Derandomizing Polynomial Identity Testing for ... - Semantic Scholar
Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae Matthew Anderson1 Dieter van Melkebeek1 Ilya Volkovich2 1 University
of Wisconsin – Madison
2 Technion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
Arithmetic Formula Identity Testing Problem (AFIT)
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
Arithmetic Formula Identity Testing Problem (AFIT) Input: F ∈ F[x1 , ..., xn ]
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Arithmetic Formula Identity Testing Problem (AFIT) Input: F ∈ F[x1 , ..., xn ], given as an arithmetic formula.
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Arithmetic Formula Identity Testing Problem (AFIT) Input: F ∈ F[x1 , ..., xn ], given as an arithmetic formula. Question: Is F ≡ 0?
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Arithmetic Formula Identity Testing Problem (AFIT) Input: F ∈ F[x1 , ..., xn ], given as an arithmetic formula. Question: Is F ≡ 0? + −1 ×
×
×
x1 x1 x2 x2
+
+ −1 x1 x2 x1 x2
≡ (x1 + x2)(x1 − x2) − x21 + x22 ≡ 0
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Arithmetic Formula Identity Testing Problem (AFIT) Input: F ∈ F[x1 , ..., xn ], given as an arithmetic formula. Question: Is F ≡ 0? + −1 ×
×
×
≡ (x1 + x2)(x1 − x2) − x21 + x22 ≡ 0
x1 x1 x2 x2
+
+ −1 x1 x2 x1 x2
Motivation: primality testing, circuit lower bounds, perfect matching, ...
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
Algorithms for the General Case Randomized algorithm [S80,Z79,DL78]:
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for the General Case Randomized algorithm [S80,Z79,DL78]: 1
Pick ai ∈ S uniformly, accept iff P (a1 , ..., an ) = 0
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for the General Case Randomized algorithm [S80,Z79,DL78]: 1
Pick ai ∈ S uniformly, accept iff P (a1 , ..., an ) = 0
2
Prai ∈u S [P (a1 , ..., an ) = 0|P 6≡ 0] ≤
d |S |
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae:
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several]
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11]
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11]
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae:
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae: Read-Once
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae: Read-Once Pk -Read-Once [SV08,SV09]
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae: Read-Once Pk -Read-Once [SV08,SV09] Multilinear Read-k [this]
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae: Read-Once Pk -Read-Once [SV08,SV09] Multilinear Read-k [this] Theorem (Main) O(k)
time deterministic algorithm for identity There is a s O(1) · n k testing size-s n-variable multilinear read-k formulae.
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts:
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k .
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
+ +
× +
+
x1
+ x2
x4
x11
x8
x3
×
+
× ×
x5
x6
+
× x7
x9
x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
Take
∂ ∂x7
+ +
× +
+
x1
+ x2
x4
x11
x8
x3
×
+
× ×
x5
x6
x7 x9 Median
+
× x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
Take
∂ ∂x7
+ +
× +
+ x3
x1
x2
x4
x11
x8
+
×
+
× ×
x5
x6
x7 x9 Median
+
× x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
Take
∂ ∂x7
+ +
× +
+ x3
x1
x2
x4
x11
x8
+
×
+
× ×
x5
x6
x7 x9 Median
+
× x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
Take
∂ ∂x7
+ +
× +
+ x3
x1
x2
x4
x11
x8
+
×
+
× ×
x5
x6
x7 x9 Median
+
× x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
Take
∂ ∂x7
+ +
× +
+ x3
x1
x2
x4
x11
x8
+
×
+
× ×
x5
x6
x17 x9 Median
+
× x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
-Read-k
Fragmenting Read-1 Formulae
Take
∂ ∂x7
× +
x6 x3
× x1
x2
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
A Fragmentation Lemma Lemma Let F be a read-once formula.
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
A Fragmentation Lemma Lemma Let F be a read-once formula.
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
A Fragmentation Lemma Lemma Let F be a read-once formula.
