Connectivity. Corollary. GRAPH CONNECTIVITY is not FO definable
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1 Connectivity Corollary. GRAPH CONNECTIVITY is not FO definable
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
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
4 Connectivity FO Obs 1: G(A) is connected if and only if n is odd 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
5 Connectivity FO Obs 2: G(A) is first-order definable from A 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
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 φ 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).
7 Connectivity Corollary. GRAPH CONNECTIVITY is not FO definable This result can be proved directly by playing the EF game e.g. on graphs C n and C n + C n The reduction to EVENNESS of linear orders illustrates the technique of a first-order interpretations.
8 Set-powersets SetPow n is the structure ([n] 2 [n],,, In) where = [n] = {1,...,n}, = powerset of, In = {(i,x) i X}.
9 Set-powersets SetPow n is the structure ([n] 2 [n],,, In) where = [n] = {1,...,n}, = powerset of, In = {(i,x) i X}. A set-powerset is any structure A with relations {,, In} which is isomorphic to SetPow n for some n > 0. It is said to be EVEN/ODD according to the parity of n.
10 Set-powersets Obs. The class of set-powersets is FO definable. We cannot say (in first-order logic): X S x, x X In(x,S) Instead, we say: " " S x "S {x} " This formula exploits finiteness in an essential way.
11 Set-powersets Theorem The class of EVEN set-powersets is not FO definable.
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 = SetPow 2^k and B = SetPow 2^k+1
13 Winning strategy (by picture) A B
14 Winning strategy (by picture) A B
15 Winning strategy (by picture) A B
16 Winning strategy (by picture) A B
17 Winning strategy (by picture) A B
18 Winning strategy (by picture) A B
19 Winning strategy (by picture) A B
20 Winning strategy (by picture) A B
21 Winning strategy (by picture) A B
22 Winning strategy (by picture) A B
23 Winning strategy (by picture) A B
24 Winning strategy (by picture) A B
25 Winning strategy (by picture) A B
26 Winning strategy (by picture) A B
27 Winning strategy (by picture) A B
28 Winning strategy (by picture) A B
29 Winning strategy (by picture) A B At this point, Spoiler wins.
30 Winning strategy (by picture) A Duplicator has a winning strategy for k rounds provided both B structures have 2 k atoms. At this point, Spoiler wins.
31 Winning strategy (by picture) A Duplicator has a winning strategy for k rounds provided both B structures have 2 k atoms. Duplicator s At winning this point, strategy: Spoiler wins. in round j, preserve the cardinality up to 2 k j of every Boolean combination of the chosen sets and atoms
32 0-1 Laws
33 G(n,p) The Erdos-Renyi random graph G(n,p) vertex set {1,...,n} indep. edge probability p
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
35 0-1 Law for FO Theorem [Fagin 1976, Glebskii et al 1969] For every FO sentence ϕ, lim n Pr[ G(n,½) ϕ ] {0,1}
![0-1 Law for FO Theorem [Fagin 1976, Glebskii et al 1969] For every FO sentence ϕ, lim n](/docs-images/92/109726381/images/36-2.jpg "Pr[ G(n,½) ϕ ] {0,1} That is, ϕ is either almost surely true or almost surely false in")
36 0-1 Law for FO Theorem [Fagin 1976, Glebskii et al 1969] For every FO sentence ϕ, lim n Pr[ G(n,½) ϕ ] {0,1} That is, ϕ is either almost surely true or almost surely false in G(n,½)
37 0-1 Law for FO Theorem [Fagin 1976, Glebskii et al 1969] For every FO sentence ϕ and p (0,1), lim n Pr[ G(n,p) ϕ ] {0,1}
38 0-1 Law for FO Theorem [Spencer and Shelah 1988] For every FO sentence ϕ and irrational c (0,1), lim n Pr[ G(n,n c ) ϕ ] {0,1}
![0-1 Law for FO Theorem [Spencer and Shelah 1988] For every FO sentence ϕ and](/docs-images/92/109726381/images/39-0.jpg "irrational c (0,1), lim n Pr[ G(n,n c ) ϕ ] {0,1} No 0-1 law for rational c")
39 0-1 Law for FO Theorem [Spencer and Shelah 1988] For every FO sentence ϕ and irrational c (0,1), lim n Pr[ G(n,n c ) ϕ ] {0,1} No 0-1 law for rational c (0,1)
40 Proof of The Zero-One Law
41 0-1 Law for FO Theorem [Fagin 1976, Glebskii et al 1969] For every FO sentence ϕ, lim n Pr[ G(n,½) ϕ ] {0,1}
42 0-1 Law for FO The 2-extension property EXT 2 says: Every vertex has a neighbor and a non-neighbor
43 0-1 Law for FO The 3-extension property EXT 3 says: Every two vertices and have a common neighbor a common non-neighbor a neighbor of each one, but not the other
44 0-1 Law for FO The 4-extension property EXT 4 says: For all distinct vertices, there exist 8 vertices witnessing each possible set of adjacencies
45 0-1 Law for FO The k-extension property EXT k says: for all k 1 distinct vertices, there exist 2 k witnesses.
46 0-1 Law for FO Lemma If G and H satisfy EXT k, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)
47 0-1 Law for FO Lemma If G and H satisfy EXT k, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)
48 0-1 Law for FO Lemma If G and H satisfy EXT k, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)
49 0-1 Law for FO Lemma If G and H satisfy EXT k, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)
50 0-1 Law for FO Lemma If G and H satisfy EXT k, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)
51 0-1 Law for FO Lemma If G and H satisfy EXT k, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)
52 0-1 Law for FO Lemma If G and H satisfy EXT k, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)
53 0-1 Law for FO Lemma If G and H satisfy EXT k, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)
54 0-1 Law for FO Lemma If G and H satisfy EXT k, then Duplicator has a winning strategy in the k-round EF game on G and H (simply by maintaining a partial isomorphism)
55 0-1 Law for FO Lemma For every k, the uniform random graph G ~ G(n,½) satisfies EXT k a.a.s.
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