The Parametric Complexity of Lossy Counter Machines Sylvain Schmitz - - PowerPoint PPT Presentation

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

The Parametric Complexity of Lossy Counter Machines

Sylvain Schmitz ICALP , July 12, 2019, Patras

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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:

Fd-hard, in Fd+1 complexity using well-quasi-orders (wqo)

◮ controlled bad sequences ◮ length function theorem

• n the length of controlled bad sequences

◮ Fd+1 upper bounds for LCM reachability

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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:

Fd-hard, in Fd+1 complexity using well-quasi-orders (wqo)

◮ amortised controlled bad sequences ◮ length function theorem

• n the length of controlled bad sequences

◮ Fd+1 upper bounds for LCM reachability

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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:

Fd-hard, in Fd+1 complexity using well-quasi-orders (wqo)

◮ strongly controlled bad sequences ◮ antichain factorisation ◮ width function theorem

• n the length of controlled antichains

◮ Fd upper bounds for LCM reachability

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Main Result

F0(x) = x + 1 F1(x) = x+1 times

• F0 ◦ ··· ◦ F0(x) = 2x + 1

F2(x) = x+1 times

• F1 ◦ ··· ◦ F1(x) ≈ 2x

F3(x) = x+1 times

• F2 ◦ ··· ◦ F2(x) ≈ tower(x)

. . . Fω(x) = Fx+1(x) ≈ ackermann(x) Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Upper Bound Theorem LCM Reachability is Fd-complete in fixed dimension d 3.

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Main Result

F0(x) = x + 1 F1(x) = x+1 times

• F0 ◦ ··· ◦ F0(x) = 2x + 1

F2(x) = x+1 times

• F1 ◦ ··· ◦ F1(x) ≈ 2x

F3(x) = x+1 times

• F2 ◦ ··· ◦ F2(x) ≈ tower(x)

. . . Fω(x) = Fx+1(x) ≈ ackermann(x) Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Upper Bound Theorem LCM Reachability is Fd-complete in fixed dimension d 3.

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Main Result

F0(x) = x + 1 F1(x) = x+1 times

• F0 ◦ ··· ◦ F0(x) = 2x + 1

F2(x) = x+1 times

• F1 ◦ ··· ◦ F1(x) ≈ 2x

F3(x) = x+1 times

• F2 ◦ ··· ◦ F2(x) ≈ tower(x)

. . . Fω(x) = Fx+1(x) ≈ ackermann(x) Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Upper Bound Theorem LCM Reachability is Fd-complete in fixed dimension d 3.

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Lossy Counter Machines

Example q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Lossy Counter Machines

Example q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Lossy Semantics q1(0,2) q2(1,1) q1(0,0)

c1++ c2

?

= 0

ℓ ℓ

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Lossy Counter Machines

Example q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Lossy Semantics q1(0,2) q2(1,1) q1(0,0)

c1++ c2

?

= 0

ℓ ℓ

q1(0,2) q2(1,2)

• c1++
• 4/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Lossy Counter Machines

Example q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Lossy Semantics q1(0,2) q2(1,1) q1(0,0)

c1++ c2

?

= 0

ℓ ℓ

q2(1,0) q1(1,0)

• c2

?

= 0

• 4/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Reachability and Coverability

Reachability Problem input an LCM M, initial configuration q0(v0), target configuration qf(vf) question q0(v0) →∗

ℓ qf(vf)?

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Reachability and Coverability

Reachability Problem input an LCM M, initial configuration q0(v0), target configuration qf(vf) question q0(v0) →∗

ℓ qf(vf)?

Remark equivalent to coverability: question ∃v vf . q0(v0) →∗

ℓ qf(v)?

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Reachability and Coverability

Reachability Problem input an LCM M, initial configuration q0(v0), target configuration qf(vf) question q0(v0) →∗

ℓ qf(vf)?

Lower Bound Theorem (Urquhart’99; Schnoebelen’02,’10) LCM Reachability is Ackermann-hard. Upper Bound Theorem (McAloon’84,Clote’86) LCM Reachability is in Ackermann.

