Logical limit laws in combinatorics Marc Noy Universitat Polit` - PowerPoint PPT Presentation

Logical limit laws in combinatorics Marc Noy Universitat Polit` ecnica de Catalunya Barcelona

First order logic ∀ , ∃ , x , y , . . . , ∧ , ∨ , ¬ , = formulas involving relations of arbitrary arity ◮ ∀ x ∃ y x ∗ y = 1 (inverse element in groups) ◮ ∃ x ∀ y x �∼ y (isolated vertex in graphs) ◮ ∃ x ∃ y ∃ z ( x ∼ y ) ∧ ( x ∼ z ) ∧ ( y ∼ z ) (existence of a triangle) ◮ ∀ x f ( x ) � = x (no fixed points in mappings) ‘A graph is connected’ is not expressible in FO Bipartite, Hamiltonian. . . not in FO

Zero-one laws A combinatorial class A n objects of size n Probability distribution on A n for each n A | = φ means A satisfies φ Definition The zero-one law holds in A if for every formula φ in FO n →∞ P ( A | lim = φ : A ∈ A n ) ∈ { 0 , 1 } With high probability (whp) every object satisfies φ or Whp every object does not satisfy φ

Classical example |G n | = 2( n 2 ) G class of all labelled graphs 1 Uniform distribution P ( G ) = 2 ) , G ∈ G n 2 ( n Theorem Glebski, Kogan, Liagonkii, Talanov (1969) Fagin (1976) The zero-one law holds for labelled graphs

Further examples M class of mappings Relation f ( x ) = y There is no zero-one law: P ( ¬∃ x f ( x ) = x ) → e − 1 But there is a convergence law: Lynch (1985) For every formula φ n →∞ P ( M | lim = φ : M ∈ M n ) exists Some combinatorial classes having a zero-one law ◮ Set partitions (labelled equivalence relations) ◮ Integer partitions (unlabelled equivalence relations)

Compton’s method Kevin Compton 1987 A logical approach to asymptotic combinatorics I Defines a notion of connected components in relational structures Class A is admissible if: A ∈ A ⇔ each component of A is in A Theorem (Compton) Let A be admissible, a n = |A n | ◮ Unlabelled Zero-one law ⇔ lim a n − 1 = 1 a n ◮ Labelled Zero-one law ⇔ lim na n − 1 = ∞ a n Uses generating functions, asymptotic estimates, Hayman’s admissiblity. . .

Page 467: A beautiful theorem of Lynch [426], much in line with the global aims of analytic combinatorics. . . The proof of the theorem is based on Ehrenfeucht games supplemented by ingenous inclusion-exclusion arguments. . .

Page 467: A beautiful theorem of Lynch [426], much in line with the global aims of analytic combinatorics. . . The proof of the theorem is based on Ehrenfeucht games supplemented by ingenous inclusion-exclusion arguments. . .

Proof of the zero-one law holds for labelled graphs Rank of φ = maximum number of nested quantifiers If φ is quantifier-free then rk( φ ) = 0 If φ = ∀ x ψ ( x ) then rk( φ ) = rk( ψ ) + 1 If φ = ∃ x ψ ( x ) then rk( φ ) = rk( ψ ) + 1 ∃ x ∀ y x �∼ y Rank 2 ∃ x ∃ y ∃ z ( x ∼ y ) ∧ ( x ∼ z ) ∧ ( y ∼ z ) Rank 3 Definition G ≡ k H ⇔ satisfy the same formulas of rank ≤ k Lemma The relation ≡ k has finitely many equivalence classes Proof Induction on k

Logic through combinatorial games Ehrenfeucht-Fra¨ ıss´ e game Ehr k ( G , H ) ◮ Spoiler and Duplicator play k rounds on two graphs G , H ◮ At each round Spoiler picks a vertex (from any graph) and Duplicator picks a vertex from the other graph ( a 1 , . . . , a k ) vertices selected from G ( b 1 , . . . , b k ) vertices selected from H Duplicator wins iff a i ↔ b i is a partial isomorphism Lemma (Ehrenfeucht-Fra¨ ıss´ e) G ≡ k H ⇔ Duplicator has a winning strategy for Ehr k ( G , H )

The G ( n , p ) model ◮ Class: Labelled graph with n edges ◮ Every possible edge xy independently with probability p P ( G ) = p | E | (1 − p )( n 2 ) −| E | G ( n , 1 / 2) ≡ uniform distribution

Extension property in G ( n , p ) for constant p For all r , s ≥ 0 For all disjoint A , B ⊂ { 1 , . . . , n } with | A | = r , | B | = s ∃ z ( ∀ x ∈ A z ∼ x ) ∧ ( ∀ x ∈ B z �∼ x ) Theorem G ( n , p ) satisfies extension property whp Proof � n �� n − r � (1 − p r (1 − p ) s ) n − r − s → 0 P ( G n | = ¬ E r , s ) ≤ r s

Theorem Duplicator wins Ehr k ( G n , H n ) for random ( G n , H n ) w.h.p. Proof By the extension property, duplicator can always find a vertex mantaining the partial isomorphism Theorem The zero-one law holds in G ( n , p ) for constant p Proof Let φ be of rank k With high probability Duplicator wins Ehr k ( G n , H n ) ⇒ Whp G n | = φ ⇔ H n | = φ Corollary The zero-one law holds for labelled graphs

