Finite Model Theory Lecture 19

1. Finite Model TheoryLecture 19 Summary on 0/1 Laws

2. Random Graphs • G(n, p): each edge (i,j) has probability p • Think of p as p(n) • Examples: • p = 1/2 Fagin’s framework • p = S/n2 expected graph size = S • p = 1/n3/2 • Etc.

3. Evolution • Erdos and Reny: evolution of random graphs, as p(n) “grows” from 0 to 1 • Spencer, Lynch, and others: evolution of FO sentences as p(n) “grows” from 0 to 1 • Will discuss next, following J. Spencer 0 … … … 1 p(n)

4. Proving a 0/1 Law Suppose a 0/1 law holds for some p(n) • Let T1 = { f | limn Pr(f) = 1} • T1 is a complete theory: 8f. f2 T1 or :f2 T1[ WHY ??] Hence: 0/1 law holds iff there exists a theory T s.t. • Th(T) = { f | T |= f} is complete (e.g w-categorical) • 8f2 T, limn Pr(f) = 1 (here we use the particular p(n))

5. 1. The Void p(n) ¿ 1/n2 • Expected graph size: p(n) £ n2! 0 • Hence limn Pr(G= ;) = 1 • What is the theory T here ? T = { Ø$ x.$ y.R(x,y) }

6. 1 1 0 0 2. On the k’th Day… 1/n1+1/(k-1)¿ p(n) ¿ 1/n1+1/k Let’s try some examples, for k = 4: 1/n4/3¿ p(n) ¿ 1/n5/4 H1 = Limn Pr(H1) = H2 = Limn Pr(H2) = H3 = Limn Pr(H3) = H4 = Limn Pr(H4) =

7. 2. On the k’th Day… The theory T: • There are no cycles • For every tree H with · k edges and 8 r >0H appears as a CC at least r times • There are no trees H with k+1 edges :9 x1…9 xk. R(x1,x2)Æ…ÆR(xk,x1) T is w categorical [ WHY ?? WHAT IS R HERE ?? ]

8. 3. On Day w 1/n1+e¿ p(n) ¿ 1/n, 8e Example: p(n) = 1/(n ln(n)) The theory T is: • There are no cycles • For every finite tree H and 8 r > 0H appears as a cc component at least r times T is NOT w categorical [ WHY ?? ] Still, it is complete [ WHY ?]

9. 4. Past the Double Jump 1/n ¿ p(n) ¿ ln(n)/n The theory T: • 8 k, there are no k nodes with ¸ k+1 edges • 8 k ¸ 3, 8 r: there are ¸ r copies of the cycle Ck • 8 k ¸ 3, s, d: there is no Ck and a vertex of degree d at distance s from Ck • For every tree H and 8 r > 0H appears as a cc at least r times Again, T is not w-categorical, but still complete

10. 5. Beyond Connectivity Ln(n)/n ¿ p(n) ¿ 1/n1-e, 8e Now the random graph G(n,p) becomes connected The theory T is: • 8 k: there are no k nodes with ¸ k+1 edges • For every d: all vertices have at least d neighbors • For every r and k ¸ 3: there exists at least r copies of Ck And, again, T is not w-categorical, but still complete

11. 6. a Irrational P(n) = 1/na, a2 (0,1) irrational Then FO has a 0/1 law The theory T is: • For every graph H with v vertices e edges s.t. v < e a, there does not exists a copy of H • “Generic extension axioms” There are complicated, but mimic extension axioms.

12. 7. a Rational P(n) = 1/na, a2 (0,1) irrational Then FO has no 0/1 law