Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Ackermann and Primitive-Recursive
Bounds with Dickson’s Lemma
D. Figueira
S. Figueira
S. Schmitz Ph. Schnoebelen
LSV, ENS Cachan & CNRS
LaBRI: Graphes et logique, November 23, 2010
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
Outline
well quasi orderings
generic tools for termination arguments
this talk
beyond termination: complexity upper bounds
contents
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Dickson’s Lemma
I
I
I
wqo (S, ⩽): well-founded quasi-ordering with
no infinite antichains
every infinite sequence x = x0 , x1 , x2 , . . . of Sω
contains an infinite increasing subsequence
xi1 ⩽ xi2 ⩽ xi3 ⩽ ⋅ ⋅ ⋅ for some i1 < i2 < i3 < ⋅ ⋅ ⋅
Dickson’s Lemma: (Nk , ⩽) with ⩽ denoting the
product ordering is a wqo.
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Good and Bad Sequences
I
a sequence of S∞ is r-good if it contains an
increasing subsequence xi1 ⩽ xi2 ⩽ ⋅ ⋅ ⋅ ⩽ xir+1 of
length r + 1
I
otherwise it is r-bad
I
for r = 1: good and bad sequences
I
wqo: every bad sequence is finite
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Well-structured transition systems
I
transition systems (Q, →, q0 ) with a wqo ⩽ on Q
compatible with transitions:
a
a
∀p, q, p ′ ∈ Q, (p →
− q∧p ⩽ p ′ ) ⇒ ∃q ′ , (q ⩽ q ′ ∧p ′ →
− q ′)
I
I
a generic framework for decidability results:
safety, termination, EF model checking, ...
many classes of concrete systems are WSTS:
I
I
I
over (Nk , ⩽): vector addition systems,
resets/transfer Petri nets, increasing counter systems,
...
over (Σ∗ , ⊑): lossy channel systems, ...
beyond: data nets, ...
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Example: (Non) Termination
I
given (Q, →, q0 ), decide whether there exists an
infinite run q0 → q1 → ⋅ ⋅ ⋅
I
holds iff there exists qi ⩽ qj with q0 →∗ qi →+ qj
I
thanks to wqo, termination is both r.e. and co-r.e.
I
what is the complexity?
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Example: (Non) Termination
I
given (Q, →, q0 ), decide whether there exists an
infinite run q0 → q1 → ⋅ ⋅ ⋅
I
holds iff there exists qi ⩽ qj with q0 →∗ qi →+ qj
I
thanks to wqo, termination is both r.e. and co-r.e.
I
what is the complexity?
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
Controlled Sequences
I
bound the length of bad sequences
I
but: choose any N, and consider the bad
sequence N, N − 1, . . . , 0 over N
similarly:
⟨3, 3⟩, ⟨3, 2⟩, ⟨3, 1⟩, ⟨3, 0⟩, ⟨2, N⟩, ⟨2, N − 1⟩, . . .
I
References
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
Controlled Sequences
I
bound the length of bad sequences
