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 An objects of size n Probability distribution on An for each n A | = φ means A satisfies φ Definition The zero-one law holds in A if for every formula φ in FO lim

n→∞ P(A |

= φ : A ∈ An) ∈ {0, 1} With high probability (whp) every object satisfies φ or Whp every object does not satisfy φ

Classical example G class of all labelled graphs |Gn| = 2(n

2)

Uniform distribution P(G) =

1 2(n

2) ,

G ∈ Gn 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 lim

n→∞ P(M |

= φ : M ∈ Mn) 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, an = |An|

◮ Unlabelled Zero-one law ⇔ lim an−1

an = 1

◮ Labelled Zero-one law ⇔ lim nan−1

an = ∞ 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 Ehrk(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 (a1, . . . , ak) vertices selected from G (b1, . . . , bk) vertices selected from H Duplicator wins iff ai ↔ bi is a partial isomorphism Lemma (Ehrenfeucht-Fra¨ ıss´ e) G ≡k H ⇔ Duplicator has a winning strategy for Ehrk(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 P(Gn | = ¬Er,s) ≤ n r n − r s

• (1 − pr(1 − p)s)n−r−s → 0

Theorem Duplicator wins Ehrk(Gn, Hn) for random (Gn, Hn) 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 Ehrk(Gn, Hn) ⇒ Whp Gn | = φ ⇔ Hn | = φ Corollary The zero-one law holds for labelled graphs

Rephrasing the proof A theory T is complete if for each formula φ T | = φ

• r

T | = ¬φ

◮ 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 | = φ

• r

T | = ¬φ

◮ 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 Ehrk(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 C3k ≡k C3k ∪ C3k Bipartiteness not expressible in FO C4k ≡k C4k+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 Ehrk(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 C3k ≡k C3k ∪ C3k Bipartiteness not expressible in FO C4k ≡k C4k+1

No zero-law in G

• n, p = n−1

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˝

• s, Kleitman, Rothschild (1976)

Almost all triangle-free graphs bipartite

◮ Kt+1-free graphs

Kolaitis, Pr¨

• mmel, Rothschild (1987)

Almost all Kt+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 ≈ δn by equivalence with G(n, p constant)

◮ Trees

McColm (2002) Zero-one law in Monadic Second Order logic

◮ Triangle-free graphs

Erd˝

• s, Kleitman, Rothschild (1976)

Almost all triangle-free graphs bipartite

◮ Kt+1-free graphs

Kolaitis, Pr¨

• mmel, Rothschild (1987)

Almost all Kt+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 ≈ δn 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 labelled trees |Tn| = nn−2 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 Uk such that

• 1. A random tree contains Uk w.h.p.
• 2. If T, T ′ both contain Uk then T ≡k T ′

Proof

• 1. A random tree contains Θ(n) copies of any fixed tree
• 2. T1, . . . , Tm representatives of all ≡k types of trees

Uk : take k copies of each Ti and glue them by adding a new root Duplicator wins Ehrk(T, T ′) by playing in suitable subtrees of Uk

Theorem The zero-one law holds for trees Proof Let φ be of rank k With high probability Duplicator wins Ehrk(Tn, T ′

n) ⇒

Whp Gn | = φ ⇔ Hn | = 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 Ehrk(Tn, T ′

n) ⇒

Whp Gn | = φ ⇔ Hn | = 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 Ehrk(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 G ≡MSO

k

H ⇔ Duplicator wins the extended game EhrMSO

k

(G, H) Theorem There are finitely many ≡k classes

Forests

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

◮ E(|Rn|) is constant ◮ P(Rn ≃ H) → µH

lim P(Fn | = φ) =

• H∈A(φ)

µH

Planar graphs

For each k there exists a planar graph Uk

◮ If G, G ′ planar contain Uk then G ≡k G ′ ◮ W.h.p. a random planar graph contains Uk

McDiarmid, Steger, Welsh 2005 Gim´ enez, N. 2009 Theorem A zero-one MSO law holds for connected planar graphs A convergence MSO law holds for arbitrary planar graphs

