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

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 covering graphs Pushdown hierarchy 2 Definition Iteration of exponentiation Higher-order stacks presentation 3 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 . . . . ω 0 1 2 3 4 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 ( γ i ) such that α = ω γ 0 + · · · + ω γ n . It is enough to define ordinals with addition operation α �→ ω α . Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa , 8 January 2010 4 / 27

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

Addition α + β α β 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 . . . 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 . . . = 0 → 1 2 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 . . . = 0 → 1 2 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 . Notation : ω ⇑ n = ω 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 . . . = 0 → 1 2 decreasing sequences = ( 1, . . . , 1, 0, . . . , 0 ) . . . 0 1 2 . . . ω ω + 1 ω + 2 . . . ω .2 + 1 ω .2 + 2 ω .2 . . . We restrict to ordinals < ε 0 = ω ε 0 . Notation : ω ⇑ n = ω ω ... ω � n . Laurent Braud (IGM, Univ. Paris-Est) 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 ) ∀ p , q ( p < q ∨ q < p ∨ p = q ) total order ∀ X , ∃ z ∈ X ⇒ well order ∃ 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 [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

MSO-logics and ordinals [B¨ uchi, Shelah] MTh ( S ) = { ϕ | S | = ϕ } . Theorem For any countable α , MTh ( α ) is decidable. � γ 0 , . . . , γ k ≥ ω α = ω γ 0 + · · · + ω γ k + ω γ k + 1 + ω γ n where γ k + 1 , . . . , γ n < ω � �� � � �� � β δ Theorem MTh ( α ) only depends on δ and whether β > 0 . MTh ( ω ω ) = MTh ( ω ω ω ) . . . Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa , 8 January 2010 8 / 27

Simplifying graphs 3 . . . 0 1 2 ω ω + 1 Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa , 8 January 2010 9 / 27

Simplifying graphs 3 . . . 0 1 2 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

Fundamental sequence [Cantor] Let α = ω γ 0 + · · · + ω γ k − 1 + ω γ k � �� � δ If γ n � = 0, α a limit ordinal. There is an ω -sequence of limit α . � δ + ω γ ′ . ( n + 1 ) if γ k = γ ′ + 1 α [ n ] = δ + ω γ k [ n ] otherwise. Laurent Braud (IGM, Univ. Paris-Est) Covering of ordinals Liafa , 8 January 2010 10 / 27

Fundamental sequence [Cantor] Let α = ω γ 0 + · · · + ω γ k − 1 + ω γ k � �� � δ If γ n � = 0, α a limit ordinal. There is an ω -sequence of limit α . � δ + ω γ ′ . ( n + 1 ) if γ k = γ ′ + 1 α [ n ] = δ + ω γ k [ 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 3 . . . 0 1 2 G ω + 2 ω ω + 1 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 ) 3 . . . 0 1 2 G ω 2 + 1 ω + 1 . . . ω . . . ω .2 ω .2 + 1 . . . ω .3 . . . ω 2 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 � = β , then MTh ( G α ) � = 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 G α 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 G α 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 [M¨ uller-Schupp, Carayol-W¨ ohrle], unfolding [Caucal] or treegraph [Carayol-W¨ ohrle] + 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

Pushdown hierarchy Many definitions : higher-order pushdown automata [M¨ uller-Schupp, Carayol-W¨ ohrle], unfolding [Caucal] or treegraph [Carayol-W¨ ohrle] + 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