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Reachability and Coverability

Reachability Problem input an LCM M, initial configuration q0(v0), target configuration qf(vf) question q0(v0) →∗

ℓ qf(vf)?

Lower Bound Theorem (S.’17) LCM Reachability is Fd-hard in fixed dimension d 3. Upper Bound Theorem (Figueira & al.’11, S. & Schnoebelen ’12) LCM Reachability is in Fd+1 in fixed dimension d 3.

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

Uk

def

= {q(v) | ∃v′ (1,1) . q(v) →k

ℓ

q2(v′)}

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U0

def

= {q(v) | ∃v′ (1,1) . q(v) →0

ℓ

q2(v′)}

q2(1,1) q1 q2 q3

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U0

def

= {q(v) | ∃v′ (1,1) . q(v) →0

ℓ

q2(v′)}

U0 ↑q2(1,1) q1 q2 q3

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U1

def

= {q(v) | ∃v′ (1,1) . q(v) →1

ℓ

q2(v′)}

U0 ↑q2(1,1) q1(0,1)

ℓ

q1 q2 q3

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U1

def

= {q(v) | ∃v′ (1,1) . q(v) →1

ℓ

q2(v′)}

U1 U0 ↑q2(1,1) ↑q1(0,1)

ℓ

q1 q2 q3

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U2

def

= {q(v) | ∃v′ (1,1) . q(v) →2

ℓ

q2(v′)}

U2 U1 U0 ↑q2(1,1) ↑q1(0,1)

ℓ

↑q3(0,0)

ℓ

q1 q2 q3

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U3

def

= {q(v) | ∃v′ (1,1) . q(v) →3

ℓ

q2(v′)}

U3 U2 U1 U0 ↑q2(1,1) ↑q1(0,1)

ℓ

↑q3(0,0)

ℓ

↑q1(1,0)

ℓ

q1 q2 q3

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U4

def

= {q(v) | ∃v′ (1,1) . q(v) →4

ℓ

q2(v′)}

U4 U3 U2 U1 U0 ↑q2(1,1) ↑q1(0,1)

ℓ

↑q3(0,0)

ℓ

↑q1(1,0)

ℓ

↑q2(1,0)

ℓ

q1 q2 q3

6/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U4

def

= {q(v) | ∃v′ (1,1) . q(v) →4

ℓ

q2(v′)}

U4 U3 U2 U1 U0 ↑q2(1,1) ↑q1(0,1)

ℓ

↑q3(0,0)

ℓ

↑q1(1,0)

ℓ

↑q2(1,0)

ℓ

q3(1,0)

ℓ

q1 q2 q3

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U5

def

= {q(v) | ∃v′ (1,1) . q(v) →5

ℓ

q2(v′)}

U5 U4 U3 U2 U1 U0 ↑q2(1,1) ↑q1(0,1)

ℓ

↑q3(0,0)

ℓ

↑q1(1,0)

ℓ

↑q2(1,0)

ℓ

q3(1,0)

ℓ

↑q1(0,0)

ℓ

q1 q2 q3

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U6

def

= {q(v) | ∃v′ (1,1) . q(v) →6

ℓ

q2(v′)}

U6 U5 U4 U3 U2 U1 U0 ↑q2(1,1) ↑q1(0,1)

ℓ

↑q3(0,0)

ℓ

↑q1(1,0)

ℓ

↑q2(1,0)

ℓ

q3(1,0)

ℓ

↑q1(0,0)

ℓ

↑q2(0,0)

ℓ

q1 q2 q3

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U6

def

= {q(v) | ∃v′ (1,1) . q(v) →∗

ℓ q2(v′)} U6 U5 U4 U3 U2 U1 U0 ↑q2(1,1) ↑q1(0,1)

ℓ

↑q3(0,0)

ℓ

↑q1(1,0)

ℓ

↑q2(1,0)

ℓ

q3(1,0)

ℓ

↑q1(0,0)

ℓ

↑q2(0,0)

ℓ ℓ

q1 q2 q3

6/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Backward Coverability

(Arnold & Latteux’78)

q1 q3 q2

c1++ c2

?