; ∂ ∂x
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
A Fragmentation Lemma Lemma Let F be a read-once formula.
×
; ∂ ∂x
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
A Fragmentation Lemma Lemma Let F be a read-once formula.
×
; ∂ ∂x
≤ 12 n variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x1
× x2
x1
x3
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x1
× x2
x1
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x1
× x2
x1
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x1
× x2
x1
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x1
× x2
x1
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x11
× x2
x11
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
× +
x4 x2
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
-Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
; ∂ ∂x
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
×
; ∂ ∂x
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
×
; ∂ ∂x
OR
×
+
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
×
; ∂ ∂x
OR
×
≤ 12 n variables
+
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
×
; ∂ ∂x
OR
×
≤ 12 n variables
> 21 n variables
+
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
×
; ∂ ∂x
OR
×
> 21 n variables
+
≤ 12 n variables read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1 (Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P2 2 (Shattering) – Reduce testing -read-k to testing read-k .
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1 (Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P2 2 (Shattering) – Reduce testing -read-k to testing read-k .
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1 (Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P2 2 (Shattering) – Reduce testing -read-k to testing read-k .
×
≤ 21 n variables |
+
P2
{z
-read-k
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1 (Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P2 2 (Shattering) – Reduce testing -read-k to testing read-k .
×
≤ 21 n variables |
+
P2
{z
-read-k
}
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w },
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that:
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that: 1. closed under zero-substitutions, and
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that: 1. closed under zero-substitutions, and 2. does not contain any monomial of degree d ≥ w .
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that: 1. closed under zero-substitutions, and 2. does not contain any monomial of degree d ≥ w . Let F consist of F (¯ x +σ ¯ ) and all its zero-substitutions.
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that: 1. closed under zero-substitutions, and 2. does not contain any monomial of degree d ≥ w . Let F consist of F (¯ x +σ ¯ ) and all its zero-substitutions. Some simple conditions on σ ¯ give property 2 for F.
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that: 1. closed under zero-substitutions, and 2. does not contain any monomial of degree d ≥ w . Let F consist of F (¯ x +σ ¯ ) and all its zero-substitutions. Some simple conditions on σ ¯ give property 2 for F. For such a σ ¯ , Hw + σ ¯ hits F .
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where 1
no variable divides any Fi ,
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where 1 2
no variable divides any Fi , factors of each Fi depend on at most
n R(m)
variables,
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where 1 2
no variable divides any Fi , factors of each Fi depend on at most
n R(m)
+ × ···
× ···
···
× ···
≤
variables,
n R(m)
variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where 1 2
no variable divides any Fi , factors of each Fi depend on at most
n R(m)
+ × ··· 3
× ···
···
a few other minor conditions.
× ···
≤
variables,
n R(m)
variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where 1 2
no variable divides any Fi , factors of each Fi depend on at most
n R(m)
+ × ··· 3
× ···
···
×
≤
variables,
n R(m)
variables
···
a few other minor conditions.
⇒ F does not compute a monomial.
Conclusion
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof.
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒ F1
F2
≡ Mn
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒ Shatter( F1
F2
≡ Mn )
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
×
··· ···
× ···
≡ Mn′
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
≥1
×
··· ···
× ···
≡ Mn′
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
≥1
×
··· ···
× ···
≡ Mn′
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
×
··· ···
×
≡ Mn′
···
≥ 1, by Lemma, some branch is divisible by a variable xj .
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
×
··· ···
×
≡ Mn′
···
≥ 1, by Lemma, some branch is divisible by a variable xj .
⇒ xj = 0 is a root of that branch.
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
×
··· ···
×
≡ Mn′
···
≥ 1, by Lemma, some branch is divisible by a variable xj .
⇒ xj = 0 is a root of that branch. Pick σ ¯ to be a common nonzero of nonzero partial derivatives of all subformulae of the Fi .
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
×
··· ···
×
≡ Mn′
···
≥ 1, by Lemma, some branch is divisible by a variable xj .
