The first order definability of finite graphs Oleg Verbitsky - - PowerPoint PPT Presentation

The first order definability

• f finite graphs

Oleg Verbitsky

Humboldt Universit¨ at IAPMM Berlin, Germany and Lviv, Ukraine

Bertinoro, October 2009 Based on joint work with Oleg Pikhurko and Joel Spencer, with important contributions by Tom Bohman, Alan Frieze, Martin Grohe, Jeong Han Kim, Tomasz Luczak, Clifford Smyth, and Helmut Veith.

Outline

• Basic concepts

(the logical depth, width, and length of a graph)

• Main tools

(the Ehrenfeucht game and the Weisfeiler-Lehman algorithm)

• Upper bounds

– trees and graphs with bounded treewidth – planar graphs – general case (Ramsey?)

• Random graph

– applications to the 0-1 law

• Succinctly definable graphs

– definitions with no quantifier alternation (Ramsey!)

1

Language of the first order theory of graphs

• variables (x, y, y1, etc), ranging through the vertex set of a

graph;

• the relations = (equality) and ∼ (vertex adjacency);
• the quantifiers ∀ (universality) and ∃ (existence);
• the Boolean connectives ∧ (and), ∨ (or), and ¬ (negation).

Example. The following first order formula ∆n(x, y) says that vertices x any y lie at distance no more than n: ∆1(x, y)

def

= x ∼ y ∨ x = y ∆n(x, y)

def

= ∃z1 . . . ∃zn−1 “ ∆1(x, z1) ∧

n−2

^

i=1

∆1(zi, zi+1) ∧ ∆1(zn−1, y) ”

2

Basics

A sentence Φ distinguishes a graph G from another graph H if Φ is true on G but false on H. Example.

• 1. The sentence ∀x∀y ∆1(x, y) distinguishes a complete graph Kn

from any other graph H that is not complete.

• 2. The sentence ∀x∀y ∆n−1(x, y) distinguishes Pn,

a path on n vertices, from any longer path Pm, m > n.

3

Basics

A sentence Φ defines a graph G (up to isomorphism) if Φ distinguishes G from every non-isomorphic graph H.

• Example. Pn is defined by

∀x∀y∆n−1(x, y) ∧ ¬∀x∀y∆n−2(x, y) to say that the diameter = n − 1 ∧ ∀x¬∃y1∃y2∃y3 “^

i=1,2,3 x ∼ yi ∧

^

i=j yi = yj

” to say that the maximum degree ≤ 2 ∧ ∃x¬∃y1∃y2 “^

i=1,2 x ∼ yi ∧ y1 = y2

” to say that the minimum degree ≤ 1 (thereby distinguishing from cycles C2n−2 and C2n−1)

4

Basics

Succinctness measures of a formula Φ: the length L(Φ), the quantifier depth D(Φ), and the width W(Φ) Definition. W(Φ) is the number of variables used in Φ (different occurrences

• f the same variable are not counted!)

Example. W(∆n) = n + 1. However, rewriting it as ∆′

n(x, y)

def

= ∃z(∆1(x, z) ∧ ∆′

n−1(z, y)), where

∆′

n−1(z, y)

def

= ∃x(∆1(z, x) ∧ ∆′

n−2(x, y)) and so on,

we get W(∆′

n) = 3.

5

Basics

Definition. D(Φ), the quantifier depth of Φ, is the maximum number of nested quantifiers in Φ. Example. D(∆n) = n − 1. However, rewriting it as ∆′′

n(x, y)

def

= ∃z “ ∆′′

⌊n/2⌋(x, z) ∧ ∆′′ ⌈n/2⌉(z, y)

” , we get D(∆′′

n) = log n + O(1).

6

Main Definition

Definition (the logical length, depth, and width of a graph). L(G) (resp. D(G), W(G)) is the minimum L(Φ) (resp. D(Φ), W(Φ))

• ver all Φ defining G.

Example. W(Pn) ≤ 4, D(Pn) ≤ log n + O(1).

7

Basics

Definition. Let G ∼ = H. Then D(G, H) (resp. W(G, H)) is the minimum D(Φ) (resp. W(Φ)) over all Φ distinguishing G from H. Proposition.

• 1. D(G) = maxH D(G, H)
• 2. W(G) = maxH W(G, H)

8

Variations of a logic

Fragments of first order logic

• Bounded number of quantifier alternations (later).
• Bounded number of variables.

Dk(G) denotes the logical depth of G in the k-variable logic An extension of first order logic: Counting quantifiers ∃mxΨ(x) means that there are at least m vertices x having property Ψ. D#(G) and W#(G) will denote the logical depth and width of a graph G in the counting logic. Dk

#(G) denotes the variant of Dk(G) for the k-variable counting

logic.