= 0 c1−− c2++

Example: coverability of q2(1,1) in

U6

def

= {q(v) | ∃v′ (1,1) . q(v) →∗

ℓ q2(v′)} U6 U5 U4 U3 U2 U1 U0 ↑q2(1,1) ↑q1(0,1)

ℓ

↑q3(0,0)

ℓ

↑q1(1,0)

ℓ

↑q2(1,0)

ℓ

↑q1(0,0)

ℓ

↑q2(0,0)

ℓ

q1 q2 q3 The sequence q2(1,1), q1(0,1), q3(0,0), q1(1,0), q2(1,0), q1(0,0), q2(0,0) is bad

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

(0,2)

7/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

(0,2), (2,1)

7/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

(0,2), (2,1), (0,1)

7/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

(0,2), (2,1), (0,1), (6,0)

7/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

(0,2), (2,1), (0,1), (6,0), (5,0)

7/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

(0,2), (2,1), (0,1), (6,0), (5,0), (4,0)

7/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

(0,2), (2,1), (0,1), (6,0), (5,0), (4,0), (3,0)

7/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

(0,2), (2,1), (0,1), (6,0), (5,0), (4,0), (3,0), (2,0)

7/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

(0,2), (2,1), (0,1), (6,0), (5,0), (4,0), (3,0), (2,0), (1,0)

7/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Bad Sequences

Over a qo (X,)

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ (X,) wqo if all bad sequences are

finite... ... but can be of arbitrary length

Example (in N2)

(0,2), (2,1), (0,1), (6,0), (5,0), (4,0), (3,0), (2,0), (1,0), (0,0)

7/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] 8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] g0(2) = 2

Example (in N2 with n0 = 2 and g(n) = n + 1)

(0,2)

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] g1(2) = 3

Example (in N2 with n0 = 2 and g(n) = n + 1)

(0,2), (2,1)

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] g2(2) = 4

Example (in N2 with n0 = 2 and g(n) = n + 1)

(0,2), (2,1), (0,1)

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] g3(2) = 5

Example (in N2 with n0 = 2 and g(n) = n + 1)

(0,2), (2,1), (0,1), (5,0)

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] g4(2) = 6

Example (in N2 with n0 = 2 and g(n) = n + 1)

(0,2), (2,1), (0,1), (5,0), (4,0)

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] g5(2) = 7

Example (in N2 with n0 = 2 and g(n) = n + 1)

(0,2), (2,1), (0,1), (5,0), (4,0), (3,0)

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] g6(2) = 8

Example (in N2 with n0 = 2 and g(n) = n + 1)

(0,2), (2,1), (0,1), (5,0), (4,0), (3,0), (2,0)

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] g7(2) = 9

Example (in N2 with n0 = 2 and g(n) = n + 1)

(0,2), (2,1), (0,1), (5,0), (4,0), (3,0), (2,0), (1,0)

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98] g8(2) = 10

Example (in N2 with n0 = 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

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98]

Proposition

In a wqo (X,), if ∀n {x ∈ X | x n} is finite, then amortised (g,n0)-controlled bad sequences have a maximal length, denoted La

g,X(n0).

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Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98]

Proposition

In a wqo (X,), if ∀n {x ∈ X | x n} is finite, then amortised (g,n0)-controlled bad sequences have a maximal length, denoted La

g,X(n0).

Theorem (S. & Schnoebelen’12)

For LCM Reachability, g(x)

def

= x + 1 and n0

def

= vf fit, and La

g,Q×Nd(n0) ≈ Fd+1(n).

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98]

◮ x0,x1,... is strongly controlled by

g:N → N and n0 ∈ N if x0 n0 and ∀i.xi+1 g(xi)

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98]

◮ x0,x1,... is strongly controlled by

g:N → N and n0 ∈ N if x0 n0 and ∀i.xi+1 g(xi)

Corollary

In a wqo (X,), if ∀n {x ∈ X | x n} is finite, then strongly (g,n0)-controlled bad sequences have a maximal length, denoted Ls

g,X(n0).

