Connectivity Corollary. GRAPH CONNECTIVITY is not FO definable - - PowerPoint PPT Presentation

▶
connectivity
SMART_READER_LITE
LIVE PREVIEW

Connectivity Corollary. GRAPH CONNECTIVITY is not FO definable - - PowerPoint PPT Presentation

Dec 29, 2023 347 likes •909 views

Connectivity Corollary. GRAPH CONNECTIVITY is not FO definable Connectivity Corollary. GRAPH CONNECTIVITY is not FO definable If A is a linear order of size n, let G( A ) be the graph with edges { i, i+2 mod n } for all a A Connectivity

slide-1
SLIDE 1

Connectivity

  • Corollary. GRAPH CONNECTIVITY is not FO definable
slide-2
SLIDE 2

Connectivity

  • Corollary. GRAPH CONNECTIVITY is not FO definable
  • If A is a linear order of size n, let G(A) be the graph

with edges { i, i+2 mod n } for all a ∈ A

slide-3
SLIDE 3

Connectivity

  • Corollary. GRAPH CONNECTIVITY is not FO definable
  • If A is a linear order of size n, let G(A) be the graph

with edges { i, i+2 mod n } for all a ∈ A

slide-4
SLIDE 4

Connectivity ∉ FO

  • Corollary. GRAPH CONNECTIVITY is not FO definable
  • If A is a linear order of size n, let G(A) be the graph

with edges { i, i+2 mod n } for all a ∈ A Obs 1: G(A) is connected if and

  • nly if n is odd
slide-5
SLIDE 5

Connectivity ∉ FO

  • Corollary. GRAPH CONNECTIVITY is not FO definable
  • If A is a linear order of size n, let G(A) be the graph

with edges { i, i+2 mod n } for all a ∈ A Obs 2: G(A) is first-order definable from A

slide-6
SLIDE 6

Connectivity

  • Corollary. GRAPH CONNECTIVITY is not FO definable
  • If A is a linear order of size n, let G(A) be the graph

with edges { i, i+2 mod n } for all a ∈ A

  • If 𝜒 were a first-order formula defining GRAPH

CONNECTIVITY, then by replacing each sub-formula E(x,y) with a formula “x and y have cyclic distance 2 in the linear order A”, we could define EVENNESS of A (which we showed is impossible by the EF game).

slide-7
SLIDE 7

Connectivity

  • Corollary. GRAPH CONNECTIVITY is not FO definable
  • This result can be proved directly by playing the EF

game e.g. on graphs Cn and Cn + Cn

  • The reduction to EVENNESS of linear orders illustrates

the technique of a first-order interpretations.

slide-8
SLIDE 8

Set-powersets

  • SetPown is the structure ([n]∪ 2[n], Atoms, Sets, In)

where Atoms = [n] = {1,...,n}, Sets = powerset of Atoms, In = {(i,X) ∈ Atoms × Sets | i ∈ X}.

slide-9
SLIDE 9

Set-powersets

  • SetPown is the structure ([n]∪ 2[n], Atoms, Sets, In)

where Atoms = [n] = {1,...,n}, Sets = powerset of Atoms, In = {(i,X) ∈ Atoms × Sets | i ∈ X}.

  • A set-powerset is any structure A with relations

{Atoms, Sets, In} which is isomorphic to SetPown for some n > 0. It is said to be EVEN/ODD according to the parity of n.

slide-10
SLIDE 10

Set-powersets

  • Obs. The class of set-powersets is FO definable.
  • We cannot say (in first-order logic):

∀X ⊆ Atoms ∃S ∈ Sets ∀x ∈ Atoms, x ∈ X ⇔ In(x,S)

  • Instead, we say:

"⦰ ∈ Sets" ∧ ∀S ∈ Sets ∀x ∈ Atoms "S ∪ {x} ∈ Sets"

  • This formula exploits finiteness in an essential way.
slide-11
SLIDE 11

Set-powersets

Theorem The class of EVEN set-powersets is not FO definable.

slide-12
SLIDE 12

Set-powersets

Theorem The class of EVEN set-powersets is not FO definable. Proof For every k, we show that Duplicator has a winning strategy in the k-round Ehrenfeucht-Fraisse game on A = SetPow2^k and B = SetPow2^k+1

slide-13
SLIDE 13

Winning strategy (by picture)

A Atoms Sets B Atoms Sets

slide-14
SLIDE 14

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-15
SLIDE 15

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-16
SLIDE 16

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-17
SLIDE 17

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-18
SLIDE 18

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-19
SLIDE 19

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-20
SLIDE 20

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-21
SLIDE 21

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-22
SLIDE 22

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-23
SLIDE 23

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-24
SLIDE 24

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-25
SLIDE 25

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-26
SLIDE 26

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-27
SLIDE 27

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-28
SLIDE 28

Winning strategy (by picture)