⇒ xj = 0 is a root of that branch. Pick σ ¯ to be a common nonzero of nonzero partial derivatives of all subformulae of the Fi . P F = F1 + F2 ∈ 2 -read-k
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
×
··· ···
×
≡ Mn′
···
≥ 1, by Lemma, some branch is divisible by a variable xj .
⇒ xj = 0 is a root of that branch. Pick σ ¯ to be a common nonzero of nonzero partial derivatives of all subformulae of the Fi . P F = F1 + F2 ∈ 2 -read-k ⇒σ ¯ can be computed efficiently using a read-k identity test!
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Introduction
Read-(k + 1) ≤
P2
-Read-k
Shattering Read-once Formulae
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Shattering Read-once Formulae
×
; ∂ ∂x
≤ 12 n variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Shattering Read-once Formulae
×
; ∂ ∂x
;
≤ 12 n variables
× ×
∂ ∂y
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Shattering Read-once Formulae
×
; ∂ ∂x
;
≤ 12 n variables ∂ ∂y
× × ≤ 14 n variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Shattering Read-once Formulae
×
; ∂ ∂x
;
≤ 12 n variables
···
;
×
∂ ∂P
∂ ∂y
× × ≤ 14 n variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Shattering Read-once Formulae
×
; ∂ ∂x
;
≤ 12 n variables
×
;
···
∂ ∂P
≤ αn variables
∂ ∂y
× × ≤ 14 n variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
A Shattering Lemma Lemma For any read-once formula F on n variables and α ∈ [0, 1] there exists a sets of variables P , with |P | = O ( α1 ), such that ∂F ∂P can be written as
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
A Shattering Lemma Lemma For any read-once formula F on n variables and α ∈ [0, 1] there exists a sets of variables P , with |P | = O ( α1 ), such that ∂F ∂P can be written as
× ···
≤ αn variables
Introduction
Read-(k + 1) ≤
P2
-Read-k
Shattering Read-k Formulae
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
;
> 12 n variables
+
∂ ∂x
read-(k − 1)
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
+
> 12 n variables
+
≡
read-(k − 1)
×
×
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
+
> 12 n variables
+
≡
read-(k − 1)
×
× read-(k − 1)
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡ |
read-(k − 1)
+ ×
× {z
6∈ read-k
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡ |
read-(k − 1)
+ ×
× {z
6 read-k ∈ ∈ read-k
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡ |
read-(k − 1)
+ ×
× {z
6 read-k ∈ ∈ read-k ∈ read-k1 + read-k2
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡ |
read-(k − 1)
+ ×
× {z
6 read-k ∈ ∈ read-k ∈ read-k1 + read-k2 where k1 + k2 ≤ k
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡
read-(k − 1)
|
+ ×
× {z
6 read-k ∈ ∈ read-k ∈ read-k1 + read-k2 where k1 + k2 ≤ k and | | ≥ Ω nk
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡
read-(k − 1)
|
+ ×
× {z
6 read-k ∈ ∈ read-k ∈ read-k1 + read-k2 where k1 + k2 ≤ k and | | ≥ Ω nk
At most k iterations are required to successfully shatter the formula.
}
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Shattering Lemma Lemma (Shattering Lemma) For any read-k formula F on n variables, there exist disjoint sets of variables P and V , with |P | = O (k 2 R(k )) and |V | = such that
∂F ∂P
n (kR(k ))O(k )
can be written as
Conclusion
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Shattering Lemma Lemma (Shattering Lemma) For any read-k formula F on n variables, there exist disjoint sets of variables P and V , with |P | = O (k 2 R(k )) and |V | = such that
∂F ∂P
can be written as
+ × ···
n (kR(k ))O(k )
× ···
···
k ′ ≤ k branches × |V | ≤ R(k ′ ) variables in V ···
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Conclusion Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Case Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae: Read-Once Pk -Read-Once [SV08,SV09] Multilinear Read-k [this] Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae.