I
fix a control function f : N → N
x = x0 , x1 , . . . over Nk is f-controlled if
I
∀i = 0, 1, . . . , ∀1 ⩽ j ⩽ k, xi [j] < f(i)
I
for fixed r, k, f, there are finitely many r-bad
f-controlled sequences over Nk
References
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Controlled Sequences
I
bound the length of controlled bad sequences
I
distinguish x0 : consider (λx.f(x + t))-controlled
sequences instead (thus xi [j] < f(i + t))
bound the maximal length of a r-bad
(λx.f(x + t))-controlled sequence over Nk as a
function of t:
Lr,k,f (t)
I
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Example
k = 2, t = 1, f(x) = x + 3
i
xi [1]
xi [2]
f(i + t)
0
3
3
4
1
3
2
5
2
3
1
6
3
3
0
7
4
2
7
8
5 ⋅ ⋅ ⋅ 10 11 12 13 ⋅ ⋅ ⋅ 26 27 28 29 ⋅ ⋅ ⋅ 58 59
2 ⋅⋅⋅ 2 2 1 1 ⋅⋅⋅ 1 1 0 0 ⋅⋅⋅ 0 0
6 ⋅ ⋅ ⋅ 1 0 15 14 ⋅ ⋅ ⋅ 1 0 31 30 ⋅ ⋅ ⋅ 1 0
9 ⋅ ⋅ ⋅ 14 15 16 17 ⋅ ⋅ ⋅ 30 31 32 33 ⋅ ⋅ ⋅ 62 63
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Easy Cases
L0,k,f (t) = 0
L1,0,f (t) = 1
Lr,0,f (t) = r
L1,1,f (t) = f(t)
(by convention)
the latter sequence being
f(t) − 1, f(t) − 2, . . . , 1, 0
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
A More General Problem
I
I
disjoint sums A1 ⊕ A2
wqo for the sum ordering :
(
)
def
x ⩽ x ′ ⇔ x, x ′ ∈ A1 ∧ x ⩽1 x ′
(
)
∨ x, x ′ ∈ A2 ∧ x ⩽2 x ′
I
I
multiset notation: τ = {|k1 , k2 , . . . |}, Nτ =
shift to Lτ,f (t)
⊕
iN
ki
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Decomposition for Nk
A bad sequence x = x0 , x1 , . . . , xl over Nk :
I control: x0 ⩽ ⟨f(t) − 1, . . . , f(t) − 1⟩
I
badness: ∀i > 0, ∃1 ⩽ j ⩽ k, xi [j] < x0 [j]
I
∀i > 1, ∃1 ⩽ j ⩽ k, 0 ⩽ s ⩽ f(t) − 1, xi ∈ Rj,s
where Rj,s = {x ∈ Nk | x[j] = s}
I
there are Nk,f (t) = k ⋅ (f(t) − 1) regions in total
def
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Decomposition for Nk
A bad sequence x = x0 , x1 , . . . , xl over Nk :
I control: x0 ⩽ ⟨f(t) − 1, . . . , f(t) − 1⟩
I
badness: ∀i > 0, ∃1 ⩽ j ⩽ k, xi [j] < x0 [j]
I
∀i > 1, ∃1 ⩽ j ⩽ k, 0 ⩽ s ⩽ f(t) − 1, xi ∈ Rj,s
where Rj,s = {x ∈ Nk | x[j] = s}
I
there are Nk,f (t) = k ⋅ (f(t) − 1) regions in total
def
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Decomposition for Nk
A bad sequence x = x0 , x1 , . . . , xl over Nk :
I control: x0 ⩽ ⟨f(t) − 1, . . . , f(t) − 1⟩
I
badness: ∀i > 0, ∃1 ⩽ j ⩽ k, xi [j] < x0 [j]
I
∀i > 1, ∃1 ⩽ j ⩽ k, 0 ⩽ s ⩽ f(t) − 1, xi ∈ Rj,s
where Rj,s = {x ∈ Nk | x[j] = s}
I
there are Nk,f (t) = k ⋅ (f(t) − 1) regions in total
def
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Decomposition for Nk
A bad sequence x = x0 , x1 , . . . , xl over Nk :
