THE STRANGE LOGIC OF RANDOM GRAPHS. Joel Spencer. Courant Institute

Transcription

1 WoLLIC 2005 THE STRANGE LOGIC OF RANDOM GRAPHS Joel Spencer Courant Institute 1

2 What is a Graph? Vertex Set V Edge Set E of {v, w} V Equivalently: Areflexive Symmetric Relation Write: v w Read: v, w are adjacent Note to cognescenti: no loops, multiple edges Usually: n := V, number of vertices 2

3 First Order Language Relations = (equality), (adjacency) Usual Boolean,,,,... Universal v, w NOTE: Quantification only over vertices! There is a triangle: v w u (u v) (v w) (u w) Diameter at most two: v w [v = w v w u (v u u w)] Diameter at most k (k fixed) Connectivity: NO! 3

4 The Random Graph G(n, p) n vertices p = adjacency probability Usually p = p(n) p = 3 n ; p = n 1/2 ; p = ln n n 5 n Allow defined for n sufficiently large 4

5 Glebskii et. al. [1969]; Fagin [1976]: Set p = 1 2. Then for all first order sentences A lim Pr[G(n, p) = A] =0or1 n Extension Statement A r,s : For all distinct x 1,...,x r,y 1,...,y s there exists distinct z adjacent to all x i and no y j. Probability: Pr[ A r,s ] ( n )( n r) (1 2 r s ) n r s 0 r s Combinatorics: There is a unique countable model satisfying all A r,s. Logic: Therefore T = {A r,s } is complete. If T = A then lim n Pr[A] = 1 by compactness. Otherwise T = A and lim n Pr[A] =0. 5

6 Given Distribution µ n over n-point models Language Possible Outcomes Zero-One Law: Pr[A] 0or1forallA Convergence: lim Pr[A] exists for all A Slow Oscillation: Pr n+1 [A] Pr n [A] 0 Nonseparability: No oracle separating A with lim Pr[A] =1 froma with lim Pr[A] =0 6

7 Erdős-Rényi On the Evolution of Random Graphs Threshold Function r(n) is threshold function for A if p r(n) Pr[A] 0 p r(n) Pr[A] 1 Existence of Triangle: n 1 Existence of K 4 : n 2/3 Diameter Two: n 1/2 ln 1/2 n Connectivity: n 1 ln n 7

8 Shelah-Spencer [1988]: α (0, 1), irrational, lim n Pr[G(n, n α ) = A] =0or1 Zero-One Law for p(n) =n α Interpretation: n α is never a threshold function for a first order A. What happens in the evolution at p = n π/7? NOTHING! (in our First Order universe) 8

9 Lynch [1992]: p = cn 1 : Convergence. lim Pr[A] exists and is nice function of c. Pr[no triangle] e c3 /6 Pr[no isolated triangle] e c3 e 3c /6 Spencer, Thoma [1999]: p = ln n n + cn 1 : Convergence. lim Pr[A] exists and is nice function of c. Pr[no isolated vertices] e e c 9

10 Random Ordered Graph p = 1 2 Vertices 1,...,n Relations =,, < Express 1 2: x y (x<y) z (z<y z = x) (x y) Convergence does not hold Shelah: Slow Oscillation 10

11 Ehrenfeucht Game EHR[G 1,G 2 ; k] Parameters G 1,G 2 (disjoint); k = number rounds Players: Duplicator and Spoiler i-th Round Spoiler picks x i G 1 or y i G 2 Duplicator then picks y i G 2 or x i G 1 Duplicator wins if x i x j y i y j and x i = x j y i = y j E.g.: G 1 has isolated point, G 2 does not. Spoiler wins EHR[G 1,G 2 ;2] Ehrenfeucht: Duplicator wins EHR[G 1,G 2 ; k] if and only if G 1,G 2 have same first order properties of quantifier depth at most k 11

12 Ehrenfeucht Classes G 1 k G 2 if Duplicator wins EHR[G 1,G 2 ; k] Equivalence Relation Finite number of equivalence classes Very large (tower function!) number of equivalence classes 12

13 G 1 : Cycle length n G 2 : Two disjoint Cycles length n Thm: For all k if n sufficiently large Duplicator wins EHR[G 1,G 2 ; k] Proof Idea: With s moves remaining Duplicator assures that 3 s -neighborhoods of points chosen are isomorphic. Corollary: Connectivity not first order 13

14 Ehrenfeucht and Zero-One Law n-point random H n THM: Zero-One Law if and only if for all k lim m,n Pr[Dupl wins EHR[H m,h n ; k]] = 1 For arbitrary first order language Duplicator must preserve all relations 14

15 p = 1 2 Zero-One Law With Pr 1, H m,h n have all extension statements up to k points. Duplicator Strategy: Find point with proper adjacencies With Pr 1 strategy succeeds Why doesn t this always work?? p = n α, 1 2 <α<1, k =3 Some, not all v, w have common neighbor u Spoiler picks x 1,x 2 H m with common neighbor Duplicator needs foresight to pick y 1,y 2 H n with common neighbor 15

16 (R, H) extensions H on a 1,...,a r,b 1,...,b v with designated roots a 1,...a r. Assume no edges between roots. Ext(R, H): For all x 1,...,x r there exist y 1,...,y s with the edges (maybe more) of H. Every point in triangle Every two points joined by path of length seven Every two points x 1,x 2 in K 4 except maybe {x 1,x 2 } v = number nonroots; e = number of edges Dense: v eα < 0 Sparse: v eα > 0(dichotomy!) Rigid: All (R, H )dense,h H Safe: All (R,H) sparse, R R 16

