Covering of ordinals Laurent Braud IGM, Univ. Paris-Est Liafa , 8 - - PowerPoint PPT Presentation

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covering of ordinals

Covering of ordinals Laurent Braud IGM, Univ. Paris-Est Liafa , 8 - - PowerPoint PPT Presentation

Feb 10, 2024 141 likes •796 views

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 Covering graphs 1 Ordinals MSO logic Fundamental sequence MSO-theory of

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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

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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

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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

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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

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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

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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 1 2 3 4 . . . ω

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 3 / 27

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Theorem (Cantor normal form, 1897)

For α < ε0, there is a unique decreasing sequence (γi) such that α = ωγ0 + · · · + ωγn. It is enough to define ordinals with addition

peration α → ωα.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 4 / 27

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Addition

α β

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 5 / 27

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Addition

α β α + β

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 5 / 27

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Addition

ω 2 . . .

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 5 / 27

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Addition

. . . ω + 2

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 5 / 27

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Addition

. . . ω + 2 2 + ω = ω . . .

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 5 / 27

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Addition

. . . ω + 2 2 + ω = ω . . .

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 5 / 27

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Exponentiation

ωα ≃ ({decreasing finite sequences of ordinals < α}, <lex)

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 6 / 27

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Exponentiation

ωα ≃ ({decreasing finite sequences of ordinals < α}, <lex) For instance, ω2 = 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

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Exponentiation

ωα ≃ ({decreasing finite sequences of ordinals < α}, <lex) For instance, ω2 = 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 < ε0 = ωε0. Notation : ω ⇑ n = ωω...ω n.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 6 / 27

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Exponentiation

ωα ≃ ({decreasing finite sequences of ordinals < α}, <lex) For instance, ω2 = ω + ω + ω + ω . . . 2 = 0 → 1 decreasing sequences = (1, . . . , 1, 0, . . . , 0) . . . . . . . . . . . . 1 2 ω ω + 1 ω + 2 ω.2 ω.2 + 1 ω.2 + 2 We restrict to ordinals < ε0 = ωε0. Notation : ω ⇑ n = ωω...ω n.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 6 / 27

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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

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Monadic second-order logic

first-order variables x,y. . . the structure : < set variables X,Y. . . and formulas x ∈ Y ∧, ∨, ¬, ∀, ∃ strict order antisymmetry ∀p, q(¬(p < q ∧ q < p)) 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

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MSO-logics and ordinals [B¨ uchi, 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

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MSO-logics and ordinals [B¨ uchi, Shelah]

MTh(S) = {ϕ | S | = ϕ}.

Theorem

For any countable α, MTh(α) is decidable. α = ωγ0 + · · · + ωγk

β

+ ωγk+1 + ωγn

δ

where γ0, . . . , γk ≥ ω γk+1, . . . , γn < ω

Theorem

MTh(α) only depends on δ and whether β > 0. MTh(ωω) = MTh(ωωω). . .

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 8 / 27

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Simplifying graphs

1 2 3 . . . ω ω + 1

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 9 / 27

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Simplifying graphs

1 2 3 . . . ω ω + 1 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

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Fundamental sequence [Cantor]

Let α = ωγ0 + · · · + ωγk−1

δ

+ωγk If γn = 0, α a limit ordinal. There is an ω-sequence of limit α. α[n] = δ + ωγ′.(n + 1) if γk = γ′ + 1 δ + ωγk[n]

therwise.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 10 / 27

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Fundamental sequence [Cantor]

Let α = ωγ0 + · · · + ωγk−1

δ

+ωγk If γn = 0, α a limit ordinal. There is an ω-sequence of limit α. α[n] = δ + ωγ′.(n + 1) if γk = γ′ + 1 δ + ωγk[n]

therwise.

Successor ordinals are a degenerate case : α ≺ β if α = β[n] α + 1 = β.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 10 / 27

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Covering graph of ω + 2

ω[n] = n + 1 Gω+2 1 2 3 . . . ω ω + 1

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 11 / 27

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Covering graph of ω2 + 1

ω2[n] = ω.(n + 1) Gω2+1 1 2 3 . . . ω ω + 1 . . . ω.2 ω.2 + 1 . . . ω.3 . . . . . . ω2

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 12 / 27

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Covering graph of ωω

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 13 / 27

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First result

Proposition

< is the transitive closure of ≺.