I control: x0 ⩽ ⟨f(t) − 1, . . . , f(t) − 1⟩
I
badness: ∀i > 0, ∃1 ⩽ j ⩽ k, xi [j] < x0 [j]
I
∀i > 1, ∃1 ⩽ j ⩽ k, 0 ⩽ s ⩽ f(t) − 1, xi ∈ Rj,s
where Rj,s = {x ∈ Nk | x[j] = s}
I
there are Nk,f (t) = k ⋅ (f(t) − 1) regions in total
def
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Decomposition for Nk
Example
x = ⟨2, 2⟩, ⟨1, 5⟩, ⟨4, 0⟩, ⟨1, 1⟩, ⟨0, 100⟩, ⟨0, 99⟩, ⟨3, 0⟩
⎡
⎢ ⟨1, 5⟩,
⟨2, 2⟩,⎢
⎣
⟨0, 100⟩,⟨0, 99⟩,
⟨1, 1⟩,
⟨4, 0⟩,
(R1,0
(R1,1
⟨3, 0⟩ (R2,0
(R2,1
⎤
: x[1] = 0)
: x[1] = 1) ⎥
⎥
: x[2] = 0) ⎦
: x[2] = 1)
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Decomposition for Nk
Example
x = ⟨2, 2⟩, ⟨1, 5⟩, ⟨4, 0⟩, ⟨1, 1⟩, ⟨0, 100⟩, ⟨0, 99⟩, ⟨3, 0⟩
⎡
⎢ ⟨∗, 5⟩,
⟨2, 2⟩,⎢
⎣
⟨∗, 100⟩,⟨∗, 99⟩,
⟨∗, 1⟩,
⟨4, ∗⟩,
(R1,0
(R1,1
⟨3, ∗⟩ (R2,0
(R2,1
⎤
: x[1] = 0)
: x[1] = 1) ⎥
⎥
: x[2] = 0) ⎦
: x[2] = 1)
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Decomposition for Nk
Example
x = ⟨2, 2⟩, ⟨1, 5⟩, ⟨4, 0⟩, ⟨1, 1⟩, ⟨0, 100⟩, ⟨0, 99⟩, ⟨3, 0⟩
⎡
⟨∗, 100⟩,⟨∗, 99⟩,
⎢ ⟨∗, 5⟩,
⟨2, 2⟩,⎢
⎣
⟨∗, 1⟩,
⟨4, ∗⟩,
(R1,0
(R1,1
⟨3, ∗⟩ (R2,0
(R2,1
⎤
: x[1] = 0)
: x[1] = 1) ⎥
⎥
: x[2] = 0) ⎦
: x[2] = 1)
L{k},f (t) ⩽ 1 + LNk,f (t)×{k−1},f (t + 1)
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Decomposition for
Applications
References
ki
N
i
⊕
Example
τ = {|1, 2, 2|}:
⎡
⟨5⟩,
⟨3⟩
⎣ ⟨2, 2⟩,
⟨1, 5⟩,⟨4, 0⟩,
⎤
⟨1, 1⟩,
⟨12, 1⟩,
⎡
⎢
⎢
⎢
⟨2, 2⟩ ⎢
⎢
⎣
⟨5⟩,
⟨0, 100⟩,⟨0, 99⟩,⟨3, 0⟩ ⎦
⟨3, 5⟩
⟨3⟩
⎤
⟨∗, 100⟩,⟨∗, 99⟩,
⟨∗, 5⟩,
⟨∗, 1⟩,
⟨4, ∗⟩,
⟨12, 1⟩,
⟨3, 5⟩
⎥
⎥
⎥
⎥
⟨3, ∗⟩ ⎥
⎦
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Decomposition for
Applications
References
ki
N
i
⊕
Example
τ = {|1, 2, 2|}:
⎡
⟨5⟩,
⟨3⟩
⎣ ⟨2, 2⟩,
⟨1, 5⟩,⟨4, 0⟩,
⎤
⟨1, 1⟩,
⟨12, 1⟩,
⎡
⎢
⎢
⎢
⟨2, 2⟩ ⎢
⎢
⎣
⟨5⟩,
⟨0, 100⟩,⟨0, 99⟩,⟨3, 0⟩ ⎦
⟨3, 5⟩
⟨3⟩
⎤
⟨∗, 100⟩,⟨∗, 99⟩,
⟨∗, 5⟩,
⟨∗, 1⟩,
⟨4, ∗⟩,
⟨12, 1⟩,
⟨3, 5⟩
⎥
⎥
⎥
⎥
⟨3, ∗⟩ ⎥
⎦
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Decomposition for
Applications
References
ki
N
i
⊕
Example
⎡
⎢
⎢
⎢
⟨2, 2⟩ ⎢
⎢
⎣
⟨5⟩,
⟨3⟩
⎤
⟨∗, 100⟩,⟨∗, 99⟩,
⟨∗, 5⟩,
⟨∗, 1⟩,
⟨4, ∗⟩,
⟨12, 1⟩,
⟨3, 5⟩
τ⟨k,t,f⟩ = τ − {k} + Nk,f (t) × {k − 1}
= τ − {k} + k(f(t) − 1) × {k − 1}
{
}
Lτ,f (t) ⩽ max 1 + Lτ⟨k,t,f⟩ ,f (t + 1)
def
k∈τ
⎥
⎥
⎥
⎥
⟨3, ∗⟩ ⎥
⎦
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
A Bounding Function
Mτ,f (t) = max{1 + Mτ⟨k,t,f⟩ ,f (t + 1)} .
def
k∈τ
Then for all τ and t
Lτ,f (t) ⩽ Mτ,f (t)
Proposition (Maximizing Strategy)
For all τ , ∅, t ⩾ 0, Mτ,f (t) = 1 + Mτ⟨min τ,t,f⟩ ,f (t + 1).
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
A Bounding Function
Proposition (Maximizing Strategy)
For all τ , ∅, t ⩾ 0, Mτ,f (t) = 1 + Mτ⟨min τ,t,f⟩ ,f (t + 1).
