1. The First-Order Variable Hierachy on Ordered Graphs Benjamin Rossman MIT

2. Bounded Variable Logics The variable complexity of a first-order formula is the maximum number of free variables in a subformula of. FO m = { first-order formulas with variable complexity m } Example: (in the language {E,<} of ordered graphs) 9x 1 9x 2 ( x 1 <x 2 Æ Ex 1 x 2 Æ 9x 3 ( x 2 <x 3 Æ Ex 2 x 3 Æ 9x 4 ( x 3 <x 4 Æ Ex 3 x 4 Æ 9x 5 ( x 4 <x 5 Æ Ex 4 x 5 ). This sentence expresses "there is an increasing path of length 5". Its variable complexity is 2.

3. Bounded Variable Logics The variable complexity of a first-order formula is the maximum number of free variables in a subformula of. FO m = { first-order formulas with variable complexity Example: 9x 9y ( x<y Æ Exy Æ 9x ( y<x Æ Eyx Æ 9y ( x<y Æ Exy Æ 9x ( y<x Æ Eyx ). This sentence expresses "there is an increasing path of length 5". Its variable complexity is 2. Variable complexity 2 means we can rewrite the formula using only 2 distinct variables (quantified multiple times). Hence, FO m is called the m-variable fragment of first-order logic.

4. Bounded Variable Logics The variable hierarchy refers to FO 1 ½ FO 2 ½ ½ FO m ½ where FO = m¸1 FO m. Question: Is the variable hierarchy strict/non- collapsing (in terms of expressive power) on a given class of structures? The answer is YES on the class of all structures (or all finite structures): "the universe contains at least m elements" is expressible in FO m but not FO m-1.

5. Bounded Variable Logics Some collapse results: FO ´ FO 2 on finite linear orders. FO ´ FO 3 on finite linear orders with any number of unary relations [Poizat '82]. Question: What about ordered graphs (= finite simple graphs with a linear order)? k-CLIQUE is a natural candidate property for proving the separation FO k-1 < FO k on ordered graphs.

6. Bounded Variable Logics One can show 3-CLIQUE is not definable in FO 2 by playing the 2-pebble Ehrenfeucht- Fraisse game on (seq. of) ordered graphs: It had been an open question (since [Immerman '82]) whether FO ´ FO 3 on ordered graphs. G1G1 G2G2

7. Bounded Variable Logics One can show 3-CLIQUE is not definable in FO 2 by playing the 2-pebble Ehrenfeucht- Fraisse game on (seq. of) ordered graphs: It had been an open question (since [Immerman '82]) whether FO ´ FO 3 on ordered graphs. G1G1 G2G2 Games on ordered graphs become difficult with ¸ 3 pebbles/variables. One explanation may be that every finite ordered graph is defined up to isomorphism by a sentence of FO 3.

8. Bounded Variable Logics Theorem [R. '08]. k-CLIQUE is not definable in FO bk/4c. Corollary. FO bk/4c < FO k on ordered graphs. Lemma [Immerman '08]. On ordered graphs, FO k-1 ´ FO k ) FO k ´ FO k+1 (i.e., if the variable hierarchy does not collapse, then it is strict). Corollary. The variable hierarchy is strict (i.e., FO k < FO k+1 ) on ordered graphs.

9. Bounded Variable Logics Theorem [R. '08]. k-CLIQUE is not definable in FO bk/4c. Corollary. FO bk/4c < FO k on ordered graphs. Lemma [Immerman '08]. On ordered graphs, FO k-1 ´ FO k ) FO k ´ FO k+1 (i.e., if the variable hierarchy does not collapse, then it is strict). Corollary. The variable hierarchy is strict (i.e., FO k < FO k+1 ) on ordered graphs. These results hold not only for ordered graphs, but for classes of finite graphs with arbitrary numerical predicates (e.g., arithmetic + and £).