9

Ehrenfeucht’s game

Immerman-Poizat: G and H are distinguishable with k variables and quantifier depth r iff Spoiler wins the Ehrenfeucht game with k pebbles in r moves.

Rules of the Game

G H

Players: Spoiler and Duplicator Resources: k pebbles, each in duplicate A round: Spoiler puts a pebble on a vertex in G

• r H

Duplicator puts the other copy on a vertex in the other graph Duplicator’s objective: after each round the pebbling should determine a partial isomorphism between G and H

10

k-dimensional Weisfeiler-Lehman algorithm

1-dim WL = color refinement procedure

1 1 1 1 1 1 1 1 1 1 1 1

Initial coloring

11

1-dim WL = color refinement procedure

1,11 1,111 1,1 1,111 1,1111 1,1 1,11 1,1111 1,1 1,111 1,111 1,1

Refine coloring: for each vertex New Color = Old Color + Old Colors of all neighbors

12

1-dim WL = color refinement procedure

1 1 1 1 2 3 3 4 2 3 4 3

Simplify color names

13

1-dim WL = color refinement procedure

2,33 3,234 1,4 3,234 4,1133 1,4 2,34 3,234 4,1233 3,134 1,4 1,3

Refine coloring again. The multisets of colors differ, hence the graphs are non-isomorphic.

14

k-dim WL

k-dim WL = the same idea, but now we color V k instead of V . The initial coloring of (v1, . . . , vk) is the isomorphism type of the subgraph induced on v1, . . . , vk. Theorem (Cai, F¨ urer, and Immerman) The r-round k-dim WL works correctly on any pair (G, H) if k = W#(G) − 1 and r = Dk+1

# (G) − 1.

On the other hand, it is wrong for some (G, H) if k < W#(G) − 1, whatever r.

• Theorem. Let k ≥ 2 be a constant.
• 1. Let C be a class of graphs G with Dk

#(G) = O(log n).

Then Graph Isomorphism for C is solvable in TC1.

• 2. Let C be a class of graphs G with Dk(G) = O(log n).

Then Graph Isomorphism for C is solvable in AC1.

15

Example: Trees

• Theorem. D3

#(T) ≤ 3 log n + 2 for every tree T on n vertices.

Proof (a separator strategy): Let T ′ ∼ = T (and assume T ′ is a tree too). We need to show that Spoiler wins the 3-pebble game on T and T ′ in 3 log n + 2 moves. Step 1. Spoiler pebbles a separator v in T (every component of T − v has ≤ n/2 vertices). Step 2. Spoiler ensures pebbling u ∈ Γ(v) and u′ ∈ Γ(v′) so that the corresponding components are non-isomorphic rooted trees.

v u T T´ v´ u´

Spoiler forces further play

• n these components and

applies the same strategy.

16

Proof - continuation

A complication: the strategy is now applied to a graph with one vertex pebbled and we may need more than 3 pebbles.

v T T´ v´ u0 u´ v u v u

1 1

v´ u´ v´ u´

1 1

Step 3. If T −v and T ′ − v′ differ

• nly

by the components with pebbled verti- ces u0 and u′

0, then

Spoiler pebbles a v1 in the v-u0-path such that T − v1 and T ′ − v′

1

differ by components with no pebble. (The case that d(v, u0) = d(v′, u′

0)

• r d(v, v1) = d(v′, v′

1)

is even easier for Spoiler.)

17

Isomorphism of trees (a revision of the history)

GI for trees is solvable in

• in LOG SPACE

Lindell 92

• in AC1

Miller-Reif 91

• in AC1 if ∆ = O(log n)

Ruzzo 81

• in LIN TIME by 1-WL (W#(T) = 2)

Aho-Hopcroft-Ullman 74 Miller and Reif [SIAM J. Comput. 91]: “No polylogarithmic parallel algorithm was previously known for isomorphism of unbounded-degree trees.” However, the 3 log n-round 2-WL solves TREE ISO in TC1 ⊆ NC2 and is known since 68 !

18

Estimates for particular classes

• Theorem. If a graph G on n vertices has treewidth k, then

D4k+4

#

(G) < 2(k + 1) log n + 8k + 9. Consequently, isomorphism of graphs whose treewidth does not exceed k is recognizable by the (4k + 3)-dim WL in TC1 ⊆ NC2.

• Theorem. For a 3-connected planar graph G on n vertices we

have D15(G) < 11 log2 n + 45. Consequently, the isomorphism problem for 3-connected planar graphs is solvable by the 14-dim WL in AC1.

19

General bounds

Consider G ∼ = H, both with n vertices. It is easy to find example where W(G, H) ≥ n + 1 2 . Moreover: Theorem (Cai, F¨ urer, and Immerman) There are pairs of graphs such that W#(G, H) = Ω(n). On the other hand: How to show that D(G, H) < n for all such pairs?