Example
t = 1, f(x + t) = x + t, τ = {|2, 1|}: compare
⟨0, 0⟩,⟨1⟩, ⟨0⟩
⟨0⟩,⟨1, 1⟩, ⟨1, 0⟩, ⟨0, 3⟩, ⟨0, 2⟩, ⟨0, 1⟩, ⟨0, 0⟩
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Complexity Results
Proposition
Let k, r ⩾ 1 be natural numbers and γ ⩾ 1 an ordinal. If
f is a monotone unary function of Fγ with
f(x) ⩾ max(1, x) for all x, then Mr×{k},f is in Fγ+k−1 .
Proposition
Let k, r ⩾ 1 be natural numbers and γ ⩾ 0 an ordinal
with γ + k ⩾ 3. Then Lr×{k},Fγ is bounded below by a
function which is not in Fγ+k−2 .
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Fast Growing Hierarchy: (Fα)α
Hierarchy of functions (Fα )α indexed by ordinals; we
only need the finite fragment.
def
F0 (x) = x + 1
def
Fn+1 (x) = Fnx+1 (x)
F1 (x) = 2x + 1
F2 (x) = (x + 1) ⋅ 2x+1 − 1
F3 is non elementary
def
Fω = λx.Fx (x) is non primitive-recursive
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Fast Growing Hierarchy: (Fα)α
Hierarchy of functions (Fα )α indexed by ordinals; we
only need the finite fragment.
def
F0 (x) = x + 1
def
Fn+1 (x) = Fnx+1 (x)
F1 (x) = 2x + 1
F2 (x) = (x + 1) ⋅ 2x+1 − 1
F3 is non elementary
def
Fω = λx.Fx (x) is non primitive-recursive
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Fast Growing Hierarchy: (Fα)α
Hierarchy of classes of functions: Fα is the closure of
{λx.0, λxy.x + y, λx̄.xi } ∪ {Fβ | β ⩽ α} under
substitution if h0 , h1 , . . . , hn ∈ Fα then so does f if
f(x1 , . . . , xn ) = h0 (h1 (x1 , . . . , xn ), . . . , hn (x1 , . . . , xn ))
limited recursion if h1 , h2 , h3 ∈ Fα then so does f if
f(0, x1 , . . . , xn ) = h1 (x1 , . . . , xn )
f(y + 1, x1 , . . . , xn ) = h2 (y, x1 , . . . , xn , f(y, x1 , . . . , xn ))
f(y, x1 , . . . , xn ) ⩽ h3 (y, x1 , . . . , xn )
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Fast Growing Hierarchy: (Fα)α
I
for k ⩾ 1, Fk ( Fk+1 , because Fk+1 < Fk
I
F0 = F1 : linear functions, like λx.x + 3 or λx.2x,
I
F2 : elementary functions, like λx.22 ,
x
I
2.
..
2
F3 : tetration functions, like λx. 2|{z} , etc.
x times
I
∪
k Fk :
primitive-recursive functions
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Affine Counter Systems
I
I
I
I
𝒞 = ⟨Q, k, δ, m0 ⟩
transitions (q, g, q ′ ) where g(x) = Ax + B an
affine function, A ∈ Nk×k , B ∈ Zk
m 0 ∈ Nk
generalize reset/transfer Petri nets, broadcast
protocols,. . .
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Termination for ACS
Given ⟨𝒞⟩ a k-ACS, does every run of 𝒞 terminate?
I
exponential control in F2
I
t < |m0 | < |𝒞|
I
upper bound: Fk+1
I
lower bound: Fk−O(1) (Schnoebelen, 2010)
I
if k is not fixed, non-primitive recursive, with an
upper bound in Fω
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Concluding Remarks
I
practical applications of wqo’s yield upper
bounds!
I
out-of-the-box upper bounds
I
“essentially” matching lower bounds for
decision problems on monotone counter
systems (lossy counter systems, reset or transfer
Petri nets)
I
the future: Higman’s Lemma
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Concluding Remarks
I
practical applications of wqo’s yield upper
bounds!
I
out-of-the-box upper bounds
I
“essentially” matching lower bounds for
decision problems on monotone counter
systems (lossy counter systems, reset or transfer
Petri nets)
I
the future: Higman’s Lemma
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
References: Upper Bounds for WQO
Dickson’s Lemma
McAloon, K., 1984. Petri nets and large finite sets. Theor. Comput. Sci., 32
(1–2):173–183. doi:10.1016/0304-3975(84)90029-X.
Clote, P., 1986. On the finite containment problem for Petri nets. Theor.