A B Atoms Sets Atoms Sets

slide-29
SLIDE 29

Winning strategy (by picture)

A B At this point, Spoiler wins. Atoms Sets Atoms Sets

slide-30
SLIDE 30

Winning strategy (by picture)

A B At this point, Spoiler wins. Atoms Sets Atoms Sets Duplicator has a winning strategy for k rounds provided both structures have ≥ 2k atoms.

slide-31
SLIDE 31

Winning strategy (by picture)

A B At this point, Spoiler wins. Atoms Sets Atoms Sets Duplicator has a winning strategy for k rounds provided both structures have ≥ 2k atoms. Duplicator’s winning strategy: in round j, preserve the cardinality up to 2k−j of every Boolean combination of the chosen sets and atoms

slide-32
SLIDE 32

0-1 Laws

slide-33
SLIDE 33

G(n,p)

  • The Erdos-Renyi random graph G(n,p)

– vertex set {1,...,n} – indep. edge probability p

slide-34
SLIDE 34

G(n,p)

  • The Erdos-Renyi random graph G(n,p)

– vertex set {1,...,n} – indep. edge probability p G(n,½) is known as the uniform random graph

slide-35
SLIDE 35

0-1 Law for FO

Theorem [Fagin 1976, Glebskii et al 1969] For every FO sentence ϕ, limn→∞ Pr[ G(n,½) ⊨ ϕ ] ∈ {0,1}

slide-36
SLIDE 36

0-1 Law for FO

Theorem [Fagin 1976, Glebskii et al 1969] For every FO sentence ϕ, limn→∞ Pr[ G(n,½) ⊨ ϕ ] ∈ {0,1} That is, ϕ is either almost surely true

  • r almost surely false in G(n,½)
slide-37
SLIDE 37

0-1 Law for FO

Theorem [Fagin 1976, Glebskii et al 1969] For every FO sentence ϕ and p ∈ (0,1), limn→∞ Pr[ G(n,p) ⊨ ϕ ] ∈ {0,1}

slide-38
SLIDE 38

0-1 Law for FO

Theorem [Spencer and Shelah 1988] For every FO sentence ϕ and irrational c ∈ (0,1), limn→∞ Pr[ G(n,n−c) ⊨ ϕ ] ∈ {0,1}

slide-39
SLIDE 39

0-1 Law for FO

Theorem [Spencer and Shelah 1988] For every FO sentence ϕ and irrational c ∈ (0,1), limn→∞ Pr[ G(n,n−c) ⊨ ϕ ] ∈ {0,1} No 0-1 law for rational c ∈ (0,1)

slide-40
SLIDE 40

Proof of The Zero-One Law

slide-41
SLIDE 41

0-1 Law for FO

Theorem [Fagin 1976, Glebskii et al 1969] For every FO sentence ϕ, limn→∞ Pr[ G(n,½) ⊨ ϕ ] ∈ {0,1}

slide-42
SLIDE 42
  • The 2-extension property EXT2 says:

Every vertex has a neighbor and a non-neighbor

0-1 Law for FO

slide-43
SLIDE 43

0-1 Law for FO

  • The 3-extension property EXT3 says:

Every two vertices and have

– a common neighbor – a common non-neighbor – a neighbor of each one, but not the other

slide-44
SLIDE 44

0-1 Law for FO

  • The 4-extension property EXT4 says:

For all distinct vertices , there exist 8 vertices witnessing each possible set of adjacencies

slide-45
SLIDE 45

0-1 Law for FO

  • The k-extension property EXTk says:

for all k−1 distinct vertices, there exist 2k witnesses.

……

slide-46
SLIDE 46

Lemma If G and H satisfy EXTk, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)

0-1 Law for FO

slide-47
SLIDE 47

0-1 Law for FO

Lemma If G and H satisfy EXTk, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)

slide-48
SLIDE 48

0-1 Law for FO

Lemma If G and H satisfy EXTk, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)

slide-49
SLIDE 49

0-1 Law for FO

Lemma If G and H satisfy EXTk, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)

slide-50
SLIDE 50

0-1 Law for FO

Lemma If G and H satisfy EXTk, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)

slide-51
SLIDE 51

0-1 Law for FO

Lemma If G and H satisfy EXTk, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)

slide-52
SLIDE 52

0-1 Law for FO

Lemma If G and H satisfy EXTk, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)

slide-53
SLIDE 53

0-1 Law for FO

Lemma If G and H satisfy EXTk, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)

slide-54
SLIDE 54

0-1 Law for FO

Lemma If G and H satisfy EXTk, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)

slide-55
SLIDE 55

0-1 Law for FO

Lemma For every k, the uniform random graph G ～ G(n,½) satisfies EXTka.a.s.