Hardness vs Randomness for Bounded Depth Arithmetic Circuits - PowerPoint PPT Presentation

NW generator - reducing #variables q ∈ C n f f f … m y | S 1 y | S 2 y | S n S n S 2 S 1 m Nisan- ` Wigderson y Design 14

NW generator - reducing #variables q ∈ C n ` y 14

NW generator - reducing #variables q ∈ C n ` y � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) 14

NW generator - reducing #variables q ∈ C n ` y � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) q 6⌘ 0 Q 6⌘ 0 Want : If then . 14

Key lemma 15

Key lemma q ∈ C y � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) q 6⌘ 0 Q 6⌘ 0 Goal : If , then . 15

Key lemma q ∈ C y � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) q ∈ Depth- ∆ Lemma : Let non-zero and a m-variate f multilinear polynomial of degree . If d � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) ≡ 0 15

Key lemma q ∈ C y � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) q ∈ Depth- ∆ Lemma : Let non-zero and a m-variate f multilinear polynomial of degree . If d � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) ≡ 0 √ d ) poly( n, d Then, f can be computed by a size and depth circuit. ∆ + 5 15

Key lemma q ∈ C y � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) q ∈ Depth- ∆ Lemma : Let non-zero and a m-variate f multilinear polynomial of degree . If d � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) ≡ 0 √ d ) poly( n, d Then, f can be computed by a size and depth circuit. ∆ + 5 f / Q ( y ) 6⌘ 0 , 8 q 2 Depth- ∆ ∈ Depth- ∆ + 5 15

Key lemma q ∈ C y � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) q ∈ Depth- ∆ Lemma : Let non-zero and a m-variate f multilinear polynomial of degree . If d � � Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) ≡ 0 √ d ) poly( n, d Then, f can be computed by a size and depth circuit. ∆ + 5 f / Q ( y ) 6⌘ 0 , 8 q 2 Depth- ∆ ∈ Depth- ∆ + 5 l e p p Z i - z t r a w h c S 15

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 16

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 � � If Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) ≡ 0 16

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 � � If Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) ≡ 0 q ∈ C 6⌘ 0 x n x 2 x 1 16

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 � � If Q ( y ) = q f ( y | S 1 ) , f ( y | S 2 ) , . . . , f ( y | S n ) ≡ 0 q ∈ C 6⌘ 0 x n x 2 x 1 q ∈ C ≡ 0 y 16

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 17

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 By hybrid argument, there exists q ∈ C x n x i f f y | S 1 y | S 2 17

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 By hybrid argument, there exists ˜ Q ( z , x i ) z = { x 1 , . . . , x i − 1 , x i +1 , . . . , x n , y } q ∈ C x n x i f f y | S 1 y | S 2 17

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 By hybrid argument, there exists ˜ Q ( z , x i ) z = { x 1 , . . . , x i − 1 , x i +1 , . . . , x n , y } q ∈ C x n x i f f y | S 1 y | S 2 ˜ • Q ( z , x i ) 6⌘ 0 17

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 By hybrid argument, there exists ˜ Q ( z , x i ) z = { x 1 , . . . , x i − 1 , x i +1 , . . . , x n , y } q ∈ C x n f f f y | S 1 y | S 2 y | S i ˜ • Q ( z , x i ) 6⌘ 0 ˜ • Q ( z , f ( z )) ≡ 0 17

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 By hybrid argument, there exists ˜ Q ( z , x i ) q ∈ C x n Fixed Fixed f y | S i ˜ • Q ( z , x i ) 6⌘ 0 ˜ • Q ( z , f ( z )) ≡ 0 17

∃ q ∈ Depth- ∆ , Q ( y ) ≡ 0 Proof sketch of the key lemma f ∈ Depth- ∆ + 5 By hybrid argument, there exists ˜ Q ( z , x i ) q ∈ C x n Fixed Fixed f y | S i * ˜ Q ( z , x i ) ∈ Depth- ∆ + 1 ˜ • Q ( z , x i ) 6⌘ 0 ˜ • Q ( z , f ( z )) ≡ 0 17

Proof sketch of the key lemma ˜ • Q ( z , x i ) 6⌘ 0 ˜ Q ( z , x i ) ∈ Depth- ∆ + 1 ˜ • Q ( z , f ( z )) ≡ 0 18

Proof sketch of the key lemma ˜ • Q ( z , x i ) 6⌘ 0 ˜ Q ( z , x i ) ∈ Depth- ∆ + 1 ˜ • Q ( z , f ( z )) ≡ 0 ˜ x i − f ( z ) Q ( z , x i ) divides 18

Proof sketch of the key lemma ˜ • Q ( z , x i ) 6⌘ 0 ˜ Q ( z , x i ) ∈ Depth- ∆ + 1 ˜ • Q ( z , f ( z )) ≡ 0 ˜ x i − f ( z ) Q ( z , x i ) divides Reducing to polynomial factorization! 18

Outline • Arithmetic circuits and algebraic complexity classes • Polynomial identity testing (PIT) • Hardness vs Randomness for arithmetic circuits • Polynomial factorization • Open problems 19

Polynomial factorization (Simplified setting) P ( z , f ( z )) = 0 Goal : For any such that . P ( z , y ) ∈ C f ∈ C 0 Show that . 20

Polynomial factorization (Simplified setting) P ( z , f ( z )) = 0 Goal : For any such that . P ( z , y ) ∈ C f ∈ C 0 Show that . C 0 C [Kal89] VP VP 20