Rephrasing the proof A theory T is complete if for each formula φ T | T | = ¬ φ = φ or ◮ The extension axioms E form a complete theory for graphs ◮ Every ψ ∈ E holds with asymptotic probability 1 Compton’s complete theory for an admissible class A : in terms of the number of components isomorphic to a fixed connected object B

Rephrasing the proof A theory T is complete if for each formula φ T | T | = ¬ φ = φ or ◮ The extension axioms E form a complete theory for graphs ◮ Every ψ ∈ E holds with asymptotic probability 1 Compton’s complete theory for an admissible class A : in terms of the number of components isomorphic to a fixed connected object B

The zero-one law for partitions revisited For each fixed t , a random partition has an unbounded number of parts of size t Duplicator wins Ehr k ( P , P ′ ) whp This is not the case for permutations: the number of cycles of size t is O (1) Connectedness not expressible in FO C 3 k ≡ k C 3 k ∪ C 3 k Bipartiteness not expressible in FO C 4 k ≡ k C 4 k +1

The zero-one law for partitions revisited For each fixed t , a random partition has an unbounded number of parts of size t Duplicator wins Ehr k ( P , P ′ ) whp This is not the case for permutations: the number of cycles of size t is O (1) Connectedness not expressible in FO C 3 k ≡ k C 3 k ∪ C 3 k Bipartiteness not expressible in FO C 4 k ≡ k C 4 k +1

n , p = n − 1 � � No zero-law in G Threshold for appearance of a triangle Number of triangles ⇒ Poisson law Shelah, Spencer 1988 Zero-one law in G ( n , p = n − α ) α ∈ [0 , 1] irrational Spencer The strange logic of random graphs (2001)

Constrained classes ◮ H -free graphs ◮ Regular graphs ◮ Trees ◮ Planar graphs In all cases uniform distribution on labelled graphs with n vertices

◮ Triangle-free graphs Erd˝ os, Kleitman, Rothschild (1976) Almost all triangle-free graphs bipartite ◮ K t +1 -free graphs Kolaitis, Pr¨ ommel, Rothschild (1987) Almost all K t +1 -free are t -partite ◮ d -regular graphs ◮ Lynch (2005) Convergence law for fixed d using the configuration model Number of triangles ⇒ Poisson law ◮ Haber, Krivelevich (2010) Zero-one law for d ≈ δ n by equivalence with G ( n , p constant) ◮ Trees McColm (2002) Zero-one law in Monadic Second Order logic

◮ Triangle-free graphs Erd˝ os, Kleitman, Rothschild (1976) Almost all triangle-free graphs bipartite ◮ K t +1 -free graphs Kolaitis, Pr¨ ommel, Rothschild (1987) Almost all K t +1 -free are t -partite ◮ d -regular graphs ◮ Lynch (2005) Convergence law for fixed d using the configuration model Number of triangles ⇒ Poisson law ◮ Haber, Krivelevich (2010) Zero-one law for d ≈ δ n by equivalence with G ( n , p constant) ◮ Trees McColm (2002) Zero-one law in Monadic Second Order logic

Joint work with Peter Heinig Tobias M¨ uller Anusch Taraz

Random trees |T n | = n n − 2 T labelled trees T contains S if ∃ e ∈ E ( T ) S is a component of T − e Theorem (McColm 2002) For each fixed k there exists a tree U k such that 1. A random tree contains U k w.h.p. 2. If T , T ′ both contain U k then T ≡ k T ′ Proof 1. A random tree contains Θ( n ) copies of any fixed tree 2. T 1 , . . . , T m representatives of all ≡ k types of trees U k : take k copies of each T i and glue them by adding a new root Duplicator wins Ehr k ( T , T ′ ) by playing in suitable subtrees of U k

Theorem The zero-one law holds for trees Proof Let φ be of rank k With high probability Duplicator wins Ehr k ( T n , T ′ n ) ⇒ G n | ⇔ H n | Whp = φ = φ Theorem (McColm) The zero-one law holds for trees in Monadic Second Order logic

Theorem The zero-one law holds for trees Proof Let φ be of rank k With high probability Duplicator wins Ehr k ( T n , T ′ n ) ⇒ G n | ⇔ H n | Whp = φ = φ Theorem (McColm) The zero-one law holds for trees in Monadic Second Order logic

Monadic second order logic MSO = FO + quantification over sets of vertices Connected : ( A � = ∅ , A � = V ) → ( ∃ x ∈ A ∃ y �∈ A x ∼ y ) Bipartite : ∃ A , B ( V = A ∪ B , A ∩ B = ∅ ) ∧ xy ∈ E → ( x ∈ A , y ∈ B ) ∨ ( x ∈ B , y ∈ A ) Extended Ehr k ( G , H ) games: vertex moves and set moves Duplicator wins if there is a partial isomorphism between the selected vertices that respects membership in the selected sets Lemma H ⇔ Duplicator wins the extended game Ehr MSO G ≡ MSO ( G , H ) k k Theorem There are finitely many ≡ k classes

Forests There is no zero-one law in the class F of forests P ( F n has an isolated vertex ) → e − 1 Theorem A convergence law in MSO holds for forests For each formula φ P ( F n | = φ ) → p ( φ ) ∈ [0 , 1] Proof Type of the components determines type of the forest The giant component has size n − O (1) R n = fragment: complement of largest component ◮ E ( | R n | ) is constant ◮ P ( R n ≃ H ) → µ H � lim P ( F n | = φ ) = µ H H ∈A ( φ )