Comput. Sci., 43:99–105. doi:10.1016/0304-3975(86)90169-6.
Higman’s Lemma
Cichoń, E.A. and Tahhan Bittar, E., 1998. Ordinal recursive bounds for
Higman’s Theorem. Theor. Comput. Sci., 201(1–2):63–84.
doi:10.1016/S0304-3975(97)00009-1.
Kruskal’s Theorem
Weiermann, A., 1994. Complexity bounds for some finite forms of Kruskal’s
Theorem. J. Symb. Comput., 18(5):463–488. doi:10.1006/jsco.1994.1059.
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
References
Fast Growing Hierarchy
Löb, M. and Wainer, S., 1970. Hierarchies of number theoretic functions, I.
Arch. Math. Logic, 13:39–51. doi:10.1007/BF01967649.
WSTS
Abdulla, P.A., Čerāns, K., Jonsson, B., and Tsay, Y.K., 2000. Algorithmic
analysis of programs with well quasi-ordered domains. Inform. and
Comput., 160(1–2):109–127. doi:10.1006/inco.1999.2843.
Finkel, A. and Schnoebelen, Ph., 2001. Well-structured transition systems
everywhere! Theor. Comput. Sci., 256(1–2):63–92.
doi:10.1016/S0304-3975(00)00102-X.
Lower Bounds
Schnoebelen, Ph., 2010. Revisiting Ackermann-hardness for lossy counter
machines and reset Petri nets. In Hliněný, P. and Kučera, A., editors,
MFCS 2010, volume 6281 of LNCS, pages 616–628. Springer.
doi:10.1007/978-3-642-15155-2 54.
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Lower Bound
Specific sequence, bad for (Nk , ⩽lex ), of length ℓk,f (t).
Example
k = 2, t = 1, f(x) = x + 3:
i
xi [1]
xi [2]
f(i + t)
0
3
3
4
1
3
2
5
2
3
1
6
3
3
0
7
4
2
7
8
5 ⋅ ⋅ ⋅ 10 11 12 13 ⋅ ⋅ ⋅ 26 27 28 29 ⋅ ⋅ ⋅ 58 59
2 ⋅⋅⋅ 2 2 1 1 ⋅⋅⋅ 1 1 0 0 ⋅⋅⋅ 0 0
6 ⋅ ⋅ ⋅ 1 0 15 14 ⋅ ⋅ ⋅ 1 0 31 30 ⋅ ⋅ ⋅ 1 0
9 ⋅ ⋅ ⋅ 14 15 16 17 ⋅ ⋅ ⋅ 30 31 32 33 ⋅ ⋅ ⋅ 62 63
5 = 1 + 4 = 1 + ℓ1,f (1)
13 = 5 + 8 = 5 + ℓ1,f (5)
29 = 16 + 13 = 13 + ℓ1,f (13)
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
Lower Bound
References
Specific sequence, bad for (Nk , ⩽lex ), of length ℓk,f (t).
In general, on the k + 1th coordinate:
f(t) − 1 f(t) − 1 ⋅ ⋅ ⋅ f(t) − 1
|
{z
}
ℓk,f (t) times
⋅⋅⋅
f(t) − 2, f(t) − 2, ⋅ ⋅ ⋅ , f(t) − 2
|
{z
}
ℓk,f (ok,f (t)) times
0, 0, ⋅ ⋅ ⋅ , 0
( | {z )}
f(t)−1
ℓk,f ok,f (t) times
def
ok,f (t) = t + ℓk,f (t)
f(t)
(
)
∑
j−1
ℓk+1,f (t) =
ℓk,f ok,f (t)
j=1
Dickson’s Lemma
Decomposing Bad Sequences
Subrecursive Hierarchies
Applications
References
Lower Bound
Specific sequence, bad for (Nk , ⩽lex ), of length ℓk,f (t).
One can have ℓk,f (t) < L({k}, t): let f(x) = 2x and
t = 1,
⟨1, 1⟩,⟨1, 0⟩,⟨0, 5⟩,⟨0, 4⟩,⟨0, 3⟩,⟨0, 2⟩,⟨0, 1⟩, ⟨0, 0⟩
⟨1, 1⟩,⟨0, 3⟩,⟨0, 2⟩,⟨0, 1⟩,⟨9, 0⟩,⟨8, 0⟩,⟨7, 0⟩,⟨6, 0⟩,⟨5, 0⟩,. . . ,⟨0, 0⟩
ℓ2,f (1) = 8
L{2},f (1) ⩾ 14