Polynomial factorization (Simplified setting) P ( z , f ( z )) = 0 Goal : For any such that . P ( z , y ) ∈ C f ∈ C 0 Show that . C 0 C [Kal89] VP VP Depth- ∆ [DSY09] Depth- ∆ + 3 with bounded individual degree 20

Polynomial factorization (Simplified setting) P ( z , f ( z )) = 0 Goal : For any such that . P ( z , y ) ∈ C f ∈ C 0 Show that . C 0 C [Kal89] VP VP Depth- ∆ [DSY09] Depth- ∆ + 3 with bounded individual degree VF( n log n )) VF( n log n )) [DSS18] (resp. VBP( n log n ), VNP( n log n )) (resp. VBP( n log n ), VNP( n log n )) 20

Polynomial factorization (Simplified setting) P ( z , f ( z )) = 0 Goal : For any such that . P ( z , y ) ∈ C f ∈ C 0 Show that . C 0 C [Kal89] VP VP Depth- ∆ [DSY09] Depth- ∆ + 3 with bounded individual degree VF( n log n )) VF( n log n )) [DSS18] (resp. VBP( n log n ), VNP( n log n )) (resp. VBP( n log n ), VNP( n log n )) Depth- ∆ Our result Depth- ∆ + 3 O (log 2 n/ log 2 log n ) with degree 20

Polynomial factorization (Simplified setting) P ( z , f ( z )) = 0 Goal : For any such that . P ( z , y ) ∈ C f ∈ C 0 Show that . non-deterministic (existential) C 0 C [Kal89] VP VP Depth- ∆ [DSY09] Depth- ∆ + 3 with bounded individual degree VF( n log n )) VF( n log n )) [DSS18] (resp. VBP( n log n ), VNP( n log n )) (resp. VBP( n log n ), VNP( n log n )) Depth- ∆ Our result Depth- ∆ + 3 O (log 2 n/ log 2 log n ) with degree 20

Factorization for bounded depth circuits Goal : For any s.t. . P ( z , f ( z )) = 0 P ( z , y ) ∈ Depth- ∆ f ∈ Depth- ∆ + O (1) Show that . 21

Factorization for bounded depth circuits Goal : For any s.t. . P ( z , f ( z )) = 0 P ( z , y ) ∈ Depth- ∆ f ∈ Depth- ∆ + O (1) Show that . Newton iteration 21

Factorization for bounded depth circuits Goal : For any s.t. . P ( z , f ( z )) = 0 P ( z , y ) ∈ Depth- ∆ f ∈ Depth- ∆ + O (1) Show that . Newton iteration Structure lemma 21

Factorization for bounded depth circuits Goal : For any s.t. . P ( z , f ( z )) = 0 P ( z , y ) ∈ Depth- ∆ f ∈ Depth- ∆ + O (1) Show that . Newton iteration Structure lemma Depth reduction 21

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . 22

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . Def : (Homogeneous components) 22

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . Def : (Homogeneous components) The degree i homogeneous component is the collection of monomials of degree i . 22

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . Def : (Homogeneous components) The degree i homogeneous component is the collection of monomials of degree i . Example : f ( x 1 , x 2 , x 3 ) = x 3 1 x 2 + x 1 x 2 x 3 + x 2 2 + x 1 x 3 + x 4 3 22

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . Def : (Homogeneous components) The degree i homogeneous component is the collection of monomials of degree i . f ( x 1 , x 2 , x 3 ) = x 3 1 x 2 + x 1 x 2 x 3 + x 2 2 + x 1 x 3 + x 4 Example : 3 • H 0 [ f ] = 0 22

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . Def : (Homogeneous components) The degree i homogeneous component is the collection of monomials of degree i . f ( x 1 , x 2 , x 3 ) = x 3 1 x 2 + x 1 x 2 x 3 + x 2 2 + x 1 x 3 + x 4 Example : 3 • H 0 [ f ] = 0 • H 1 [ f ] = 0 22

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . Def : (Homogeneous components) The degree i homogeneous component is the collection of monomials of degree i . f ( x 1 , x 2 , x 3 ) = x 3 1 x 2 + x 1 x 2 x 3 + x 2 2 + x 1 x 3 + x 4 Example : 3 • H 0 [ f ] = 0 • H 1 [ f ] = 0 H 2 [ f ] = x 2 • 2 + x 1 x 3 22

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . Def : (Homogeneous components) The degree i homogeneous component is the collection of monomials of degree i . f ( x 1 , x 2 , x 3 ) = x 3 1 x 2 + x 1 x 2 x 3 + x 2 2 + x 1 x 3 + x 4 Example : 3 • H 0 [ f ] = 0 • H 1 [ f ] = 0 H 2 [ f ] = x 2 • 2 + x 1 x 3 • H 3 [ f ] = x 1 x 2 x 3 22

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . Def : (Homogeneous components) The degree i homogeneous component is the collection of monomials of degree i . f ( x 1 , x 2 , x 3 ) = x 3 1 x 2 + x 1 x 2 x 3 + x 2 2 + x 1 x 3 + x 4 Example : 3 • H 0 [ f ] = 0 • H 1 [ f ] = 0 H 2 [ f ] = x 2 • 2 + x 1 x 3 • H 3 [ f ] = x 1 x 2 x 3 H 4 [ f ] = x 3 1 x 2 + x 4 • 3 22

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . 22

Newton iteration (Sloppy Hensel Lifting) Goal : H ≤ i [ h i ] = H ≤ i [ f ] . h i = h i − 1 − H i [ P ( z , h i − 1 ( z ))] Update : . δ 22