1. A Well-Mixed Function with Circuit Complexity 5n ±o(n) - Tightness of the Lachish-Raz-type Bounds - Kazuyuki Amano (Gunma Univ., Japan) Jun Tarui (Univ. of Electro-Comm., Japan)

2. Circuit Complexity Goal Give a “good” lower bound on size(f) for an explicitly defined Boolean function f size(f): min. # of gates in a Boolean circuit that computes f xxx 12n U 2 := all 16 Boolean functions on 2 vars - {, ≡ } Boolean circuit: combinational circuit consisting of gates in U 2

3. Brief History Explicit Lower Bounds 4n 4.5n 5n [Zwick SICOMP 91] [Lachish, Raz STOC 01] [Iwama, Morizumi MFCS 02] Current best lower bound for a function in NP ・ No Super-linear lower bounds are known for a function in NP ・ All results are shown by “Gate-Elimination Method” ??? ・ Target of 4.5n and 5n bounds is “k-mixed” function, which we will explain next...

4. Partial Assignment ρ: { x 1,x 2,...,x n } → { 0, 1, ＊ } f|ρ := function obtained from f by f(x 1,x 2,..., x n ) : Boolean function on n vars Def. Ex.: f = x1 x2 ∨ x3 ρ: ( x1, x2, x3 ) → ( 1, ＊, 0 ) f|ρ = x2 xixi ρ(x i ) if ρ(x i ) = 0 or 1, x i remains free if ρ(x i ) = ＊

5. k-mixed f: Boolean function on { x 1,x 2,...,x n } is k-mixed f|α ≠ f|β ∀ V ⊆ { x 1,x 2,...,x n } with |V| = k ∀ α≠β s.t. α and β fix all variables in V k-mixed = any two distinct partial assignments to the same set of k variables yield different subfunctions on n-k variables Def. [Jukna ’88]

6. Ex.: f = x 1 x 2 x n 1-mixed ? ∀ i f| xi = 0 ≠ f| xi = 1 Yes ! 2-mixed ? f| xi=0,xj=0 ＝ f| xi=1,xj=1 No ! k-mixed f: Boolean function on { x 1,x 2,...,x n } is k-mixed f|α ≠ f|β ∀ V ⊆ { x 1,x 2,...,x n } with |V| = k ∀ α≠β s.t. α and β fix all variables in V Def. [Jukna ’88]

7. k-mixed = any two distinct partial assignments to the same set of k variables yield different subfunctions on n-k variables ！ Highly mixed function may have high complexity... k-mixed f: Boolean function on { x 1,x 2,...,x n } is k-mixed f|α ≠ f|β ∀ V ⊆ { x 1,x 2,...,x n } with |V| = k ∀ α≠β s.t. α and β fix all variables in V Def. [Jukna ’88]

8. Motivation and... Every n-o(n)-mixed function on n variables has circuit complexity at least 5n-o(n) Theorem [Iwama,Lachish,Morizumi,Raz ’01+’02] Such a function with circuit complexity O(n log n) is known. [Savicky,Zak, ’96] Can we improve the lower bound, or...

9. Motivation and Result Every n-o(n)-mixed function on n variables has circuit complexity at least 5n-o(n) Theorem [Iwama,Lachish,Morizumi,Raz ’01+’02] Such a function with circuit complexity O(n log n) is known. [Savicky,Zak, ’96] Theorem [Today] There is an n-o(n)-mixed function on n variables whose circuit complexity is at most 5n+o(n) Can we improve the lower bound, or...

10. Construction (1 of 2) X = { x 1, x 2,..., x n } f(X) := x w(X) ( just outputs w(X)-th input variable ) X B1B1 B2B2 BbBb size of each block ( B i ) = log 2 n # of blocks ( b ) = n / log 2 n PAR(B i ) = Parity over all variables in B i w ~ (X) = ∑ i ・ PAR(B i ) i=1..b Def. of w(X) ( ~ weighted sum of block parities )...

11. Construction (2 of 2) f(X) = x w(X) ( just outputs w(X)-th input variable ) w ~ (X) = ∑ i ・ PAR(B i ) i=1..b p(n) := smallest prime with p(n) ≧ n (note: p(n) ≦ 2n) w(X) = k w(X) ≡ k (mod p(n)) & k = 1 ~ n 1 otherwise 1. size(f) = 5n+o(n) 2. f is (n – c √n log 2 n)-mixed for some const. c ~ Theorem (Main)

12. Circuit w ~ (X) = ∑ i ・ PAR(B i ) i=1..b 1. Compute PAR(B i ) for each i 2. Compute bin. rep. of i ・ PAR(B i ) for each i 3. Compute bin. rep. of 4. Compute bin. rep. of w(X) from w(X) 5. Output x w(X) ~

13. Circuit w ~ (X) = ∑ i ・ PAR(B i ) i=1..b 1. Compute PAR(B i ) for each i 2. Compute bin. rep. of i ・ PAR(B i ) for each i 3. Compute bin. rep. of 4. Compute bin. rep. of w(X) from w(X) 5. Output x w(X) size(x y)=3 3n # of gates ~