Theorem

For α, β < ε0, if α = β, then MTh(Gα) = MTh(Gβ).

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 14 / 27

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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

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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

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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

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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

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Pushdown hierarchy

Many definitions : higher-order pushdown automata [M¨ uller-Schupp, Carayol-W¨

hrle],

unfolding [Caucal] or treegraph [Carayol-W¨

hrle]

+ MSO-interpretations or rational mappings prefix-recognizable relations [Caucal-Knapik,Carayol], term grammars [Dam, Knapik-Niwi´ nski-Urzyczyn]. . .

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 17 / 27

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Pushdown hierarchy

Many definitions : higher-order pushdown automata [M¨ uller-Schupp, Carayol-W¨

hrle],

unfolding [Caucal] or treegraph [Carayol-W¨

hrle]

+ MSO-interpretations or rational mappings prefix-recognizable relations [Caucal-Knapik,Carayol], term grammars [Dam, Knapik-Niwi´ nski-Urzyczyn]. . .

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 17 / 27

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Pushdown hierarchy

Graph0 (finite)

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 18 / 27

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Pushdown hierarchy

Graph0 (finite) Graph1 (prefix-recognizable) I◦Treegraph

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 18 / 27

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Pushdown hierarchy

Graph0 (finite) Graph1 (prefix-recognizable) I◦Treegraph Graph2 I◦Treegraph

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 18 / 27

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Pushdown hierarchy

Graph0 (finite) Graph1 (prefix-recognizable) I◦Treegraph Graph2 I◦Treegraph . . . Each graph in Graphn has a decidable MSO-theory.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 18 / 27

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MSO-interpretations

MSO-interpretation : I = {ϕa(x, y)}a∈Γ where ϕa is a formula over G. I(G) = {x

a

− → y | G | = ϕa(x, y)}

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 19 / 27

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MSO-interpretations

MSO-interpretation : I = {ϕa(x, y)}a∈Γ where ϕa is a formula over G. I(G) = {x

a

− → y | G | = ϕa(x, y)} Exemple : transitive closure. ϕ<(x, y) := ∀X (x ∈ X ∧ closed≺(X) ⇒ y ∈ X) ∧ x = y closed≺(X) := ∀z ∈ X, ∀z′(z ≺ z′ ⇒ z′ ∈ X)

Proposition

For α < ε0, there is an interpretation I(Gα) = α.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 19 / 27

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MSO-interpretations

MSO-interpretation : I = {ϕa(x, y)}a∈Γ where ϕa is a formula over G. I(G) = {x

a

− → y | G | = ϕa(x, y)} Exemple : transitive closure. ϕ<(x, y) := ∀X (x ∈ X ∧ closed≺(X) ⇒ y ∈ X) ∧ x = y closed≺(X) := ∀z ∈ X, ∀z′(z ≺ z′ ⇒ z′ ∈ X)

Proposition

For α < ε0, there is an interpretation I(Gα) = α.

Proposition

There is an interpretation from Gα to Gβ with β ≤ α < ε0.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 19 / 27

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Treegraph

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 20 / 27

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Treegraph

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 20 / 27

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Main result

Theorem (Bloom, ´ Esik)

If α < ω ⇑ 3 = ωωω then α ∈ Graph2.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 21 / 27

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Main result

Theorem (Bloom, ´ Esik)

If α < ω ⇑ 3 = ωωω 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

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Main result

Theorem (Bloom, ´ Esik)

If α < ω ⇑ 3 = ωωω 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

Gε0 does not belong to the hierarchy.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 21 / 27

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Treegraph on covering graphs

Gω . . .

Proposition

There is a interpretation I such that I ◦ Treegraph(Gα) = Gωα for each α.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 22 / 27

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Treegraph on covering graphs

. . . . . .

# # #

. . .

# # #

. . .

# # #

Proposition

There is a interpretation I such that I ◦ Treegraph(Gα) = Gωα for each α.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 22 / 27

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Treegraph on covering graphs

() . . .

(0)

. . .

(0, 0) (0, 1) # # #

(1)

. . .

(1, 0) (1, 1) # # #

(2)

. . .