17 ...and G(n, n α ) Expected number of extensions of x 1,...,x r is Θ(n v p e )=Θ(n v eα ) Dense. v eα < 0. Most x 1,...,x r have no (R, H) extension. E.g.: α = π/ Most pairs have no common neighbor Safe. v eα > 0 and no dense parts Thm: All x 1,...,x r have Θ(n v eα ) extensions. E.g.: α = π/7, all pairs joined by Θ(n 2 3α ) paths of length three. 17

18 t-closure cl t (X) ing For any 1 u t and any rigid (R, H) extension with u roots and any x 1,...,x u X with (R, H) extension to y 1,...,y v Add y 1,...,y v to X Iterate E.g: α = π/ t =1. X = {x 1,x 2 } Add common neighbors to any pair of X. Iterate 18

19 Bounded Closure Size E.g.: cl 1 (X) 44 for all X =2 n 2 choices of X Bounded number of pictures np 2 = n factor for each extension n 2 (np 2 ) 42 = o(1) 19

20 Duplicator Look-Ahead Strategy Constants 0 = a 0 <a 1 =1<...<a k Select so cl ai (x 1,...,x k i ) < (k i)+a i+1 After i rounds Duplicator assures that x s and y s have same a k i -closures. a = a i, b = a i+1, X =(x 1,...,x i ), Y =(y 1,...,y i ), x = x i+1,y = y i+1 Need: If cl a (X) = cl a (Y ) then after one round Duplicator can assure cl b (X, x) = cl b (Y,y) 20

21 Assume cl a (X) = cl a (Y ) WLOG Spoiler picks x G 1 Inside: x cl a (X). Duplicator picks isomorphic y cl a (Y ) Outside: Not Inside H =cl b (X, x), R = cl b (X, x) cl a (X) x H, x R, (R, H) safe Safe extensions always exist, find y 21

22 Zero-One Law Complete Theory Countable Model(s) p = n α,0<α<1 irrational. Countable list of safe (R, H) Countable list of witness requests E.g.: y1,y y 1 y 1 y 2 y Use new vertices to satisfy each witness request minimally. Get countable G Thm: G is Countable Model Thm: G independent of order of requests Thm: Theory not ℵ 0 -categorical 22

23 The Very Sparse Cases p<<n 2 : No Edge! n 2 p(n) n 3/2 No tree (or more) on 3 vertices (for all r) r (or more) isolated vertices r (or more) isolated edges ℵ 0 -Categorical n (k+1)/k p(n) n (k+2)/(k+1) No trees (or more) on k + 2 vertices All trees on k + 1 vertices ℵ 0 -Categorical p = n 1+o(1) and p n 1 All finite trees. No cycles Not ℵ 0 -Categorical: May have infinite trees! 23

24 p = c n Theory of A with Pr[A] 1: All trees as components No bicyclic (or more) subgraphs Open: Cycles and their Neighborhoods Countable Models: All tree components infinitely often Maybe infinite trees Maybe unicyclic graphs 24

25 Binary Strings Models {0, 1} = finite strings Set {1,...,n}; unary predicates U 0,U 1 U α (x) : x-th position α =;<; U α,α=0, 1 There exist two consecutive ones: x y [U 1 (x) U 1 (y) (x <y) z (x<z z <y)] Random String U(n, p): Pr[U 1 (x) =p] 25

26 Ehrenfeucht Semigroup σ k τ: Duplicator wins EHR[σ, τ ; k] Equivalence Relation E = set of equivalence classes E finite, though very large! σ k σ, τ k τ implies σ + τ k σ + τ E forms Semigroup under concatenation e = empty string mσ = nσ if m, n 3 k 26

27 Convergence for U(n, p) Ehrenfeucht: lim Pr[A] exists k = quantifier depth, E = equivalence classes Markov Chain! Initial State e =emptychain Pr[x x1] = p; Pr[x x0] = 1 p NonPeriodic Therefore: Stationary Distribution on E lim Pr[A] = lim Pr[x] over x with A. 27

28 Persistent Strings Following equivalent for x E k : y z x + y + z = x y z z + y + x = x p s y p + y + s = x x persistent in Markov Chain x called persistent. There exist (many) persistent x (very long!) Persistency not dependent on edge effects x persistent implies p + x + s persistent lim n Pr[persistent] = 1 28

29 Circular Strings Over Z n with C(x, y, z) = clockwise No edge effects Zero-One Law for p constant Thm (Shelah/JS): Zero-One Law if n 1/k p(n) n 1/(k+1) Countable Models (StJohn/JS): p n 1 Line Z All 0 n 1 n 1 n 1/2 Page Z 2 One 1 on each line n 1/2 p(n) n 1/3 Book Z 3 Each page with one line with two 1 s Volume, Library,... 29

30 Coming Attractions Thursday, 1:30 Analytic Questions Given Zero-One Law A with quantifier depth k Asymptotics of n(k) so that n n(k) Pr[A] < 0.01 or Pr[A] > 0.99 G(n, p) withp = 1 2 : ( ) n k 2 k (1 2 k ) n k 0 n =Θ(2 k k 2 ) 30

31 Succint Definitions General First Order Structure Def: D(G) = smallest quantifier depth of A that defines G What is D(G) for random n-element model? Kim/Pikhurko/Verbitsky/JS G(n, 1 2 ): Θ(lnn) StJohn/JS: G < (n, 1 2 ): Θ(ln n) BitString U(n, 1 2 ): Θ(ln ln n) 31