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Kazuyuki Amano (Gunma Univ., Japan) Jun Tarui (Univ. of Electro-Comm., Japan)

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). Circuit Complexity. x. x. x. 1. 2. n.

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Kazuyuki Amano (Gunma Univ., Japan) Jun Tarui (Univ. of Electro-Comm., Japan)

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  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 x x x 1 2 n U2 := all 16 Boolean functions on 2 vars - { , ≡} Boolean circuit: combinational circuit consisting of gates in U2 size(f): min. # of gates in a Boolean circuit that computes f Goal Give a “good” lower bound on size(f) for an explicitly defined Boolean function f

  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 f(x1,x2, ..., xn) : Boolean function on n vars ρ: { x1,x2,...,xn } → { 0, 1,＊ } Def. f|ρ := function obtained from f by ρ(xi) if ρ(xi) = 0 or 1, xi xi remains free if ρ(xi) = ＊ Ex.: f = x1 x2 ∨ x3 ρ: ( x1, x2, x3 ) → ( 1, ＊, 0 ) f|ρ = x2

  5. k-mixed Def. [Jukna ’88] f: Boolean function on { x1,x2,...,xn } is k-mixed ∀V ⊆ { x1,x2,...,xn } with |V| = k ∀α≠β s.t. α and β fix all variables in V f|α ≠ f|β k-mixed = any two distinct partial assignments to the same set of k variables yield different subfunctions on n-k variables

  6. k-mixed Def. [Jukna ’88] f: Boolean function on { x1,x2,...,xn } is k-mixed ∀V ⊆ { x1,x2,...,xn } with |V| = k ∀α≠β s.t. α and β fix all variables in V f|α ≠ f|β Ex.: f = x1 x2 xn 1-mixed ? ∀i f|xi = 0 ≠ f|xi = 1 Yes ! 2-mixed ? f|xi=0,xj=0＝ f|xi=1,xj=1 No !

  7. k-mixed Def. [Jukna ’88] f: Boolean function on { x1,x2,...,xn } is k-mixed ∀V ⊆ { x1,x2,...,xn } with |V| = k ∀α≠β s.t. α and β fix all variables in V f|α ≠ f|β 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...

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

  10. Construction (1 of 2) X = { x1, x2, ..., xn } f(X) := xw(X) ( just outputs w(X)-th input variable ) Def. of w(X) ( ~ weighted sum of block parities ) X ... B2 B1 Bb size of each block ( Bi ) = log2n # of blocks ( b ) = n / log2n PAR(Bi) = Parity over all variables in Bi ~ (X) = ∑ i・PAR(Bi) w i=1..b

  11. Construction (2 of 2) f(X) = xw(X) ( just outputs w(X)-th input variable ) ~ (X) = ∑ i・PAR(Bi) w 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 Theorem (Main) 1. size(f) = 5n+o(n) 2. f is (n – c √n log2n)-mixed for some const. c

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

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

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

  15. ~ w (X) = ∑ i・PAR(Bi) i=1..b Circuit # of gates 1. Compute PAR(Bi) for each i 2. Compute bin. rep. of i・PAR(Bi) for each i 3. Compute bin. rep. of 4. Compute bin. rep. of w(X) from w(X) 5. Output xw(X) 3n o(n) o(n) ~ ~ w(X) is a sum of b(=n/log2n) numbers each has log n digits, which can be computed in O(n/log n) size

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

  17. ~ w (X) = ∑ i・PAR(Bi) i=1..b xw(X) Construction of size 2n+o(n) is known. MULTIPLEXER x1x2 xn bin. rep. of w(X) Circuit # of gates 1. Compute PAR(Bi) for each i 2. Compute bin. rep. of i・PAR(Bi) for each i 3. Compute bin. rep. of 4. Compute bin. rep. of w(X) from w(X) 5. Output xw(X) 3n o(n) o(n) ~ o(n) 2n + o(n) [Klein, Paterson ’80]

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

  19. Proof sketch for “f is well-mixed” α ＊１１０＊ ０１＊＊１ ００１１＊ ．．． β ．．． ＊１１０＊ ０１＊＊１ １１１１＊ α，β: partial assignments with c √n log2n ＊’s Note : at least c √n blocks contain at least one ＊ Goal Find an assignment x* to ＊-variables such that f|α(x*) ≠ f|β(x*)

  20. More detail... w(α０) = w(β０) α ＊１１０＊ ０１＊＊１ ００１１＊ ．．． β ．．． ＊１１０＊ ０１＊＊１ １１１１＊ α０，β０ : every ＊ is assigned by ０ in α，β Simple Case w(α０) = w(β０) ( f|α(０)=　　，f|β(０)=)

  21. More detail... w(α０) = w(β０) α ＊１１０＊ ０１＊＊１ ００１１＊ ．．． β ．．． ＊１１０＊ ０１＊＊１ １１１１＊ ・assigning odd １’s to ＊-variables in i-th block moves index by i

  22. More detail... w(α０) = w(β０) α ＊１１０＊ ０１＊＊１ ００１１＊ ．．． １ β ．．． ＊１１０＊ ０１＊＊１ １１１１＊ １ ・assigning odd １’s to ＊-variables in i-th block moves index by i

  23. More detail... w(α０) = w(β０) values of α and β differ α ＊１１０＊ ０１＊＊１ ００１１＊ ．．． β ．．． ＊１１０＊ ０１＊＊１ １１１１＊ ・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., f|α(x*) = , f|β(x*) =

  24. Key lemma Theorem [da Silva,Hamidoune ’94] 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 Intuitively, if there are at least c√n blocks which has a ＊-variable then we can move to an arbitrary position...

  25. More detail... w(α０) = w(β０) values of α and β differ α ＊１１０＊ ０１＊＊１ ００１１＊ ．．． β ．．． ＊１１０＊ ０１＊＊１ １１１１＊ ・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., f|α(x*) = , f|β(x*) =

  26. Yet more detail... w(α０) ≠ w(β０) values of α and β differ α ＊１１０＊ ０１＊＊１ ００１１＊ ．．． β ．．． ＊１１０＊ ０１＊＊１ １１１１＊ General Case w(α０) ≠ w(β０) ( f|α(０)=　　，f|β(０)=) ・find a good assignment x* to ＊-variables that moves to , i.e., f|α(x*) = , f|β(x*)= q.e.d

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

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