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

Proposition

There is a interpretation I such that I ◦ Treegraph(Gα) = Gωα for each α.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 22 / 27

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Treegraph on covering graphs

() . . .

(0)

. . .

(0, 0) # # #

(1)

. . .

(1, 0) (1, 1) # # #

(2)

. . .

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

Proposition

There is a interpretation I such that I ◦ Treegraph(Gα) = Gωα for each α.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 22 / 27

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Treegraph on covering graphs

. . .

1

. . .

2 # # #

ω

. . .

ω + 1 ω.2 # # #

ω2

. . .

ω2 + 1 ω2 + ω # # #

Proposition

There is a interpretation I such that I ◦ Treegraph(Gα) = Gωα for each α.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 22 / 27

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Treegraph on covering graphs

Gωω

1

2

ω

ω + 1 ω.2

ω2

ω2 + 1 ω2 + ω

Proposition

There is a interpretation I such that I ◦ Treegraph(Gα) = Gωα for each α.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 22 / 27

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Higher-order stacks [Carayol]

b a → a b a a b a → b a pusha pop1

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 23 / 27

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Higher-order stacks [Carayol]

b a → a b a a b a → b a pusha pop1 c b a b a → c b a b a a b a c b a b a c a → c b a b a copy2 pop2

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 23 / 27

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Higher-order stacks [Carayol]

Ops1 = {pusha, pop1} for n > 1, Opsn = {copyn, popn} ∪ Opsn−1 Opsn is a mono¨ ıd for composition. Let L ∈ Ops∗

n.

A graph in Graphn can be represented with vertices in Stacksn. s0 . . . sk

L

− → s′

0 . . . s′ k′

with si, s′

i ∈ Stacksn−1.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 24 / 27

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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 ∈ pop+

1 (sβ) ⇐

⇒ sβ ∈ push+(sα) Let dec1 = pop+

1 , inc1 = push+.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 25 / 27

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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 ∈ pop+

1 (sβ) ⇐

⇒ sβ ∈ push+(sα) Let dec1 = pop+

1 , inc1 = push+.

For infinite α, α = ωγ0 + · · · + ωγk. 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

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For n > 1, if α < β, sα = sγ0 . . . sγi . . . sγk

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 26 / 27

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For n > 1, if α < β, sα = sγ0 . . . sγi . . . sγk either sβ = sγ0 . . . sγi . . . sγksγk+1 . . . sγh sβ ∈ (copyn.(id + decn−1))+ (sα)

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 26 / 27

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For n > 1, if α < β, sα = sγ0 . . . sγi . . . sγk either sβ = sγ0 . . . sγi . . . sγksγk+1 . . . sγh sβ ∈ (copyn.(id + decn−1))+ (sα) = tail+

n (sα)

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 26 / 27

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For n > 1, if α < β, sα = sγ0 . . . sγi . . . sγk either sβ = sγ0 . . . sγi . . . sγksγk+1 . . . sγh sβ ∈ (copyn.(id + decn−1))+ (sα) = tail+

n (sα)

r sβ

= sγ0 . . . sγi−1sγ′

i>γi . . . sγ′ k′

sβ ∈ pop∗

n.incn−1.(copyn.(id + decn−1))∗ (sα)

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 26 / 27

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For n > 1, if α < β, sα = sγ0 . . . sγi . . . sγk either sβ = sγ0 . . . sγi . . . sγksγk+1 . . . sγh sβ ∈ (copyn.(id + decn−1))+ (sα) = tail+

n (sα)

r sβ

= sγ0 . . . sγi−1sγ′

i>γi . . . sγ′ k′

sβ ∈ pop∗

n.incn−1.(copyn.(id + decn−1))∗ (sα)

incn = [pop∗

n.incn−1 + tailn].tail∗ n

decn = pop∗

n.[popn + decn−1].tail∗ n

Theorem

If (α, <, >) is in Graphn, then (ωα, <, >) is in Graphn+1.

Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa, 8 January 2010 26 / 27

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Future work

Obtain the stronger result α < ω ⇑(n + 1) ⇔ α ∈ Graphn. Is there a definition of fundamental sequence so that the result α = β ⇒ MTh(Gα) = MTh(Gβ) remains true for further ordinals? Ordinals greater than ωω 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