X , � x 0 � � n 0 13/15

  67. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 2: Normed Reflections r X Y Definition Let ( X , � X ) and ( Y , � Y ) be qos with norms � . � X and � . � Y . A normed reflection is a function r : X → Y such that 1. ∀ x , x ′ ∈ X . x � X x ′ implies r ( x ) � Y r ( x ′ ) 2. ∀ x ∈ X . � r ( x ) � Y � � x � X 14/15

  68. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 2: Normed Reflections r X Y r r ( x ′ ) x ′ � X r x r ( x ) Definition Let ( X , � X ) and ( Y , � Y ) be qos with norms � . � X and � . � Y . A normed reflection is a function r : X → Y such that 1. ∀ x , x ′ ∈ X . x � X x ′ implies r ( x ) � Y r ( x ′ ) 2. ∀ x ∈ X . � r ( x ) � Y � � x � X 14/15

  69. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 2: Normed Reflections r X Y r r ( x ′ ) x ′ � X � Y r x r ( x ) Definition Let ( X , � X ) and ( Y , � Y ) be qos with norms � . � X and � . � Y . A normed reflection is a function r : X → Y such that 1. ∀ x , x ′ ∈ X . x � X x ′ implies r ( x ) � Y r ( x ′ ) 2. ∀ x ∈ X . � r ( x ) � Y � � x � X 14/15

  70. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 2: Normed Reflections r X Y r r ( x ′ ) x ′ � X � Y r x r ( x ) � x � X � r ( x ) � Y Definition Let ( X , � X ) and ( Y , � Y ) be qos with norms � . � X and � . � Y . A normed reflection is a function r : X → Y such that 1. ∀ x , x ′ ∈ X . x � X x ′ implies r ( x ) � Y r ( x ′ ) 2. ∀ x ∈ X . � r ( x ) � Y � � x � X 14/15

  71. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 2: Normed Reflections r X Y r r ( x ′ ) x ′ � X � Y r x r ( x ) � x � X � r ( x ) � Y � Definition Let ( X , � X ) and ( Y , � Y ) be qos with norms � . � X and � . � Y . A normed reflection is a function r : X → Y such that 1. ∀ x , x ′ ∈ X . x � X x ′ implies r ( x ) � Y r ( x ′ ) 2. ∀ x ∈ X . � r ( x ) � Y � � x � X 14/15

  72. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 2: Normed Reflections Definition Let ( X , � X ) and ( Y , � Y ) be qos with norms � . � X and � . � Y . A normed reflection is a function r : X → Y such that 1. ∀ x , x ′ ∈ X . x � X x ′ implies r ( x ) � Y r ( x ′ ) 2. ∀ x ∈ X . � r ( x ) � Y � � x � X Fact If r : X → Y is a normed reflection, then for all g , n 0 W a g , X ( n 0 ) � W a g , Y ( n 0 ) . (if x 0 , x 1 ,... is an amortised ( g , n 0 ) -controlled antichain, then r ( x 0 ) , r ( x 1 ) ,... is as well) ◮ over-approximate W a g , X ⊥ x 0 ( g ( n 0 )) in the descent equation ◮ e.g. ( N d ) ⊥ v reflects into the disjoint sum N d − 1 · � 1 � i � d v ( i ) 14/15

  73. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 2: Normed Reflections Definition Let ( X , � X ) and ( Y , � Y ) be qos with norms � . � X and � . � Y . A normed reflection is a function r : X → Y such that 1. ∀ x , x ′ ∈ X . x � X x ′ implies r ( x ) � Y r ( x ′ ) 2. ∀ x ∈ X . � r ( x ) � Y � � x � X Fact If r : X → Y is a normed reflection, then for all g , n 0 W a g , X ( n 0 ) � W a g , Y ( n 0 ) . (if x 0 , x 1 ,... is an amortised ( g , n 0 ) -controlled antichain, then r ( x 0 ) , r ( x 1 ) ,... is as well) ◮ over-approximate W a g , X ⊥ x 0 ( g ( n 0 )) in the descent equation ◮ e.g. ( N d ) ⊥ v reflects into the disjoint sum N d − 1 · � 1 � i � d v ( i ) 14/15

  74. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Ingredient 2: Normed Reflections Definition Let ( X , � X ) and ( Y , � Y ) be qos with norms � . � X and � . � Y . A normed reflection is a function r : X → Y such that 1. ∀ x , x ′ ∈ X . x � X x ′ implies r ( x ) � Y r ( x ′ ) 2. ∀ x ∈ X . � r ( x ) � Y � � x � X Fact If r : X → Y is a normed reflection, then for all g , n 0 W a g , X ( n 0 ) � W a g , Y ( n 0 ) . (if x 0 , x 1 ,... is an amortised ( g , n 0 ) -controlled antichain, then r ( x 0 ) , r ( x 1 ) ,... is as well) ◮ over-approximate W a g , X ⊥ x 0 ( g ( n 0 )) in the descent equation ◮ e.g. ( N d ) ⊥ v reflects into the disjoint sum N d − 1 · � 1 � i � d v ( i ) 14/15

  75. Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem Outline lossy counter machines (LCM) reachability ◮ canonical Ackermann -complete problem ◮ complexity gap in fixed dimension d : F d -hard, in F d + 1 complexity using well-quasi-orders (wqo) ◮ strongly controlled bad sequences ◮ antichain factorisation ◮ width function theorem on the length of controlled antichains ◮ F d upper bounds for LCM reachability 15/15

  76. Complexity Classes Backward Coverability Subrecursive Functions Technical Appendix 16/15

  77. Complexity Classes Backward Coverability Subrecursive Functions Complexity Classes Beyond Elementary [S.’16] Multiply Recursive Fast-Growing Complexity F ω Primitive Recursive = Ackermann F 3 = Tower Elementary 17/15

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