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Extensions
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Extensions 1 Sparse substituted – quasi-poly-time
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Extensions 1 Sparse substituted – quasi-poly-time Includes depth-four multilinear formulae [KMSV10]
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Extensions 1 Sparse substituted – quasi-poly-time Includes depth-four multilinear formulae [KMSV10] 2
Blackbox – quasi-poly-time
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Extensions 1 Sparse substituted – quasi-poly-time Includes depth-four multilinear formulae [KMSV10] 2
Blackbox – quasi-poly-time Constant-depth formulae – poly-time
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
Questions?
Thanks!
P2
-Read-k ≤ Read-k
Conclusion
of Wisconsin – Madison
2 Technion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
Arithmetic Formula Identity Testing Problem (AFIT)
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
Arithmetic Formula Identity Testing Problem (AFIT) Input: F ∈ F[x1 , ..., xn ]
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Arithmetic Formula Identity Testing Problem (AFIT) Input: F ∈ F[x1 , ..., xn ], given as an arithmetic formula.
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Arithmetic Formula Identity Testing Problem (AFIT) Input: F ∈ F[x1 , ..., xn ], given as an arithmetic formula. Question: Is F ≡ 0?
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Arithmetic Formula Identity Testing Problem (AFIT) Input: F ∈ F[x1 , ..., xn ], given as an arithmetic formula. Question: Is F ≡ 0? + −1 ×
×
×
x1 x1 x2 x2
+
+ −1 x1 x2 x1 x2
≡ (x1 + x2)(x1 − x2) − x21 + x22 ≡ 0
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Arithmetic Formula Identity Testing Problem (AFIT) Input: F ∈ F[x1 , ..., xn ], given as an arithmetic formula. Question: Is F ≡ 0? + −1 ×
×
×
≡ (x1 + x2)(x1 − x2) − x21 + x22 ≡ 0
x1 x1 x2 x2
+
+ −1 x1 x2 x1 x2
Motivation: primality testing, circuit lower bounds, perfect matching, ...
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
Algorithms for the General Case Randomized algorithm [S80,Z79,DL78]:
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for the General Case Randomized algorithm [S80,Z79,DL78]: 1
Pick ai ∈ S uniformly, accept iff P (a1 , ..., an ) = 0
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for the General Case Randomized algorithm [S80,Z79,DL78]: 1
Pick ai ∈ S uniformly, accept iff P (a1 , ..., an ) = 0
2
Prai ∈u S [P (a1 , ..., an ) = 0|P 6≡ 0] ≤
d |S |
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae:
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several]
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11]
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11]
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae:
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae: Read-Once
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae: Read-Once Pk -Read-Once [SV08,SV09]
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae: Read-Once Pk -Read-Once [SV08,SV09] Multilinear Read-k [this]
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Cases Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae: Read-Once Pk -Read-Once [SV08,SV09] Multilinear Read-k [this] Theorem (Main) O(k)
time deterministic algorithm for identity There is a s O(1) · n k testing size-s n-variable multilinear read-k formulae.
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts:
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k .
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
+ +
× +
+
x1
+ x2
x4
x11
x8
x3
×
+
× ×
x5
x6
+
× x7
x9
x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
Take
∂ ∂x7
+ +
× +
+
x1
+ x2
x4
x11
x8
x3
×
+
× ×
x5
x6
x7 x9 Median
+
× x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
Take
∂ ∂x7
+ +
× +
+ x3
x1
x2
x4
x11
x8
+
×
+
× ×
x5
x6
x7 x9 Median
+
× x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
Take
∂ ∂x7
+ +
× +
+ x3
x1
x2
x4
x11
x8
+
×
+
× ×
x5
x6
x7 x9 Median
+
× x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
Take
∂ ∂x7
+ +
× +
+ x3
x1
x2
x4
x11
x8
+
×
+
× ×
x5
x6
x7 x9 Median
+
× x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-1 Formulae
Take
∂ ∂x7
+ +
× +
+ x3
x1
x2
x4
x11
x8
+
×
+
× ×
x5
x6
x17 x9 Median
+
× x10
x12
x13
Read-(k + 1) ≤
Introduction
P2
-Read-k
Fragmenting Read-1 Formulae
Take
∂ ∂x7
× +
x6 x3
× x1
x2
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
A Fragmentation Lemma Lemma Let F be a read-once formula.