10. Bounded Variable Logics Theorem [R. '08]. k-CLIQUE is not definable in FO bk/4c. Corollary. FO bk/4c < FO k on ordered graphs. Lemma [Immerman '08]. On ordered graphs, FO k-1 ´ FO k ) FO k ´ FO k+1 (i.e., if the variable hierarchy does not collapse, then it is strict). Corollary. The variable hierarchy is strict (i.e., FO k < FO k+1 ) on ordered graphs. Moreover, we show that k/4 variables cannot distinguish classes {ordered graphs with no k-cliques} and {ordered graphs with a K- clique} for all k < K.

11. Circuits Circuits are comprised of AND, OR, NOT gates (with unbounded fan-in). We consider polynomial-size, constant-depth (i.e. AC 0 ) circuits with input variables (representing potential edges in an n-vertex graph). Descriptive Complexity: {m-variable, quantifier-depth d formulas} (on structures with arbitrary numerical predicates, e.g., linear order) ´ {non-uniform AC 0 circuits with size O(n m ) and depth d}. [Immerman '82]

12. AC 0 Circuit (in normal form) Æ ÆÆ ÇÇÇÇÇÇÇÇÇÇÇÇ ÆÆÆÆÆÆ ÇÇÇ................ Ç Æ 1,2n-1,n1,21,3 1,4 i, j n-1,nn-2,n... size n O(1) depth O(1) Ç

13. AC 0 Lower Bound for k-Clique The k-clique problem (on graphs of size n) requires depth-d circuits of what size? Trivial upper bound: O(n k ) (acheived in depth 2) J. Lynch '86: (n p(k/d 3 ) ) P. Beame '90: (n k/89d 2 )

14. AC 0 Lower Bound for k-Clique The k-clique problem (on graphs of size n) requires depth-d circuits of what size? Trivial upper bound: O(n k ) (acheived by depth 2) J. Lynch '86: (n p(k/d 3 ) ) P. Beame '90: (n k/89d 2 ) These lower bounds degrade in the exponent of n as the depth d increases, becoming trivial when d > pk.

15. AC 0 Lower Bound for k-Clique The k-clique problem (on graphs of size n) requires depth-d circuits of what size? Trivial upper bound: O(n k ) (acheived in depth 2) J. Lynch '86: (n p(k/d 3 ) ) P. Beame '90: (n k/89d 2 ) We show: (n k/4 )

16. k-Clique Lower Bound The k-clique problem (on graphs of size n) requires depth-d circuits of what size? Trivial upper bound: O(n k ) (acheived in depth 2) J. Lynch '86: (n p(k/d 3 ) ) P. Beame '90: (n k/89d 2 ) We show: (n k/4 ) We eliminate dependence on d in the exponent of n. Result holds up to depth d = O(plog n) and potentially up to clique- size k = o(log n).

17. Erdos-Renyi Random Graphs The random graph ErdosRenyi(n,p), p = p(n) 2 [0,1], has n vertices. Edges are independently included with probability p. We parameterize p(n) = n – where > 0 (so small ! large p ! denser graph). = 2/(k-1) is the threshold for the appearance of k-cliques. That is, ER(n,n - ) almost surely has k-cliques if < 2/(k-1). ER(n,n - ) almost surely has no k-clique if > 2/(k-1).

18. ErdosRenyi(n,p) n = 50 p = n –0.68 (p is above the threshold n -2/3 for the appearance of 4-cliques) Erdos-Renyi Random Graphs

19. A Bit of Notation Notation 1. For an ordered graph G = hV, E, <i and a set A µ V, let G[A] = G [ (clique on A) = hV, E [, <i. Notation 2.For ordered graphs G and H,, G and H satisfy the same FO m sentences of quantifier-depth d, 9 winning strategy for Duplicator in the d-round m-pebble game on G,H

20. Random Graph + Random Cliques Let G = Erdos-Renyi(n,1/2) with linear order. Let e = random 2-element subset of {1,...,n}. Theorem. [Ajtai, Furst-Saxe-Sipser, Hastad, Boppana,...] For every (quantifier-depth) d, almost surely as n ! 1. NB.This is a disguised way of saying that AC 0 functions have average sensitivity n o(1).

