Covering of ordinals
Laurent Braud
IGM, Univ. Paris-Est
Liafa, 8 January 2010
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
1 / 27
1
Covering graphs
Ordinals
MSO logic
Fundamental sequence
MSO-theory of covering graphs
2
Pushdown hierarchy
Definition
Iteration of exponentiation
3
Higher-order stacks presentation
Definition
Ordinal presentation
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
2 / 27
Ordinals
An ordinal is a well-ordering, i.e. an order where
each set has a smallest element
each strictly decreasing sequence is finite
During this talk, we confuse ordinal with graph of the order.
5
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
3 / 27
Ordinals
An ordinal is a well-ordering, i.e. an order where
each set has a smallest element
each strictly decreasing sequence is finite
During this talk, we confuse ordinal with graph of the order.
...
ω
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
3 / 27
Ordinals
An ordinal is a well-ordering, i.e. an order where
each set has a smallest element
each strictly decreasing sequence is finite
During this talk, we confuse ordinal with graph of the order.
...
ω+1
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
3 / 27
Ordinals
An ordinal is a well-ordering, i.e. an order where
each set has a smallest element
each strictly decreasing sequence is finite
During this talk, we confuse ordinal with graph of the order.
0
1
2
3
4
...
ω
ω+1
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
3 / 27
Theorem (Cantor normal form, 1897)
For α < ε 0 , there is a unique decreasing sequence (γi ) such that
α = ω γ0 + · · · + ω γn .
It is enough to define ordinals with
addition
operation α 7→ ω α .
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
4 / 27
Addition
α
Laurent Braud (IGM, Univ. Paris-Est)
β
Covering of ordinals
Liafa, 8 January 2010
5 / 27
Addition
α+β
α
Laurent Braud (IGM, Univ. Paris-Est)
β
Covering of ordinals
Liafa, 8 January 2010
5 / 27
Addition
ω
2
...
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
5 / 27
Addition
ω+2
...
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
5 / 27
Addition
ω+2
...
2+ω = ω
...
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
5 / 27
Addition
ω+2
...
2+ω = ω
...
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
5 / 27
Exponentiation
ω α ≃ ({decreasing finite sequences of ordinals < α}, <lex )
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
6 / 27
Exponentiation
ω α ≃ ({decreasing finite sequences of ordinals < α}, <lex )
For instance, ω 2 = ω + ω + ω + ω . . .
2 = 0→1
decreasing sequences = (1, . . . , 1, 0, . . . , 0)
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
6 / 27
Exponentiation
ω α ≃ ({decreasing finite sequences of ordinals < α}, <lex )
For instance, ω 2 = ω + ω + ω + ω . . .
2 = 0→1
decreasing sequences = (1, . . . , 1, 0, . . . , 0)
()
(0)
(0,0) . . .
(1)
(1,0)
(1,0,0) . . .
(1,1)
(1,1,0)
(1,1,0,0) . . .
...
We restrict to ordinals < ε 0 = ω ε 0 .
o
...ω
Notation : ω ⇑ n = ω ω
n.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
6 / 27
Exponentiation
ω α ≃ ({decreasing finite sequences of ordinals < α}, <lex )
For instance, ω 2 = ω + ω + ω + ω . . .
2 = 0→1
decreasing sequences = (1, . . . , 1, 0, . . . , 0)
0
1
ω
ω+1
ω.2 + 1
ω.2
...
We restrict to ordinals < ε 0 = ω ε 0 .
o
...ω
Notation : ω ⇑ n = ω ω
n.
Laurent Braud (IGM, Univ. Paris-Est)
2
...
ω +2 ...
ω.2 + 2 . . .
Covering of ordinals
Liafa, 8 January 2010
6 / 27
Monadic second-order logic
first-order variables x,y. . .
the structure : <
set variables X,Y. . . and formulas x ∈ Y
∧, ∨, ¬, ∀, ∃
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
7 / 27
Monadic second-order logic
first-order variables x,y. . .
the structure : <
set variables X,Y. . . and formulas x ∈ Y
∧, ∨, ¬, ∀, ∃
antisymmetry
∀p, q (¬(p < q ∧ q < p ))
strict order
transitivity
∀p, q, r ((p < q ∧ (q < r ) ⇒ p < r )
total order
∀p, q (p < q ∨ q < p ∨ p = q )
well order
∀X , ∃z ∈ X ⇒
∃x (x ∈ X ∧ ∀y (y ∈ X ⇒ (x < y ∨ x = y )))
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
7 / 27
MSO-logics and ordinals [Büchi, Shelah]
MTh(S ) = { ϕ | S |= ϕ}.
