Structural Limits FO-limits? Going further Modeling Limits Jaroslav Nešetřil Patrice Ossona de Mendez Charles University CAMS, CNRS/EHESS LIA STRUCO Praha, Czech Republic Paris, France — IHP 2018 —
Structural Limits FO-limits? Going further Limits of Structures
Structural Limits FO-limits? Going further Classical Graph Limits Left limits Local limits Dense ( m = Ω( n 2 ) ) Sparse (bounded ∆ ) assumption Isomorphism type of Isomorphism type of sample G [ X 1 , . . . , X p ] B r ( G, X ) Exchangeable random Unimodular distribution graph distribution (Aldous ’81, Hoover ’79) (Benjamini–Schramm ’01) Graphon Graphing analytic measurable W : [0 , 1] 2 → [0 , 1] limit d measure preserving involutions object (Lovász et al. ’06) (Elek ’07)
Structural Limits FO-limits? Going further Fibonacci Sequence G 1 G 2 G 3 G 4 G 5 G 6 G 7
Structural Limits FO-limits? Going further Fibonacci Sequence 1 τ 1 τ 2 G 11
Structural Limits FO-limits? Going further Fibonacci Sequence Local Limit 1 τ In R / Z : x ⇒ x ≡ y ± 1 1 τ 2 x ∼ y ⇐ y 2 τ 2 0 , 1 � � Black( x ) ⇐ ⇒ x ∈ τ y 1 Graphing: two views D measure preserving Borel graph + Mass Transport Borel involutions � � f 1 , . . . , f d deg B ( v ) d v = deg A ( v ) d v A B
Structural Limits FO-limits? Going further Grids x ± α x ∈ R / Z x �→ x ± β
Structural Limits FO-limits? Going further High-girth Regular Graphs ( x, y ) ± ( α, 0) ( x, y ) ∈ ( R / Z ) × ( R / Z ) ( x, y ) �→ ( x, y ) ± ( β, β )
Structural Limits FO-limits? Going further How to handle unbounded degrees? Instead of the isomorphism type of the radius d ball around v , consider the local type of v for d -local formulas.
Structural Limits FO-limits? Going further Local Formulas Definition A formula φ is local if there exists r such that satisfaction of φ only depends on the r -neighborhood of the free variables: G | = φ ( v 1 , . . . , v p ) ⇐ ⇒ G [ N r ( { v 1 , . . . , v p } )] | = φ ( v 1 , . . . , v p ) . Definition A sequence ( G n ) is FO local -convergent if, for every local formula 1 φ ( x ) with one free variable, the probability that G n satisfies φ ( v ) for random v ∈ V ( G n ) converges as n → ∞ . That is: convergence of � φ, G n � := |{ v : G n | = φ ( v ) }| . | G n |
Structural Limits FO-limits? Going further Stone pairing Let φ be a first-order formula with p free variables and let G be a graph (or a structure with countable signature). The Stone pairing of φ and G is � φ, G � = Pr( G | = φ ( X 1 , . . . , X p )) , for independently and uniformly distributed X i ∈ G . That is: � φ, G � = | φ ( G ) | | G | p . Remark If φ is a sentence then � φ, G � ∈ { 0 , 1 } .
Structural Limits FO-limits? Going further Structural Limits Definition A sequence ( G n ) is X -convergent if, for every φ ∈ X , the sequence � φ, G 1 � , . . . , � φ, G n � , . . . is convergent. FO 0 Sentences Elementary limits QF Quantifier free formulas Left limits FO local Local formulas with 1 free variable Local limits 1 FO 1 Formulas with 1 free variable FO 1 -limits FO local FO local -limits Local formulas FO All first-order formulas FO-limits Remark (Sequential compactness) Every sequence has an X -convergent subsequence.
Structural Limits FO-limits? Going further Modelings Definition A totally Borel graph is a graph on a standard Borel space s.t. every first-order definable set is Borel. A modeling A is totally Borel graph with a probability measure ν A . The Stone pairing extends to modelings: � φ, A � = ν ⊗ p A ( φ ( A )) = Pr ν A [ A | = φ ( X 1 , . . . , X p )] .
Structural Limits FO-limits? Going further Modeling FO local -Limits 1 Theorem (Nešetřil, OdM 2016+) Every FO local -convergent sequence ( G n ) n ∈ N of graphs (or struc- 1 tures with countable signature) has a modeling FO local -limit L . 1
Structural Limits FO-limits? Going further Modeling FO 1 -Limits Theorem (Nešetřil, OdM 2016+) Every FO local -convergent sequence ( G n ) n ∈ N of graphs (or struc- 1 tures with countable signature) has a modeling FO local -limit L . 1 +Sentences: Theorem (Nešetřil, OdM 2016+) Every FO 1 -convergent sequence ( G n ) n ∈ N of graphs (or structures with countable signature) has a modeling FO 1 -limit L .
Structural Limits FO-limits? Going further Modeling FO ∗ 1 -Limits Theorem (Nešetřil, OdM 2016+) Every FO 1 -convergent sequence ( G n ) n ∈ N of graphs (or structures with countable signature) has a modeling FO 1 -limit L . + ∀ φ ∈ FO s.t. ( � φ, G n � ) n ∈ N converges it also holds � φ, L � = 0 ⇐ ⇒ n →∞ � φ, G n � = 0 . lim We denote this by FO ∗ 1 G n − − → L .