21. Random Graph + Random Cliques Let G = Erdos-Renyi(n,n – 2/(k – 1.5) ) with lin. order. NB. With high probability, G has no k-clique but n (1) many (k–1)-cliques. X µ {1,...,n} random of size |X| = k–1. Obs 1. For every (quantifier-depth) d, asymptotically almost surely

22. Random Graph + Random Cliques Let G = Erdos-Renyi(n,n – 2/(k – 1.5) ) with lin. order. NB. With high probability, G has no k-clique but n (1) many (k–1)-cliques. X, A µ {1,...,n} random of size |X| = k–1, |A| = k. Obs 1. For every (quantifier-depth) d, asymptotically almost surely Obs 2. a.a.s. /

23. Random Graph + Random Cliques Let G = Erdos-Renyi(n,n – 2/(k – 1.5) ) with lin. order. NB. With high probability, G has no k-clique but n (1) many (k–1)-cliques. X, A µ {1,...,n} random of size |X| = k–1, |A| = k. Obs 1. For every (quantifier-depth) d, asymptotically almost surely Obs 2. a.a.s. Main Theorem. For every (quantifier-depth) d, a.a.s. /

24. Theorem. G = Erdos-Renyi(n, n –2/(k–1.5) ), |A| = k. Then a.a.s. for every d. Proof Idea. For each k/4-tuple of vertices v = (v 1,...,v k/4 ), we identify a small subset B(v) µ A (of size · k/2, but empty for most v). Key Property: For every B(v) µ C µ A such that |C| · k/2, hG[B(v)], vi ´ d hG[C], vi.

25. Theorem. G = Erdos-Renyi(n, n –2/(k–1.5) ), |A| = k. Then a.a.s. for every d. Proof Idea. For each k/4-tuple of vertices v = (v 1,...,v k/4 ), we identify a small subset B(v) µ A (of size · k/2, but empty for most v). Key Property: For every B(v) µ C µ A such that |C| · k/2, hG[B(v)], vi ´ d hG[C], vi. In other words: Once you add the clique on B(v) to the pebbled structure hG, vi, adding a larger subclique of A up to size · k/2 does not change the ´ d -theory.

26. Theorem. G = Erdos-Renyi(n, n –2/(k–1.5) ), |A| = k. Then a.a.s. for every d. Proof Idea. For each k/4-tuple of vertices v = (v 1,...,v k/4 ), we identify a small subset B(v) µ A (of size · k/2, but empty for most v). Key Property: For every B(v) µ C µ A such that |C| · k/2, hG[B(v)], vi ´ d hG[C], vi. Duplicator's Strategy: "Mentally substitute" the pebbled structure hG[B(v)], vi for hG[A], vi.

27. Theorem. G = Erdos-Renyi(n, n –2/(k–1.5) ), |A| = k. Then a.a.s. for every d. Def. Call v = (v 1,...,v k/4 ), w = (w 1,...,w k/4 ) 2 [n] k/4 neighbors if v i w i for exactly one i 2 {1,...,k/4}. Suppose we could show (with high probability): There exist sets B(v) µ A for all v 2 [n] k/4 s.t. 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all nbrs v, w. Then the conclusion follows easily!

28. Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 u1u1 u2u2 u3u3 01234560123456 GG[A] Round

29. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 u1u1 u2u2 u3u3 xu2u2 u3u3 01234560123456 GG[A] Round Duplicator matches Spoiler exactly in Round 1. Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

30. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 u1u1 u2u2 u3u3 xu2u2 u3u3 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], x, u 2, u 3 i ´ d hG[B(x,u 2,u 3 )], x, u 2, u 3 i We have: G = G[B(u 1,u 2,u 3 )] by (1) ´ d above by (3) Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

31. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 u1u1 u2u2 u3u3 xu2u2 u3u3 01234560123456 GG[A] Round Suppose Spoiler moves pebble 2 to y in G. hG[B(u 1,u 2,u 3 )], x, u 2, u 3 i ´ d hG[B(x,u 2,u 3 )], x, u 2, u 3 i Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

32. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 u1u1 u2u2 u3u3 xu2u2 u3u3 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], x, y, u 3 i ´ d-1 hG[B(x,u 2,u 3 )], x, y', u 3 i Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w. Duplicator plays an auxiliary game to find vertex y'.

33. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], x, y, u 3 i ´ d-1 hG[B(x,u 2,u 3 )], x, y', u 3 i Duplicator replies by moving pebble 2 to y' in G[A]. Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

34. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], x, y, u 3 i ´ d-1 hG[B(x,u 2,u 3 )], x, y', u 3 i ´ d hG[B(x,y',u 3 )], x, y', u 3 i assumption (3) Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

35. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 x z'' 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], x, y, u 3 i ´ d-1 hG[B(x,u 2,u 3 )], x, y', u 3 i ´ d hG[B(x,y',u 3 )], x, y', u 3 i Suppose Spoiler moves pebble 3 to z'' in G[A]. Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

36. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 x z'' 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], x, y, zi ´ d-2 hG[B(x,u 2,u 3 )], x, y', z'i ´ d-1 hG[B(x,y',u 3 )], x, y', z''i Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w. Duplicator plays two auxiliary games to find vertex z.

37. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 xyz u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 x z'' 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], x, y, zi ´ d-2 hG[B(x,u 2,u 3 )], x, y', z'i ´ d-1 hG[B(x,y',u 3 )], x, y', z''i Duplicator replies by moving pebble 1 to z in G. Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

38. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 xyz u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 x z'' 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], x, y, zi ´ d-2 hG[B(x,u 2,u 3 )], x, y', z'i ´ d-1 hG[B(x,y',u 3 )], x, y', z''i ´ d hG[B(x,y',z'')], x, y', z''i assumption (3) Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

39. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 xyz tyz u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 x z'' 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], x, y, zi ´ d-2 hG[B(x,u 2,u 3 )], x, y', z'i ´ d-1 hG[B(x,y',u 3 )], x, y', z''i ´ d hG[B(x,y',z'')], x, y', z''i Spoiler Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

40. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 xyz tyz u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 x z'' t'''y'z'' 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], t, y, zi ´ d-3 hG[B(x,u 2,u 3 )], t', y', z'i ´ d-2 hG[B(x,y',u 3 )], t'',y', z''i ´ d-1 hG[B(x,y',z'')], t''',y',z''i ´ d hG[B(t''',y',z'')], t''',y',z''i Duplicator Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

41. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 xyz tyz u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 x z'' t'''y'z'' t'''y'j '''' 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], t, y, zi ´ d-3 hG[B(x,u 2,u 3 )], t', y', z'i ´ d-2 hG[B(x,y',u 3 )], t'',y', z''i ´ d-1 hG[B(x,y',z'')], t''',y',z''i ´ d hG[B(t''',y',z'')], t''',y',z''i Spoiler Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

42. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 xyz tyz tyj u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 x z'' t'''y'z'' t'''y'j '''' 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], t, y, ji ´ d-4 hG[B(x,u 2,u 3 )], t', y', j'i ´ d-3 hG[B(x,y',u 3 )], t'',y', j''i ´ d-2 hG[B(x,y',z'')], t''',y',j'''i ´ d-1 hG[B(t''',y',z'')], t''',y',j''''i ´ d hG[B(t''',y',j'''')], t''',y',j''''i Duplicator Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

43. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 xyz tyz tyj u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 x z'' t'''y'z'' t'''y'j '''' 01234560123456 GG[A] Round hG[B(u 1,u 2,u 3 )], t, y, ji ´ d-4 hG[B(x,u 2,u 3 )], t', y', j'i ´ d-3 hG[B(x,y',u 3 )], t'',y', j''i ´ d-2 hG[B(x,y',z'')], t''',y',j'''i ´ d-1 hG[B(t''',y',z'')], t''',y',j''''i ´ d hG[B(t''',y',j'''')], t''',y',j''''i Auxiliary games on G + various small cliques give strategy for G versus G + large clique Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

44. Winning strategy for Duplicator in hG, ui hG[A], ui (k=12) u1u1 u2u2 u3u3 xu2u2 u3u3 xyu3u3 xyz tyz tyj etcetera u1u1 u2u2 u3u3 xu2u2 u3u3 xy'u3u3 x z'' t'''y'z'' t'''y'j '''' etcetera 01234560123456 GG[A] Round...and so on, for d rounds. NB. Assumption (2) implies this is a winning strategy for Duplicator (i.e., quantifier-free types are preserved in each round). Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w.

45. Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w. Alas, this assumption is too strong! We cannot actually find sets B(v) µ A, v 2 [n] k/4, meeting these conditions.

46. Assume: There exist sets B(v) µ A, v 2 [n] k/4, such that 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ d hG[B(w)], vi for all neighbors v, w. Alas, this assumption is too strong! We cannot actually find sets B(v) µ A, v 2 [n] k/4, meeting these conditions. So we weaken this assumption (by simply restricting the notion of neighboring tuples). The weaker assumption will still imply

47. What actually works For a sufficiently small constant > 0, we fix an arbitrary tree T with vertices 1,...,n and degree n and diameter 2/ (plus all self-loops). Call v = (v 1,...,v m ), w = (w 1,...,w m ) 2 [n] m T-neighbors if 9i 2 {1,...,m} such that (v i,w i ) is an edge of T and v j = w j for all j i. T-guarded d-round Ehrenfeucht-Fraisse game on ordered graphs H 1,H 2 with vertex set [n]: Spoiler and Duplicator are constrained to move along edges of T. (Notation. )

48. We are able to show: Lemma. W.h.p. 9 sets B(v) µ A, v 2 [n] k/4, s.t. 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ T, 2d/ hG[B(w)], vi for all T-neighbors v and w. It follows that (by same argument). We conclude that since 2/ = Diam(T) T-guarded rounds simulate one unguarded round. What actually works

49. We are able to show: Lemma. W.h.p. 9 sets B(v) µ A, v 2 [n] k/4, s.t. 1. B(u) = ; for some u, 2. B(v) ¶ A Å {v 1,...,v k/4 } for all v, 3. hG[B(v)], vi ´ T, 2d/ hG[B(w)], vi for all T-neighbors v and w. It follows that (by same argument). We conclude that since 2/ = Diam(T) T-guarded rounds simulate one unguarded round. What actually works These sets B(v) have a nifty definition! Main challenge: Proving 8v, |B(v)| · k/2 with high probability. Arguments involve Hastad's Switching Lemma & new results on randomly restrictly AC 0 functions.

50. Main Theorem (circuit version) Let > 0 and k 2 N. Suppose C is a sequence of constant-depth circuits of size O(n (1+ )/2 ) on inputs. Then almost surely C has the same value on 1. a random graph G = ErdosRenyi(n,n – ), 2. the graph G [ (random k-clique).

51. Full Sensitivity A Boolean function f : n ! is fully sensitive if it depends on all n input bits. Formally: for every i 2 [n] = {1,...,n}, there exists x 2 {0,1} n such that f(x) f(x with i th bit flipped).

52. Random Restrictions Fix a Boolean function f : n ! A restriction is a function : n ! ¤. That is, for each input bit of f, the restriction either fixes a value (0 or 1) or leaves the input bit "unassigned" (¤). We can apply to f to get a function fd : -1 ¤ !.

53. Random Restrictions Suppose f : n ! is an AC 0 function (computed by poly-size constant-depth circuits). Let : n ! ¤ be a random restriction subject to 1. | -1 (¤)| = k (for a fixed constant k) 2. Pr[ (i) = 1 | (i) ¤] = 1/2 So restriction fd : k ! has k input bits.

54. Random Restrictions Suppose f : n ! is an AC 0 function (computed by poly-size constant-depth circuits). Let : n ! ¤ be a random restriction subject to 1. | -1 (¤)| = k (for a fixed constant k) 2. Pr[ (i) = 1 | (i) ¤] = 1/2 So restriction fd : k ! has k input bits. Lemma 1. Pr[fd is fully sensitive] · 1/n k - o(1)

55. Random Restrictions Suppose f : n ! is an AC 0 function (computed by poly-size constant-depth circuits). Let : n ! ¤ be a random restriction subject to 1. | -1 (¤)| = k (for a fixed constant k) 2. Pr[ (i) = 1 | (i) ¤] = 1/2. So restriction fd : k ! has k input bits. Lemma 1. Pr[fd is fully sensitive] · 1/n k - o(1) Proof uses a standard application of Hastad's Switching Lemma. In the special case k = 1, lemma says that AC 0 functions have average sensitivity n o(1).