20

An initial approach with the Ramsey theorem

D(G, H) < n − 1 4 log2 n.

Indeed, as D(G, H) = D( ¯ G, ¯ H) we can assume that G contains an independent set A with |A| > 1

2 log2 n.

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✉ ✉ ✉ ✉ ✉ ✉

A B G H

• Spoiler selects all V (G) \ A.
• Duplicator selects V (H)\B for some B, |B| = |A|.

Identify V (G)\A and V (H)\B according to the partial isomorphism established. Suppose B is independent too for else Spoiler wins in 2 moves. Given X ⊆ V (G) \ A, let mG(X) = | {v ∈ A : Γ(v) = X} | and mH(X) = | {v ∈ B : Γ(v) = X} |. Since G ∼ = H, there is X with mG(X) = mH(X). Spoiler can now win in min{mG(X), mH(X)}+1 moves. Since P

X mG(X) = P X mH(X), there are at least

two such sets X1 and X2 and for one of them min{mG(Xi), mH(Xi)} < |A|/2.

21

The final bound

• Theorem. D(G, H) ≤ n+3

2

for all non-isomorphic G and H on n vertices.

22

Average case bounds

Theorem (Babai, Erd˝

• s, and Selkow).

2 color refinements split a random graph Gn,1/2 into color classes which are singletons with probability more than 1 − 1/ 7 √n, for all large enough n. Consequently, D2

#(Gn,1/2) ≤ 4 with this probability.

• Theorem. With high probability we have

log n−2 log log n+1 < W(Gn,1/2) ≤ D(Gn,1/2) ≤ log n−log log n+ω+O(1), where ω = ω(n) is an arbitrarily slowly increasing function.

• Theorem. For infinitely many n we have

D2(Gn,1/2) ≤ log n − 2 log log n + 5 + log log e + o(1) with high probability.

23

An application: The convergency rate in the 0-1 law

Let pn(Φ) = P [Gn,1/2 | = Φ]. 0-1-law (Glebskii-et-al.–Fagin): pn(Φ) approaches 0 or 1 as n → ∞. Denote the limit by p(Φ). Define the convergency rate function for the 0-1-law by R(k, n) = max

Φ

{|pn(Φ) − p(Φ)| : D(Φ) ≤ k} . The standard version of the 0-1-law gives only the following: R(k, n) → 0 as n → ∞ for any fixed k.

• Theorem. Let k(n) = log n − 2 log log n + c.
• 1. Set c = 1. Then R(k(n), n) → 0 as n → ∞.
• 2. The claim does not hold true for c = 6.

24

Succinctly definable graphs How small can the logical depth be?

Definition (the succinctness functions). s(n) = min {D(G) : G has n vertices} s(n) → ∞ as n → ∞ but admits no recursive lower bound. Theorem. There is no general recursive function f such that f(s(n)) ≥ n for all n.

25

Definitions with no quantifier alternation

Consider the first order logic with no quantifier alternation (purely existential and universal formulas and their monotone Boolean combinations). This fragment

• f first order logic is provably weak.

FINITE SATISFIABILITY problem: Given a f.o. sentence about graphs, decide whether it is true on at least one finite graph. Lavrov (63): FINITE SATISFIABILITY is unsolvable even for sentences without equality. A sentence with no quantifier alternation is equivalent to a sentence in the Bernays-Sch¨

• nfinkel class, i.e., to some sentence

Φ = ∃x1 . . . ∃xk∀y1 . . . ∀ylΨ(¯ x, ¯ y) (∗) FINITE SATISFIABILITY for the Bernays-Sch¨

• nfinkel

class is solvable. Moreover, let Spectrum(Φ) consists of all those n such that there is a graph

• n n vertices satisfying Φ.

The logical Ramsey theorem (the case of graphs). Let Φ have form (∗). If Spectrum(Φ) contains some n ≥ 2k4l, then Spectrum(Φ) contains all n ≥ k + l.

26

Definitions with no quantifier alternation

Nevertheless,

• D0(Gn,1/2) ≤ (2 + o(1)) log2 n with high probability;
• D0(G, H) ≤ n+5

2

for all non-isomorphic graphs G and H of the same order n, where D0(G) and D0(G, H) are analogs of D(G) and D(G, H) in the logic with no quantifier alternation.

27

Definitions with no quantifier alternation

Define s0(n) = min {D0(G) : G has n vertices} .

• Theorem. For all n we have

log∗ n − log∗ log∗ n − 2 ≤ s0(n) ≤ log∗ n + 22.

• Corollary. A gap in relation D(G) ≤ D0(G) can be super-recursive:

There is no general recursive function f such that D0(G) ≤ f(D(G)).

28

Thank you!

29