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

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

x x x1 2 n

U2 := all 16 Boolean functions on 2 vars - { , ≡} Boolean circuit: combinational circuit consisting of gates in U2

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...

Partial Assignment

ρ: { x1,x2,...,xn } → { 0, 1, ＊ }

f|ρ := function obtained from f by

f(x1,x2, ..., xn) : Boolean function on n vars

Def.

Ex.: f = x1 x2 ∨ x3 ρ: ( x1, x2, x3 ) → ( 1, ＊ , 0 )

f|ρ = x2

xi ρ(xi) if ρ(xi) = 0 or 1,

xi remains free if ρ(xi) = ＊

k-mixed

f: Boolean function on { x1,x2,...,xn } is k-mixed

f|α ≠ f|β

∀V ⊆ { x1,x2,...,xn } 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]

Ex.: f = x1 x2 xn

1-mixed ?

∀i f|xi = 0 ≠ f|xi = 1Yes !

2-mixed ?

f|xi=0,xj=0 ＝ f|xi=1,xj=1 No !

k-mixed

f: Boolean function on { x1,x2,...,xn } is k-mixed

f|α ≠ f|β

∀V ⊆ { x1,x2,...,xn } with |V| = k∀α≠β s.t. α and β fix all variables in V

Def. [Jukna ’88]

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 { x1,x2,...,xn } is k-mixed

f|α ≠ f|β

∀V ⊆ { x1,x2,...,xn } with |V| = k∀α≠β s.t. α and β fix all variables in V

Def. [Jukna ’88]

Motivation and ...

Every n-o(n)-mixed function on n variables hascircuit 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 ...

Motivation and Result

Every n-o(n)-mixed function on n variables hascircuit 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 variableswhose circuit complexity is at most 5n+o(n)

Can we improve the lower bound, or ...

Construction (1 of 2)

X = { x1, x2, ..., xn }

f(X) := xw(X)( just outputs w(X)-th input variable )

X

B1 B2 Bb

size of each block ( Bi ) = log2n

# of blocks ( b ) = n / log2n

PAR(Bi) = Parity over all variables in Bi

w~ (X) = ∑ i ・ PAR(Bi)i=1..b

Def. of w(X) ( ~ weighted sum of block parities )

...

Construction (2 of 2)

f(X) = xw(X)( just outputs w(X)-th input variable )

w~ (X) = ∑ i ・ PAR(Bi)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 log2n)-mixed for some const. c

~

Theorem (Main)

Circuit

w~(X) = ∑ i ・ PAR(Bi) i=1..b

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)

~

Circuit

w~(X) = ∑ i ・ PAR(Bi) i=1..b

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)

size(x y)=3

3n

# of gates

~

Circuit

w~(X) = ∑ i ・ PAR(Bi) i=1..b

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

# of gates

~

bin. rep. of i PAR(Bi)

o(n)

Circuit

w~(X) = ∑ i ・ PAR(Bi) i=1..b

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

# of gates

~

o(n)

w(X) is a sum of b(=n/log2n) numbers each haslog n digits, which can be computed in O(n/log n) size

~

o(n)

Circuit

w~(X) = ∑ i ・ PAR(Bi) i=1..b

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

# of gates

~

o(n)

w(X) can be computed from w(X) via several arithmetic operations ( × ， ÷ ，＋，－ ) on O(log n) digits number.

o(n)

o(n)

~

Circuit

w~(X) = ∑ i ・ PAR(Bi) i=1..b

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

# of gates

o(n)

o(n)

o(n)

2n + o(n)

~

bin. rep. of w(X)x1x2 xn

xw(X)

Construction of size 2n+o(n) is known.

[Klein, Paterson ’80]

Circuit

w~(X) = ∑ i ・ PAR(Bi) i=1..b

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

# of gates

o(n)

o(n)

o(n)

2n + o(n)

~

Total : 5n + o(n)

q.e.d.

Proof sketch for “f is well-mixed”

．．．

α ， β: partial assignments with c √n log2n ＊’ sNote : 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＊＊１ ００１１＊

０１＊＊１ １１１１＊

＊１１０＊

＊１１０＊

Goal

More detail...

．．．

．．．

α

β

０１＊＊１ ００１１＊

０１＊＊１ １１１１＊

＊１１０＊

＊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

More detail...

．．．

．．．

α

β

０１＊＊１ ００１１＊

０１＊＊１ １１１１＊

＊１１０＊

＊１１０＊

　　

w(α ０ ) = w(β ０ )

・ assigning odd １’ s to ＊ -variables in i-th block moves index by i

More detail...

．．．

．．．

α

β

０１＊＊１ ００１１＊

０１＊＊１ １１１１＊

＊１１０＊

＊１１０＊

　　

w(α ０ ) = w(β ０ )

・ assigning odd １’ s to ＊ -variables in i-th block moves index by i

１

１

More detail...

．．．

．．．

α

β

０１＊＊１ ００１１＊

０１＊＊１ １１１１＊

＊１１０＊

＊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*) =

Key lemma

p: primeH: 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 ＊ -variablethen we can move to an arbitrary position...

More detail...

．．．

．．．

α

β

０１＊＊１ ００１１＊

０１＊＊１ １１１１＊

＊１１０＊

＊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*) =

Yet more detail...

．．．

．．．

α

β

０１＊＊１ ００１１＊

０１＊＊１ １１１１＊

＊１１０＊

＊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

Conclusion

So, we need to find another property to improvethe lower bound...

Every n-o(n)-mixed function on n variables hascircuit 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 variableswhose circuit complexity is at most 5n+o(n)

Thank you.