56. Lemma 1. Pr[fd is fully sensitive] · 1/n k - o(1) Proof. Consider an AC 0 circuit computing f. f Fix small > 0 and hit f with a random restriction : [n] ! {0,1,¤} such that Pr[ (i) = ¤] = n – Pr[ (i) = 1 | (i) ¤] = 1/2

57. Lemma 1. Pr[fd is fully sensitive] · 1/n k - o(1) Proof. fd Fix small > 0 and hit f with a random restriction : [n] ! {0,1,¤} such that Pr[ (i) = ¤] = n – Pr[ (i) = 1 | (i) ¤] = 1/2 expected size n 1- ¤00¤¤¤¤¤110110¤10¤¤¤¤¤1011001010¤

58. Lemma 1. Pr[fd is fully sensitive] · 1/n k - o(1) Proof. fd expected size n 1- ¤00¤¤¤¤¤110110¤10¤¤¤¤¤1011001010¤ Hastad's Switching Lemma ) there is a constant c such that fd is computed by a decision tree of depth c (and hence depends on at most 2 c = O(1) inputs) with high probability (better than 1 - 1/n k ). 2c2c

59. Lemma 1. Pr[fd is fully sensitive] · 1/n k - o(1) Proof. expected size n 1- 00¤¤¤110110101011001010 Pick : [n] ! {0,1,¤} by randomly setting all but k variables in -1 (¤) to 0 or 1. 2c2c 1101100010 fd Assuming fd has a small decision tree, we have

60. Random Restrictions Suppose f : n ! is an AC 0 function. Let : n ! ¤ be a random restriction subject to 1. | -1 (¤)| = k 2. Pr[ (i) = 1 | (i) ¤] = 1/2 So restriction fd : k ! has k input bits. Lemma 1. Pr[fd is fully sensitive] · 1/n k - o(1)

61. Random Restrictions Suppose f : n ! is an AC 0 function. Let : n ! ¤ be a random restriction subject to 1. | -1 (¤)| = k 2. Pr[ (i) = 1 | (i) ¤] = n – So restriction fd : k ! has k input bits. Lemma 2. Pr[fd is fully sensitive] · 1/n k – k – o(1)

62. Lemma 2. Pr[fd is fully sensitive] · 1/n k – k – o(1) Proof. f Apply Lemma 1 to this modified circuit, which generates an n - biased distribution on n bits from the uniform distribution on n log(n) bits. Æ log(n) Æ Æ.........................

63. Full Vertex Sensitivity A Boolean function f : ! (taking graphs as inputs) is fully vertex sensitive if every "vertex" belongs to a sensitive "edge". Formally: for every i 2 [n], there exist j 2 [n] - {i} and a graph G with vertex set [n] such that f(G) f(G © {i,j}).

64. Random Restrictions Suppose f : ! is an AC 0 function. Let : ! ¤ be a random restriction subject to -1 (¤) µ is a k-clique 2. Pr[ (i) = 1 | (i) ¤] = n – So restriction fd : ! has inputs. Lemma 3. Pr[fd is fully vertex sensitive] · 1/n k – – o(1) Proof. Same idea as in lemmas of 1 and 2.

65. Random Restrictions Suppose f : ! is an AC 0 function. Let : ! ¤ be a random restriction subject to -1 (¤) µ is a k-clique 2. Pr[ (i) = 1 | (i) ¤] = n – So restriction fd : ! has inputs. Lemma 3. Pr[fd is fully vertex sensitive] · 1/n k – – o(1) Obs: n k – is roughly the expected number of k-cliques in G(n,n – ).

66. Open Questions Is k-CLIQUE definable in k-1 variables? (We showed that k/4 are required.) Does k-CLIQUE require constant-depth circuits of size (n k– ) for every ?

67. Thank you!