Theorem
For any countable α, MTh(α) is decidable.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
8 / 27
MSO-logics and ordinals [Büchi, Shelah]
MTh(S ) = { ϕ | S |= ϕ}.
Theorem
For any countable α, MTh(α) is decidable.
α = |ω
γ0
γk
+ ·{z
· · + ω } + |ω
γk +1
β
γn
{z+ ω } where
δ
γ0 , . . . , γk ≥ ω
γk +1 , . . . , γn < ω
Theorem
MTh(α) only depends on δ and whether β > 0.
ω
MTh(ω ω ) = MTh(ω ω ). . .
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
8 / 27
Simplifying graphs
0
1
ω
ω+1
Laurent Braud (IGM, Univ. Paris-Est)
2
Covering of ordinals
3 ...
Liafa, 8 January 2010
9 / 27
Simplifying graphs
0
1
ω
ω+1
2
3 ...
Each countable limit ordinal is limit of an ω-sequence. How to define this
sequence in fixed way?
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
9 / 27
Fundamental sequence [Cantor]
γ0
Let α = ω
· + ω γk −}1 +ω γk
| + · ·{z
δ
If γn 6= 0, α a limit ordinal. There is an ω-sequence of limit α.
′
δ + ω γ .(n + 1) if γk = γ′ + 1
α [n ] =
δ + ω γk [n]
otherwise.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
10 / 27
Fundamental sequence [Cantor]
γ0
Let α = ω
· + ω γk −}1 +ω γk
| + · ·{z
δ
If γn 6= 0, α a limit ordinal. There is an ω-sequence of limit α.
′
δ + ω γ .(n + 1) if γk = γ′ + 1
α [n ] =
δ + ω γk [n]
otherwise.
Successor ordinals are a degenerate case :
α = β [n ]
α ≺ β if
α + 1 = β.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
10 / 27
Covering graph of ω + 2
ω [n ] = n + 1
0
1
ω
ω+1
2
3 ...
G ω +2
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
11 / 27
Covering graph of ω 2 + 1
ω 2 [n] = ω.(n + 1)
0
1
2
3 ...
G ω 2 +1
ω
ω +1 ...
ω.2 + 1 . . .
ω.2
ω.3 . . .
...
ω2
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
12 / 27
Covering graph of ω ω
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
13 / 27
First result
Proposition
< is the transitive closure of ≺.
Theorem
For α, β < ε 0 , if α 6= β, then MTh(Gα ) 6= MTh(G β ).
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
14 / 27
Proof sketch
Proposition
For α ≤ ω ⇑ n, the out-degree of Gα is at most n.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
15 / 27
Proof sketch
Proposition
For α ≤ ω ⇑ n, the out-degree of Gα is at most n.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
15 / 27
Proof sketch
Let σ be the sequence
0 ∈ σ,
β ∈ σ ⇒ if β′ is the largest s.t. β ≺ β′ , then β′ ∈ σ.
Degree word : sequence of out-degrees of this sequence.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
16 / 27
Proof sketch
Let σ be the sequence
0 ∈ σ,
β ∈ σ ⇒ if β′ is the largest s.t. β ≺ β′ , then β′ ∈ σ.
Degree word : sequence of out-degrees of this sequence.
Proposition
The degree word is
ultimately periodic,
MSO-definable,
injective.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
16 / 27
Pushdown hierarchy
Many definitions :
higher-order pushdown automata [Müller-Schupp, Carayol-Wöhrle],
unfolding [Caucal] or treegraph [Carayol-Wöhrle]
+ MSO-interpretations or rational mappings
prefix-recognizable relations [Caucal-Knapik,Carayol],
term grammars [Dam, Knapik-Niwiński-Urzyczyn]. . .
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
17 / 27
Pushdown hierarchy
Many definitions :
higher-order pushdown automata [Müller-Schupp, Carayol-Wöhrle],
unfolding [Caucal] or treegraph [Carayol-Wöhrle]
+ MSO-interpretations or rational mappings
prefix-recognizable relations [Caucal-Knapik,Carayol],
term grammars [Dam, Knapik-Niwiński-Urzyczyn]. . .
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
17 / 27
Pushdown hierarchy
Graph0 (finite)
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
18 / 27
Pushdown hierarchy
Graph0 (finite)
I◦Treegraph
Graph1 (prefix-recognizable)
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
18 / 27
Pushdown hierarchy
Graph0 (finite)
I◦Treegraph
Graph1 (prefix-recognizable)
I◦Treegraph
Graph2
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
18 / 27
Pushdown hierarchy
Graph0 (finite)
I◦Treegraph
Graph1 (prefix-recognizable)
I◦Treegraph
Graph2
...