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
A Fragmentation Lemma Lemma Let F be a read-once formula.
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
A Fragmentation Lemma Lemma Let F be a read-once formula.
; ∂ ∂x
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
A Fragmentation Lemma Lemma Let F be a read-once formula.
×
; ∂ ∂x
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
A Fragmentation Lemma Lemma Let F be a read-once formula.
×
; ∂ ∂x
≤ 12 n variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x1
× x2
x1
x3
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x1
× x2
x1
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x1
× x2
x1
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x1
× x2
x1
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x1
× x2
x1
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
+ ×
x4
+
x4
× x11
× x2
x11
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Fragmenting Read-(k + 1) Formulae A read-2 formula:
× +
x4 x2
x3
Algorithm: Pick largest child which contains k + 1 occurrences of some variable.
Introduction
Read-(k + 1) ≤
P2
-Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
; ∂ ∂x
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
×
; ∂ ∂x
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
×
; ∂ ∂x
OR
×
+
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
×
; ∂ ∂x
OR
×
≤ 12 n variables
+
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
×
; ∂ ∂x
OR
×
≤ 12 n variables
> 21 n variables
+
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
The Fragmentation Lemma Lemma (Fragmentation Lemma) Let F be a read-(k + 1) formula.
×
; ∂ ∂x
OR
×
> 21 n variables
+
≤ 12 n variables read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1 (Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P2 2 (Shattering) – Reduce testing -read-k to testing read-k .
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1 (Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P2 2 (Shattering) – Reduce testing -read-k to testing read-k .
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1 (Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P2 2 (Shattering) – Reduce testing -read-k to testing read-k .
×
≤ 21 n variables |
+
P2
{z
-read-k
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1 (Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P2 2 (Shattering) – Reduce testing -read-k to testing read-k .
×
≤ 21 n variables |
+
P2
{z
-read-k
}
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w },
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that:
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that: 1. closed under zero-substitutions, and
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that: 1. closed under zero-substitutions, and 2. does not contain any monomial of degree d ≥ w .
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that: 1. closed under zero-substitutions, and 2. does not contain any monomial of degree d ≥ w . Let F consist of F (¯ x +σ ¯ ) and all its zero-substitutions.
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that: 1. closed under zero-substitutions, and 2. does not contain any monomial of degree d ≥ w . Let F consist of F (¯ x +σ ¯ ) and all its zero-substitutions. Some simple conditions on σ ¯ give property 2 for F.
Conclusion
Read-(k + 1) ≤
Introduction
Testing
P2
P2
-Read-k
P2
-Read-k ≤ Read-k
-read-k ≤ Testing read-k
Let F = F1 + F2 be a nonzero multilinear
P2
-read-k formula.
Fact (SV Hitting Set [SV09]) Hw = {¯ x ∈ {0, 1}n of Hamming weight at most w }, Hw hits any class F of multilinear polynomials that: 1. closed under zero-substitutions, and 2. does not contain any monomial of degree d ≥ w . Let F consist of F (¯ x +σ ¯ ) and all its zero-substitutions. Some simple conditions on σ ¯ give property 2 for F. For such a σ ¯ , Hw + σ ¯ hits F .
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where 1
no variable divides any Fi ,
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where 1 2
no variable divides any Fi , factors of each Fi depend on at most
n R(m)
variables,
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where 1 2
no variable divides any Fi , factors of each Fi depend on at most
n R(m)
+ × ···
× ···
···
× ···
≤
variables,
n R(m)
variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where 1 2
no variable divides any Fi , factors of each Fi depend on at most
n R(m)
+ × ··· 3
× ···
···
a few other minor conditions.
