Finite Model Theory Lecture 19 Summary on 0/1 Laws

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.

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 … … … 1 p(n)

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: 8 f. f 2 T1 or : f 2 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) 8 f 2 T, limn Pr(f) = 1 (here we use the particular p(n))

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) }

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 1 H1 = Limn Pr(H1) = H2 = Limn Pr(H2) = H3 = Limn Pr(H3) = H4 = Limn Pr(H4) =

2. On the k’th Day… The theory T: There are no cycles For every tree H with · k edges and 8 r >0 H 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 ?? ]

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

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 > 0 H appears as a cc at least r times Again, T is not w-categorical, but still complete

5. Beyond Connectivity Ln(n)/n ¿ p(n) ¿ 1/n1-e, 8 e 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

6. a Irrational P(n) = 1/na, a 2 (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.

7. a Rational P(n) = 1/na, a 2 (0,1) irrational Then FO has no 0/1 law