Structural Limits FO-limits? Going further Step 1 : non standard construction (ultraproduct+Loeb measure) of a model M (not on a standard Borel space, only Fubini-like properties)
Structural Limits FO-limits? Going further Step 1 : non standard construction (ultraproduct+Loeb measure) of a model M (not on a standard Borel space, only Fubini-like properties) Step 2 : let T be the sentences (in Friedman’s L (Q m ) logic) of the form ( Q m x 1 ) . . . ( Q m x p ) φ ( x 1 , . . . , x p ) if lim n →∞ � φ, G n � > 0 ¬ ( Q m x 1 ) . . . ( Q m x p ) φ ( x 1 , . . . , x p ) if lim n →∞ � φ, G n � = 0 By Friedman-Steinhorn theorem, T has a totally Borel model L .
Structural Limits FO-limits? Going further Step 1 : non standard construction (ultraproduct+Loeb measure) of a model M (not on a standard Borel space, only Fubini-like properties) Step 2 : let T be the sentences (in Friedman’s L (Q m ) logic) of the form ( Q m x 1 ) . . . ( Q m x p ) φ ( x 1 , . . . , x p ) if lim n →∞ � φ, G n � > 0 ¬ ( Q m x 1 ) . . . ( Q m x p ) φ ( x 1 , . . . , x p ) if lim n →∞ � φ, G n � = 0 By Friedman-Steinhorn theorem, T has a totally Borel model L . Step 3 : Adjust the probability measure. λ ( X ∩ θ r i ( L )) � n →∞ � θ r π ⇐ π r , where π r ( X ) = lim i , G n � . λ ( θ r i ( L )) i ∈ λ ( θ r i ( L )) � =0
Structural Limits FO-limits? Going further FO-limits?
Structural Limits FO-limits? Going further Convergence of Bounded Degree Graphs For a sequence ( G n ) n ∈ N of graphs with degree ≤ d the following are equivalent: 1. the sequence ( G n ) n ∈ N is local convergent; 2. the sequence ( G n ) n ∈ N is FO local -convergent; 1 3. the sequence ( G n ) n ∈ N is FO local -convergent; Theorem (Nešetřil, OdM 2012) Every FO-convergent sequences of graphs with bounded degrees has a graphing FO -limit.
Structural Limits FO-limits? Going further Residual Sequences ∀ d ∈ N : | N d G n ( v n ) | lim sup = 0 . | G n | n →∞ v n ∈ G n FO FO 1 G n − − → L ⇐ ⇒ G n − − → L for a residual sequence ( G n ) . Theorem (Nešetřil, OdM 2016+) Every residual FO -convergent sequence ( G n ) n ∈ N of graphs has a modeling FO -limit L .
Structural Limits FO-limits? Going further Modeling limits? Theorem (Nešetřil, OdM 2013, 2017) If a monotone class C has modeling FO local -limits then the class C is nowhere dense. Almost wide Nowhere dense Excluded Bounded Locally bounded topological minor expansion expansion Locally excluded Excluded minor minor Locally bounded Bounded genus tree-width Bounded degree Planar
Structural Limits FO-limits? Going further Modeling limits for Nowhere dense? Conjecture (Nešetřil, OdM ) Every nowhere dense class has modeling FO -limits. • true for bounded degree graphs (Nešetřil, OdM 2012) • true for bounded tree-depth graphs (Nešetřil, OdM 2013) • true for trees (Nešetřil, OdM 2016) • true for plane trees and for graphs with bounded pathwidth (Gajarský, Hliněný, Kaiser, Kráľ, Kupec, Obdržálek, Ordy- niak, Tůma 2016)
Structural Limits FO-limits? Going further Modeling Limits of Nowhere Dense Sequences Theorem (Nešetřil, OdM 2016) A hereditary class of graphs C is nowhere dense if and only if ∀ d, ∀ ǫ > 0 , ∀ G ∈ C , ∃ S ⊆ G with | S | ≤ N ( d, ǫ ) such that | N d G − S ( v ) | sup ≤ ǫ. | G | v ∈ G − S Almost wide Nowhere dense Excluded Bounded Locally bounded topological minor expansion expansion Locally excluded Excluded minor minor Locally bounded Bounded genus tree-width Bounded degree Planar
Structural Limits FO-limits? Going further Modeling Limits of Quasi-Residual Sequences ( G n ) is quasi-residual if | N d G n − S n ( v n ) | lim lim lim inf sup = 0 . | G n | d →∞ C →∞ n →∞ | S n |≤ C v n ∈ G n − S n ↔ ǫ -close to residual by removing ≤ C ( ǫ ) vertices. Theorem (Nešetřil, OdM 2016+) Every FO -convergent quasi-residual sequence of graphs has a modeling FO -limit. Corollary A monotone class C is nowhere dense if and only if every FO-convergent sequence of graphs in C has a modeling FO -limit.
Structural Limits FO-limits? Going further Sketch of the Proof FO ∗ Marking 1 L ≈ FO I 2 1 I 1 I 2 3 I 1 I 2 I 1 FO ∗ 1 2 L ∗ ≈ FO
Structural Limits FO-limits? Going further Going further
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