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

[Cicho´ n & Tahhan Bittar’98]

◮ x0,x1,... is strongly controlled by

g:N → N and n0 ∈ N if x0 n0 and ∀i.xi+1 g(xi)

Corollary

In a wqo (X,), if ∀n {x ∈ X | x n} is finite, then strongly (g,n0)-controlled bad sequences have a maximal length, denoted Ls

g,X(n0)and a maximal norm, denoted Ns g,X(n0).

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Bad Sequences

Over a qo (X,) with norm ·

◮ x0,x1,... is bad if ∀i < j . xi xj ◮ x0,x1,... is strongly controlled by

g:N → N and n0 ∈ N if x0 n0 and ∀i.xi+1 g(xi)

Corollary

In a wqo (X,), if ∀n {x ∈ X | x n} is finite, then (g,n0)-controlled bad sequences have a maximal length, denoted Ls

g,X(n0)and a maximal norm, denoted Ns g,X(n0).

Theorem

For LCM Reachability, g(x)

def

= x + 1 and n0

def

= vf fit, and Ls

g,Q×Nd(n0) ≈ Fd(n).

8/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Antichains

Over a qo (X,)

◮ x0,x1,... is an antichain if

∀i < j . xi⊥xj (i.e. xi xj and xj xi)

9/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Antichains

Over a qo (X,)

◮ x0,x1,... is an antichain if

∀i < j . xi⊥xj (i.e. xi xj and xj xi)

◮ (X,) wqo iff it is well-founded

and all antichains are finite

9/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Antichains

Over a qo (X,) with norm ·

◮ x0,x1,... is an antichain if

∀i < j . xi⊥xj (i.e. xi xj and xj xi)

◮ (X,) wqo iff it is well-founded

and all antichains are finite

◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

9/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Controlled Antichains

Over a qo (X,) with norm ·

◮ x0,x1,... is an antichain if

∀i < j . xi⊥xj (i.e. xi xj and xj xi)

◮ (X,) wqo iff it is well-founded

and all antichains are finite

◮ x0,x1,... is amortised controlled

by g:N → N and n0 ∈ N if ∀i . xi gi(n0)

Corollary

In a wqo (X,), if ∀n {x ∈ X | x n} is finite, then amortised (g,n0)-controlled antichains have a maximal length, denoted Wa

g,X(n0).

9/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) >

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) >

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) > >

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) > >

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) (5,0) > >

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) (5,0) (1,4) > >

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) (5,0) (1,4) (3,1) > > >

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) (5,0) (1,4) (3,1) (0,4) > > >

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) (5,0) (1,4) (3,1) (0,4) (3,0) > > > >

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) (5,0) (1,4) (3,1) (0,4) (3,0) (1,1) > > > > >

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) (5,0) (1,4) (3,1) (0,4) (3,0) (1,1) > > > > >

Property

Every branch is a strongly (x → x + 1,4)-controlled antichain

• ver N2.

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) (5,0) (1,4) (3,1) (0,4) (3,0) (1,1) > > > > >

Here, height is 3, hence

◮ maximal norm at most g3(n0) = n0 + 3 = 7, ◮ length of the bad sequence at most (7 + 1)2 = 64

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) (5,0) (1,4) (3,1) (0,4) (3,0) (1,1) > > > > >

Here, height is 3, hence

◮ maximal norm at most g3(n0) = n0 + 3 = 7, ◮ length of the bad sequence at most (7 + 1)2 = 64

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Antichain Factorisation

Example (strongly (x → x + 1,4)-controlled bad sequence over N2)

(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) (4,2) (5,1) (2,3) (4,1) (5,0) (1,4) (3,1) (0,4) (3,0) (1,1) > > > > >

Here, height is 3, hence

◮ maximal norm at most g3(n0) = n0 + 3 = 7, ◮ length of the bad sequence at most (7 + 1)2 = 64

10/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Factorisation Lemma

Lemma Let (X,) be a wqo with norm .:X → N monotone. Then Ns

g,X(n0) gWs

g,X(n0)(n0).

11/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Factorisation Lemma

Lemma Let (X,) be a wqo with norm .:X → N monotone. Then Ns

g,X(n0) gWs

g,X(n0)(n0).