Minor-closed classes of graphs

G is minor-closed if closed under deletion and contraction G ∈ G, H minor of G ⇒ H ∈ G Forests, Planar, Graphs embeddable in a fixed surface S Outerplanar, Series-Parallel, Bounded tree-width G addable if closed under connected and 2-connected components Theorem (McDiarmid 2009) G addable and minor-closed, H fixed graph in G A random graph in G contains H w.h.p. Theorem A zero-one MSO law holds for connected graphs in G A convergence MSO law holds for arbitrary graphs in G

No zero-one law for caterpillars (not addable) The probability that then endpoints have given degree → constant = 0, 1

Graphs on surfaces

GS class of graphs embeddable in S Is not addable: K5 can be embedded in the torus but K5 ∪ K5 cannot B(x, r) = {y : d(x, y) ≤ r} W.h.p. in a random graph in GS

◮ All balls B(x, R) are planar for fixed R > 0

Chapuy-Fusy-Gim´ enez-Mohar-N., Bender-Gao 2011

◮ Contains any fixed planar graph McDiarmid 2008 CFGMN

Gaifman’s locality theorem

FO can express only “local” properties A local sentence is of the form ∃x1 . . . ∃xs  

i=j

d(xi, xj) > 2r   ∧

• i

ψB(xi,r)(xi)

• Gaifman’s theorem

Every FO sentence is equivalent to a Boolean combination of local sentences Enough to prove FO zero-one law for local sentences

Theorem A zero-one FO law holds for connected graphs in GS A convergence FO law holds for arbitrary graphs in GS p(φ) = lim Pr(Gn | = φ) independent of S Conjecture A zero-one MSO law holds for connected graphs in GS A convergence MSO law holds for graphs in GS If true the limiting probabilities must depend on S The existence of a minor is expressible in MSO

The set of limiting probabilities

L = {lim P(Gn | = φ) : φ MSO formula} L ⊆ [0, 1] is countable and symmetric with respect to 1/2 Theorem If G addable minor-closed class then L is a finite union of closed intervals Forests L = [0, 0.1703] ∪ [0.2231, 0.3935] ∪ [0.6065, 0.7769] ∪ [0.8297, 1] 0.6065 · · · = e−1/2 = lim P(Random forest is connected) φ a.s. true for trees ⇒ lim P(φ) ≥ 0.6065 φ a.s. false for trees ⇒ lim P(φ) ≤ 1 − 0.6065 = 0.3935

Lemma (Guthrie-Nymann 1988) p1 ≥ p2 ≥ · · · ≥ pn · · · > 0 and pn < +∞ If pn ≤

k>n pk for n ≥ n0 then

• i∈A

pi : A ⊂ N

• is a finite union of closed intervals

In our case the pi are the probabilities of the possible fragments

◮ Same L for FO and MSO ◮ At least two intervals since P(connectivity) ≥ 0.06065

Conjecture proved by Addario-Berry, McDiarmid, Reed and by Kang, Panagiotou For planar graphs L = union of 108 invertvals, of length ≈ 10−6

Lemma (Guthrie-Nymann 1988) p1 ≥ p2 ≥ · · · ≥ pn · · · > 0 and pn < +∞ If pn ≤

k>n pk for n ≥ n0 then

• i∈A

pi : A ⊂ N

• is a finite union of closed intervals

In our case the pi are the probabilities of the possible fragments

◮ Same L for FO and MSO ◮ At least two intervals since P(connectivity) ≥ 0.06065

Conjecture proved by Addario-Berry, McDiarmid, Reed and by Kang, Panagiotou For planar graphs L = union of 108 invertvals, of length ≈ 10−6

Conjecture If G is addable minor-closed then L has at least 3 gaps (4 intervals) p = lim P(connectedness) p′ = lim P(having exactly two components, one an isolated vetex) Conjecture is equivalent to 1 − p 2 < p′ equivalent to G(ρ) < 1 + 2ρ where G(z) =

• |Gn|zn

n! , ρ = radius of covergence