14. Circuit w ~ (X) = ∑ i ・ PAR(B i ) i=1..b 1. Compute PAR(B i ) for each i 2. Compute bin. rep. of i ・ PAR(B i ) for each i 3. Compute bin. rep. of 4. Compute bin. rep. of w(X) from w(X) 5. Output x w(X) 3n # of gates ~ bin. rep. of iPAR(B i ) o(n)

15. Circuit w ~ (X) = ∑ i ・ PAR(B i ) i=1..b 1. Compute PAR(B i ) for each i 2. Compute bin. rep. of i ・ PAR(B i ) for each i 3. Compute bin. rep. of 4. Compute bin. rep. of w(X) from w(X) 5. Output x w(X) 3n # of gates ~ o(n) w(X) is a sum of b(=n/log 2 n) numbers each has log n digits, which can be computed in O(n/log n) size ~ o(n)

16. Circuit w ~ (X) = ∑ i ・ PAR(B i ) i=1..b 1. Compute PAR(B i ) for each i 2. Compute bin. rep. of i ・ PAR(B i ) for each i 3. Compute bin. rep. of 4. Compute bin. rep. of w(X) from w(X) 5. Output x w(X) 3n # of gates ~ o(n) w(X) can be computed from w(X) via several arithmetic operations ( × ， ÷ ，＋，－ ) on O(log n) digits number. o(n) ~

17. Circuit w ~ (X) = ∑ i ・ PAR(B i ) i=1..b 1. Compute PAR(B i ) for each i 2. Compute bin. rep. of i ・ PAR(B i ) for each i 3. Compute bin. rep. of 4. Compute bin. rep. of w(X) from w(X) 5. Output x w(X) 3n # of gates o(n) 2n + o(n) ~ bin. rep. of w(X) x1x2x1x2 xnxn x w(X) MULTIPLEXER Construction of size 2n+o(n) is known. [Klein, Paterson ’80]

18. Circuit w ~ (X) = ∑ i ・ PAR(B i ) i=1..b 1. Compute PAR(B i ) for each i 2. Compute bin. rep. of i ・ PAR(B i ) for each i 3. Compute bin. rep. of 4. Compute bin. rep. of w(X) from w(X) 5. Output x w(X) 3n # of gates o(n) 2n + o(n) ~ Total : 5n + o(n) q.e.d.

19. Proof sketch for “f is well-mixed” ．．． α ， β: partial assignments with c √n log 2 n ＊ ’s Note : at least c √n blocks contain at least one ＊ Find an assignment x* to ＊ -variables such that f|α(x*) ≠ f|β(x*) ．．． α β ０１＊＊１００１１＊ ０１＊＊１１１１１＊ ＊１１０＊ Goal

20. More detail... ．．． α β ０１＊＊１００１１＊ ０１＊＊１１１１１＊ ＊１１０＊ w(α ０ ) = w(β ０ ) ( f|α( ０ )= ， f|β( ０ )= ) α ０， β ０ : every ＊ is assigned by ０ in α ， β w(α ０ ) = w(β ０ ) Simple Case

21. More detail... ．．． α β ０１＊＊１００１１＊ ０１＊＊１１１１１＊ ＊１１０＊ w(α ０ ) = w(β ０ ) ・ assigning odd １ ’s to ＊ -variables in i-th block moves index by i

22. More detail... ．．． α β ０１＊＊１００１１＊ ０１＊＊１１１１１＊ ＊１１０＊ w(α ０ ) = w(β ０ ) ・ assigning odd １ ’s to ＊ -variables in i-th block moves index by i １ １

23. More detail... ．．． α β ０１＊＊１００１１＊ ０１＊＊１１１１１＊ ＊１１０＊ w(α ０ ) = w(β ０ ) ・ assigning odd １ ’s to ＊ -variables in i-th block moves index by i ・ find a good assignment x* to ＊ -variables that moves to, i.e., values of α and β differ f|α(x*) =, f|β(x*) =

24. Key lemma p: prime H: subset of {0,1,...,p-1} with size ≧ c√p ∀ k ∈ {0,1,...,p-1} ∃ A ⊆ H ∑ a ≡ k (mod p) a ∈ A Theorem [da Silva,Hamidoune ’94] Intuitively, if there are at least c√n blocks which has a ＊ -variable then we can move to an arbitrary position...

25. More detail... ．．． α β ０１＊＊１００１１＊ ０１＊＊１１１１１＊ ＊１１０＊ w(α ０ ) = w(β ０ ) ・ assigning odd １ ’s to ＊ -variables in i-th block moves index by i ・ find a good assignment x* to ＊ -variables that moves to, i.e., values of α and β differ f|α(x*) =, f|β(x*) =

26. Yet more detail... ．．． α β ０１＊＊１００１１＊ ０１＊＊１１１１１＊ ＊１１０＊ w(α ０ ) ≠ w(β ０ ) ( f|α( ０ )= ， f|β( ０ )= ) ・ find a good assignment x* to ＊ -variables that moves to, i.e., f|α(x*) =, f|β(x*)= values of α and β differ w(α ０ ) ≠ w(β ０ ) General Case q.e.d

27. Conclusion So, we need to find another property to improve the lower bound... Every n-o(n)-mixed function on n variables has circuit complexity at least 5n-o(n) Theorem [Iwama,Lachish,Morizumi,Raz ’01+’02] Theorem [Today] There is an n-o(n)-mixed function on n variables whose circuit complexity is at most 5n+o(n) Thank you.