Each graph in Graphn has a decidable MSO-theory.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
18 / 27
MSO-interpretations
MSO-interpretation : I = { ϕa (x, y )}a∈Γ where ϕa is a formula over G .
a
I(G ) = {x −→ y | G |= ϕa (x, y )}
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
19 / 27
MSO-interpretations
MSO-interpretation : I = { ϕa (x, y )}a∈Γ where ϕa is a formula over G .
a
I(G ) = {x −→ y | G |= ϕa (x, y )}
Exemple : transitive closure.
ϕ< (x, y ) := ∀X (x ∈ X ∧ closed≺ (X ) ⇒ y ∈ X ) ∧ x 6= y
closed≺ (X ) := ∀z ∈ X , ∀z ′ (z ≺ z ′ ⇒ z ′ ∈ X )
Proposition
For α < ε 0 , there is an interpretation I(Gα ) = α.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
19 / 27
MSO-interpretations
MSO-interpretation : I = { ϕa (x, y )}a∈Γ where ϕa is a formula over G .
a
I(G ) = {x −→ y | G |= ϕa (x, y )}
Exemple : transitive closure.
ϕ< (x, y ) := ∀X (x ∈ X ∧ closed≺ (X ) ⇒ y ∈ X ) ∧ x 6= y
closed≺ (X ) := ∀z ∈ X , ∀z ′ (z ≺ z ′ ⇒ z ′ ∈ X )
Proposition
For α < ε 0 , there is an interpretation I(Gα ) = α.
Proposition
There is an interpretation from Gα to G β with β ≤ α < ε 0 .
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
19 / 27
Treegraph
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
20 / 27
Treegraph
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
20 / 27
Main result
Theorem (Bloom, Ésik)
ω
If α < ω ⇑ 3 = ω ω then α ∈ Graph2 .
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
21 / 27
Main result
Theorem (Bloom, Ésik)
ω
If α < ω ⇑ 3 = ω ω then α ∈ Graph2 .
Proposition
For α < ω ⇑ n, Gα ∈ Graphn−1 .
Theorem
For α < ω ⇑ n, α ∈ Graphn−1 .
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
21 / 27
Main result
Theorem (Bloom, Ésik)
ω
If α < ω ⇑ 3 = ω ω then α ∈ Graph2 .
Proposition
For α < ω ⇑ n, Gα ∈ Graphn−1 .
Theorem
For α < ω ⇑ n, α ∈ Graphn−1 .
Proposition
For n > 0 and α ≥ ω ⇑(3n + 1), Gα ∈
/ Graphn .
Corollary
Gε 0 does not belong to the hierarchy.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
21 / 27
Treegraph on covering graphs
...
Gω
Proposition
There is a interpretation I such that I ◦ Treegraph(Gα ) = Gω α for each α.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
22 / 27
Treegraph on covering graphs
...
#
#
#
...
#
#
...
#
#
...
#
#
Proposition
There is a interpretation I such that I ◦ Treegraph(Gα ) = Gω α for each α.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
22 / 27
Treegraph on covering graphs
...
()
...
(0)
#
(0, 0)
#
#
#
#
(0, 1)
...
(1)
#
(1, 0)
(2)
#
(1, 1)
#
(2, 0)
...
#
(2, 1)
Proposition
There is a interpretation I such that I ◦ Treegraph(Gα ) = Gω α for each α.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
22 / 27
Treegraph on covering graphs
...
()
...
(0)
#
#
#
#
#
(0, 0)
...
(1)
#
(1, 0)
(2)
#
(1, 1)
#
(2, 0)
...
#
(2, 1)
Proposition
There is a interpretation I such that I ◦ Treegraph(Gα ) = Gω α for each α.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
22 / 27
Treegraph on covering graphs
...
0
...
1
#
#
#
#
#
2
...
ω
#
ω+1
ω2
#
ω.2
#
ω2 + 1
...