11/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Width Function Theorem

Informal statement Wa

g,Q×Nd(n0) ≈ Fd(n) in the case of LCMs.

Proof ingredients

• 1. descent equation
• 2. normed reflections
• 3. subrecursive hierarchies

12/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Width Function Theorem

Informal statement Wa

g,Q×Nd(n0) ≈ Fd(n) in the case of LCMs.

Proof ingredients

• 1. descent equation
• 2. normed reflections
• 3. subrecursive hierarchies

12/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Width Function Theorem

Informal statement Wa

g,Q×Nd(n0) ≈ Fd(n) in the case of LCMs.

Proof ingredients

• 1. descent equation
• 2. normed reflections
• 3. subrecursive hierarchies

12/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Ingredient 1: Descent Equation

amortised (g,n0)-controlled antichain x0,x1,x2,x3,...

• ver a wqo (X,):

norms xi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = x0 x1 x2 x3 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ 13/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Ingredient 1: Descent Equation

amortised (g,n0)-controlled antichain x0,x1,x2,x3,...

• ver a wqo (X,):

norms xi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = x0 x1 x2 x3 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

• ver

the suffix x1,x2,x3,..., ∀i > 0, x0⊥xi

13/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Ingredient 1: Descent Equation

amortised (g,n0)-controlled antichain x0,x1,x2,x3,...

• ver a wqo (X,):

norms xi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = x0 x1 x2 x3 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

• ver

the suffix x1,x2,x3,..., ∀i > 0, xi ∈ X⊥x0

def

= {x ∈ X | x0⊥x}

13/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Ingredient 1: Descent Equation

amortised (g,n0)-controlled antichain x0,x1,x2,x3,...

• ver a wqo (X,):

norms xi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = x0 x1 x2 x3 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

• ver

the suffix x1,x2,x3,..., ∀i > 0, xi ∈ X⊥x0

def

= {x ∈ X | x0⊥x} xi gi−1(g(n0))

13/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Ingredient 1: Descent Equation

amortised (g,n0)-controlled antichain x0,x1,x2,x3,...

• ver a wqo (X,):

norms xi indices i g0(n0) g1(n0) g0(g(n0)) = g2(n0) g1(g(n0)) = g3(n0) g2(g(n0)) = x0 x1 x2 x3 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

• ver

the suffix x1,x2,x3,..., ∀i > 0, xi ∈ X⊥x0

def

= {x ∈ X | x0⊥x} xi gi−1(g(n0))

Wa

g,X(n0) =

max

x0∈X,x0n0

1 + Wa

g,X⊥x0(g(n0))

13/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Ingredient 2: Normed Reflections

X Y r 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 xX

14/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Ingredient 2: Normed Reflections

X x′ x Y r(x′) r(x) r r 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 xX

14/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Ingredient 2: Normed Reflections

X x′ x Y r(x′) r(x) r r 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 xX

14/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Ingredient 2: Normed Reflections

X x′ x Y r(x′) r(x) r r r X Y xX 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 xX

14/15

Lossy Counter Machines Controlled Sequences Antichain Factorisation Width Function Theorem

Ingredient 2: Normed Reflections

X x′ x Y r(x′) r(x) r r r X Y xX 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 xX

14/15

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 xX

Fact If r:X → Y is a normed reflection, then for all g,n0 Wa

g,X(n0) Wa g,Y(n0).

(if x0,x1,... is an amortised (g,n0)-controlled antichain, then r(x0),r(x1),... is as well) ◮ over-approximate Wa

g,X⊥x0(g(n0)) in the descent equation

◮ e.g. (Nd)⊥v reflects into the disjoint sum Nd−1·

1id v(i)

14/15

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 xX

Fact If r:X → Y is a normed reflection, then for all g,n0 Wa

g,X(n0) Wa g,Y(n0).

(if x0,x1,... is an amortised (g,n0)-controlled antichain, then r(x0),r(x1),... is as well) ◮ over-approximate Wa

g,X⊥x0(g(n0)) in the descent equation

◮ e.g. (Nd)⊥v reflects into the disjoint sum Nd−1·

1id v(i)

14/15

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 xX

Fact If r:X → Y is a normed reflection, then for all g,n0 Wa

g,X(n0) Wa g,Y(n0).

(if x0,x1,... is an amortised (g,n0)-controlled antichain, then r(x0),r(x1),... is as well) ◮ over-approximate Wa

g,X⊥x0(g(n0)) in the descent equation

◮ e.g. (Nd)⊥v reflects into the disjoint sum Nd−1·

1id v(i)

14/15

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:

Fd-hard, in Fd+1 complexity using well-quasi-orders (wqo)

◮ strongly controlled bad sequences ◮ antichain factorisation ◮ width function theorem

• n the length of controlled antichains

◮ Fd upper bounds for LCM reachability

15/15

Complexity Classes Backward Coverability Subrecursive Functions

Technical Appendix

16/15

Complexity Classes Backward Coverability Subrecursive Functions

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Fast-Growing Complexity

17/15

Complexity Classes Backward Coverability Subrecursive Functions

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Fast-Growing Complexity

F3

def

=

• e elementary

DTime(tower(e(n)))

17/15

Complexity Classes Backward Coverability Subrecursive Functions

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Fast-Growing Complexity 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]

17/15

Complexity Classes Backward Coverability Subrecursive Functions

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Fast-Growing Complexity

Fω

def

=

• p primitive recursive

DTime(ack(p(n)))

17/15

Complexity Classes Backward Coverability Subrecursive Functions

Complexity Classes Beyond Elementary

[S.’16]

Elementary Primitive Recursive Multiply Recursive F3 = Tower Fω = Ackermann

Fast-Growing Complexity 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]

17/15

Complexity Classes Backward Coverability Subrecursive Functions

Backward Coverability

(Arnold & Latteux’78)

Goal: Check whether q0(v0) ∈ Pre∗

∃(↑qf(vf))

def

= {q(v) | ∃v′ vf . q(v) →∗

ℓ qf(v′)}

Fixed-point computation U0

def

= ↑qf(vf) Un+1

def

= Un ∪ Pre∃(Un) where Pre∃(S)

def

= {q(v) | ∃q′(v′) ∈ S . q(v) →ℓ q′(v′)} Ascending chain U0 U1 ··· UL = UL+1 = Pre∗

∃(↑qf(vf))

18/15

Complexity Classes Backward Coverability Subrecursive Functions

Backward Coverability

(Arnold & Latteux’78)

Goal: Check whether q0(v0) ∈ Pre∗

∃(↑qf(vf))

def

= {q(v) | ∃v′ vf . q(v) →∗

ℓ qf(v′)}

Fixed-point computation U0

def

= ↑qf(vf) Un+1

def

= Un ∪ Pre∃(Un) where Pre∃(S)

def

= {q(v) | ∃q′(v′) ∈ S . q(v) →ℓ q′(v′)} Ascending chain U0 U1 ··· UL = UL+1 = Pre∗

∃(↑qf(vf))

18/15

Complexity Classes Backward Coverability Subrecursive Functions

Backward Coverability

(Arnold & Latteux’78)

Goal: Check whether q0(v0) ∈ Pre∗

∃(↑qf(vf))

def

= {q(v) | ∃v′ vf . q(v) →∗

ℓ qf(v′)}

Fixed-point computation U0

def

= ↑qf(vf) Un+1

def

= Un ∪ Pre∃(Un) where Pre∃(S)

def

= {q(v) | ∃q′(v′) ∈ S . q(v) →ℓ q′(v′)} Ascending chain U0 U1 ··· UL = UL+1 = Pre∗

∃(↑qf(vf))

18/15

Complexity Classes Backward Coverability Subrecursive Functions

Backward Coverability

(Arnold & Latteux’78)

Goal: Check whether q0(v0) ∈ Pre∗

∃(↑qf(vf))

def

= {q(v) | ∃v′ vf . q(v) →∗

ℓ qf(v′)}

Fixed-point computation U0

def

= ↑qf(vf) Un+1

def

= Un ∪ Pre∃(Un) where Pre∃(S)

def

= {q(v) | ∃q′(v′) ∈ S . q(v) →ℓ q′(v′)} Ascending chain U0 U1 ··· UL = UL+1 = Pre∗