#
ω2 + ω
Proposition
There is a interpretation I such that I ◦ Treegraph(Gα ) = Gω α for each α.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
22 / 27
Treegraph on covering graphs
Gω ω
0
2
ω2
ω
1
ω+1
ω.2
ω2 + 1
ω2 + ω
Proposition
There is a interpretation I such that I ◦ Treegraph(Gα ) = Gω α for each α.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
22 / 27
Higher-order stacks [Carayol]
a
b
→ b
a
a
pusha
Laurent Braud (IGM, Univ. Paris-Est)
a
b
b →
a
a
pop1
Covering of ordinals
Liafa, 8 January 2010
23 / 27
Higher-order stacks [Carayol]
a
b
→ b
a
a
pusha
c
b
a
b
a
→
c
b
a
b
b →
a
a
pop1
a
b
a
a
b
a
c
b
copy2
Laurent Braud (IGM, Univ. Paris-Est)
a
b
a
c
a
→
c
b
a
b
a
pop2
Covering of ordinals
Liafa, 8 January 2010
23 / 27
Higher-order stacks [Carayol]
Ops1 = {pusha , pop1 }
for n > 1, Opsn = {copyn , popn } ∪ Opsn−1
Opsn is a monoı̈d for composition. Let L ∈ Opsn∗ .
A graph in Graphn can be represented with vertices in Stacksn .
L
s0 . . . sk −
→ s0′ . . . sk′ ′
with si , si′ ∈ Stacksn−1 .
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
24 / 27
Presentation of ordinals
Integers are represented by stacks over a unary alphabet.
push(i ) = i + 1
pop1 (i + 1) = i
For finite integers,
α < β ⇐⇒ sα ∈ pop1+ (sβ ) ⇐⇒ sβ ∈ push+ (sα )
Let dec1 = pop1+ , inc1 = push+ .
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
25 / 27
Presentation of ordinals
Integers are represented by stacks over a unary alphabet.
push(i ) = i + 1
pop1 (i + 1) = i
For finite integers,
α < β ⇐⇒ sα ∈ pop1+ (sβ ) ⇐⇒ sβ ∈ push+ (sα )
Let dec1 = pop1+ , inc1 = push+ .
For infinite α,
α = ω γ0 + · · · + ω γk .
Each γi is representable by sγi ∈ Stacksn−1
sα = sγ0 . . . sγk
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
25 / 27
For n > 1, if α < β,
sα =
Laurent Braud (IGM, Univ. Paris-Est)
sγ0 . . . sγi . . . sγk
Covering of ordinals
Liafa, 8 January 2010
26 / 27
For n > 1, if α < β,
sα =
either sβ =
sβ
sγ0 . . . sγi . . . sγk
sγ0 . . . sγi . . . sγk sγk +1 . . . sγh
∈ (copyn .(id + decn−1 ))+ (sα )
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
26 / 27
For n > 1, if α < β,
sα =
either sβ =
sβ
sγ0 . . . sγi . . . sγk
sγ0 . . . sγi . . . sγk sγk +1 . . . sγh
∈ (copyn .(id + decn−1 ))+ (sα ) = tailn+ (sα )
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
26 / 27
For n > 1, if α < β,
sα =
either sβ =
sβ
sγ0 . . . sγi . . . sγk sγk +1 . . . sγh
∈ (copyn .(id + decn−1 ))+ (sα ) = tailn+ (sα )
or sβ =
sβ
sγ0 . . . sγi . . . sγk
sγ0 . . . sγi −1 sγi′ >γi . . . sγ′ ′
k
∈ popn∗ .incn−1 .(copyn .(id + decn−1 ))∗ (sα )
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
26 / 27
For n > 1, if α < β,
sα =
either sβ =
sβ
sγ0 . . . sγi . . . sγk sγk +1 . . . sγh
∈ (copyn .(id + decn−1 ))+ (sα ) = tailn+ (sα )
or sβ =
sβ
sγ0 . . . sγi . . . sγk
sγ0 . . . sγi −1 sγi′ >γi . . . sγ′ ′
k
∈ popn∗ .incn−1 .(copyn .(id + decn−1 ))∗ (sα )
incn = [popn∗ .incn−1 + tailn ].tailn∗
decn = popn∗ .[popn + decn−1 ].tailn∗
Theorem
If (α, <, >) is in Graphn , then (ω α , <, >) is in Graphn+1 .
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
26 / 27
Future work
Obtain the stronger result
α < ω ⇑(n + 1) ⇔ α ∈ Graphn .
Is there a definition of fundamental sequence so that the result
α 6= β ⇒ MTh(Gα ) 6= MTh(G β )
remains true for further ordinals?
Ordinals greater than ω ω are not selectable [Rabinovich, Shomrat].
Are covering graphs selectable?
Extend the method to other linear orderings.
Laurent Braud (IGM, Univ. Paris-Est)
Covering of ordinals
Liafa, 8 January 2010
27 / 27