∃(↑qf(vf))

18/15

Complexity Classes Backward Coverability Subrecursive Functions

Backward Coverability

(Arnold & Latteux’78)

Goal: Check whether q0(v0) ∈ Pre∗

∃(↑qf(vf))

def

= {q(v) | ∃v′ vf . q(v) →∗

ℓ qf(v′)}

Fixed-point computation U0

def

= ↑qf(vf) Un+1

def

= Un ∪ Pre∃(Un) where Pre∃(S)

def

= {q(v) | ∃q′(v′) ∈ S . q(v) →ℓ q′(v′)} Ascending chain U0 U1 ··· UL = UL+1 = Pre∗

∃(↑qf(vf))

18/15

Complexity Classes Backward Coverability Subrecursive Functions

Backward Coverability

(Arnold & Latteux’78)

Goal: Check whether q0(v0) ∈ Pre∗

∃(↑qf(vf))

def

= {q(v) | ∃v′ vf . q(v) →∗

ℓ qf(v′)}

Fixed-point computation U0

def

= ↑qf(vf) Un+1

def

= Un ∪ Pre∃(Un) where Pre∃(S)

def

= {q(v) | ∃q′(v′) ∈ S . q(v) →ℓ q′(v′)} Ascending chain U0 U1 ··· UL = UL+1 = Pre∗

∃(↑qf(vf))

18/15

Complexity Classes Backward Coverability Subrecursive Functions

Backward Coverability

(Arnold & Latteux’78)

Goal: Check whether q0(v0) ∈ Pre∗

∃(↑qf(vf))

def

= {q(v) | ∃v′ vf . q(v) →∗

ℓ qf(v′)}

Fixed-point computation B0

def

= {qf(vf)} Bn+1

def

= min(Bn ∪ Pre∃(↑Bn)) where Pre∃(S)

def

= {q(v) | ∃q′(v′) ∈ S . q(v) →ℓ q′(v′)} Ascending chain of upwards-closed sets ↑B0 ↑B1 ··· ↑BL = ↑BL+1 = Pre∗

∃(↑qf(vf))

18/15

Complexity Classes Backward Coverability Subrecursive Functions

Backward Coverability

(Arnold & Latteux’78)

Ascending chain of upwards-closed sets ↑B0 ↑B1 ··· ↑BL = ↑BL+1 = Pre∗

∃(↑qf(vf))

Pseudo-witness c0

def

= qf(vf) cn+1 ∈ Bn+1 \ ↑Bn Then ∀i < j . ci cj (Q × Nd,) is a well-quasi-order (wqo), thus finite basis property: each Bn = minUn is finite ascending chain condition: a finite length L exists finite bad sequences: c0,c1,... is a bad sequence

18/15

Complexity Classes Backward Coverability Subrecursive Functions

Backward Coverability

(Arnold & Latteux’78)

Ascending chain of upwards-closed sets ↑B0 ↑B1 ··· ↑BL = ↑BL+1 = Pre∗

∃(↑qf(vf))

Pseudo-witness c0

def

= qf(vf) cn+1 ∈ Bn+1 \ ↑Bn Then ∀i < j . ci cj (Q × Nd,) is a well-quasi-order (wqo), thus finite basis property: each Bn = minUn is finite ascending chain condition: a finite length L exists finite bad sequences: c0,c1,... is a bad sequence

18/15

Complexity Classes Backward Coverability Subrecursive Functions

Backward Coverability

(Arnold & Latteux’78)

Ascending chain of upwards-closed sets ↑B0 ↑B1 ··· ↑BL = ↑BL+1 = Pre∗

∃(↑qf(vf))

Pseudo-witness c0

def

= qf(vf) cn+1 ∈ Bn+1 \ ↑Bn Then ∀i < j . ci cj (Q × Nd,) is a well-quasi-order (wqo), thus finite basis property: each Bn = minUn is finite ascending chain condition: a finite length L exists finite bad sequences: c0,c1,... is a bad sequence

18/15

Complexity Classes Backward Coverability Subrecursive Functions

Subrecursive Functions

Definition (Cicho´ n Hierarchy) For g:N → N, define (gα:N → N)α by g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x))

19/15

Complexity Classes Backward Coverability Subrecursive Functions

Subrecursive Functions

Definition (Cicho´ n Hierarchy) For g:N → N, define (gα:N → N)α by g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x)) Px(α) denotes the predecessor at x of α > 0: “maximal

• rdinal β < α with Cantor normal form coefficients at

most x”

19/15

Complexity Classes Backward Coverability Subrecursive Functions

Subrecursive Functions

Definition (Cicho´ n Hierarchy) For g:N → N, define (gα:N → N)α by g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x)) Px(α) denotes the predecessor at x of α > 0: “maximal

• rdinal β < α with Cantor normal form coefficients at

most x”

19/15

Complexity Classes Backward Coverability Subrecursive Functions

Subrecursive Functions

Definition (Cicho´ n Hierarchy) For g:N → N, define (gα:N → N)α by g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x)) Px(α) denotes the predecessor at x of α > 0: “maximal

• rdinal β < α with Cantor normal form coefficients at

most x” Example

P3(ω2) = ω · 3 + 3 P3(ωω2) = ωω·3+3 · 3 + ωω·3+2 · 3 + ωω·3+1 · 3 + ωω·3 · 3 + ωω·2+3 · 3 + ωω·2+2 · 3 + ωω·2+1 · 3 + ωω·2 · 3 + ωω+3 · 3 + ωω+2 · 3 + ωω+1 · 3 + ωω · 3 + ω3 · 3 + ω2 · 3 + ω · 3 + 3

19/15

Complexity Classes Backward Coverability Subrecursive Functions

Subrecursive Functions

Definition (Cicho´ n Hierarchy) For g:N → N, define (gα:N → N)α by g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x)) Px(α) denotes the predecessor at x of α > 0: “maximal

• rdinal β < α with Cantor normal form coefficients at

most x” Example

P3(ω2) = ω · 3 + 3 P3(ωω2) = ωω·3+3 · 3 + ωω·3+2 · 3 + ωω·3+1 · 3 + ωω·3 · 3 + ωω·2+3 · 3 + ωω·2+2 · 3 + ωω·2+1 · 3 + ωω·2 · 3 + ωω+3 · 3 + ωω+2 · 3 + ωω+1 · 3 + ωω · 3 + ω3 · 3 + ω2 · 3 + ω · 3 + 3

19/15

Complexity Classes Backward Coverability Subrecursive Functions

Subrecursive Functions

Definition (Cicho´ n Hierarchy) For g:N → N, define (gα:N → N)α by g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x)) Definition (Hardy Hierarchy) For g:N → N, define (gα:N → N)α by g0(x)

def

= x gα(x)

def

= gPx(α)(g(x)) for α > 0

19/15

Complexity Classes Backward Coverability Subrecursive Functions

Subrecursive Functions

Definition (Cicho´ n Hierarchy) For g:N → N, define (gα:N → N)α by g0(x)

def

= 0 gα(x)

def

= 1 + gPx(α)(g(x)) Definition (Hardy Hierarchy) For g:N → N, define (gα:N → N)α by g0(x)

def

= x gα(x)

def

= gPx(α)(g(x)) for α > 0 If g(x) = x + 1, then gωα(x) = Fα(x)

19/15

Complexity Classes Backward Coverability Subrecursive Functions

Relating Norm and Length

[Cicho´ n & Tahhan Bittar’98] length: Cicho´ n function gα(n0) norm: Hardy function gα(n0) norms xi indices i g0(n0) g1(n0) g2(n0) g3(n0) x0 x1 x2 x3

gα(x) = ggα(x)(x) gα(x) gα(x) + x

20/15

Complexity Classes Backward Coverability Subrecursive Functions

Relating Norm and Length

[Cicho´ n & Tahhan Bittar’98] length: Cicho´ n function gα(n0) norm: Hardy function gα(n0) norms xi indices i g0(n0) g1(n0) g2(n0) g3(n0) x0 x1 x2 x3

gα(x) = ggα(x)(x) gα(x) gα(x) + x

20/15