× ···
≤
variables,
n R(m)
variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
A Structural Lemma Lemma ([DS07,SS10,KMSV10]) 3 There exists a function R(m) Pm = O (m log m) such that the following holds: Let F = i=1 Fi be a multilinear formula on n variables, where 1 2
no variable divides any Fi , factors of each Fi depend on at most
n R(m)
+ × ··· 3
× ···
···
×
≤
variables,
n R(m)
variables
···
a few other minor conditions.
⇒ F does not compute a monomial.
Conclusion
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof.
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒ F1
F2
≡ Mn
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒ Shatter( F1
F2
≡ Mn )
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
×
··· ···
× ···
≡ Mn′
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
≥1
×
··· ···
× ···
≡ Mn′
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
≥1
×
··· ···
× ···
≡ Mn′
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
×
··· ···
×
≡ Mn′
···
≥ 1, by Lemma, some branch is divisible by a variable xj .
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
×
··· ···
×
≡ Mn′
···
≥ 1, by Lemma, some branch is divisible by a variable xj .
⇒ xj = 0 is a root of that branch.
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
×
··· ···
×
≡ Mn′
···
≥ 1, by Lemma, some branch is divisible by a variable xj .
⇒ xj = 0 is a root of that branch. Pick σ ¯ to be a common nonzero of nonzero partial derivatives of all subformulae of the Fi .
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
×
··· ···
×
≡ Mn′
···
≥ 1, by Lemma, some branch is divisible by a variable xj .
⇒ xj = 0 is a root of that branch. Pick σ ¯ to be a common nonzero of nonzero partial derivatives of all subformulae of the Fi . P F = F1 + F2 ∈ 2 -read-k
Theorem For a “good” σ ¯ , F (¯ x +σ ¯ ) is not a monomial of degree n ≥ k O(k ) . Proof. Suppose F (¯ x +σ ¯ ) is a monomial, Mn ,
+ ⇒
× ···
⇒ If
n′
×
··· ···
×
≡ Mn′
···
≥ 1, by Lemma, some branch is divisible by a variable xj .
⇒ xj = 0 is a root of that branch. Pick σ ¯ to be a common nonzero of nonzero partial derivatives of all subformulae of the Fi . P F = F1 + F2 ∈ 2 -read-k ⇒σ ¯ can be computed efficiently using a read-k identity test!
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Outline Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Introduction
Read-(k + 1) ≤
P2
-Read-k
Shattering Read-once Formulae
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Shattering Read-once Formulae
×
; ∂ ∂x
≤ 12 n variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Shattering Read-once Formulae
×
; ∂ ∂x
;
≤ 12 n variables
× ×
∂ ∂y
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Shattering Read-once Formulae
×
; ∂ ∂x
;
≤ 12 n variables ∂ ∂y
× × ≤ 14 n variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Shattering Read-once Formulae
×
; ∂ ∂x
;
≤ 12 n variables
···
;
×
∂ ∂P
∂ ∂y
× × ≤ 14 n variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Shattering Read-once Formulae
×
; ∂ ∂x
;
≤ 12 n variables
×
;
···
∂ ∂P
≤ αn variables
∂ ∂y
× × ≤ 14 n variables
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
A Shattering Lemma Lemma For any read-once formula F on n variables and α ∈ [0, 1] there exists a sets of variables P , with |P | = O ( α1 ), such that ∂F ∂P can be written as
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
A Shattering Lemma Lemma For any read-once formula F on n variables and α ∈ [0, 1] there exists a sets of variables P , with |P | = O ( α1 ), such that ∂F ∂P can be written as
× ···
≤ αn variables
Introduction
Read-(k + 1) ≤
P2
-Read-k
Shattering Read-k Formulae
P2
-Read-k ≤ Read-k
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
;
> 12 n variables
+
∂ ∂x
read-(k − 1)
Conclusion
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
+
> 12 n variables
+
≡
read-(k − 1)
×
×
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
+
> 12 n variables
+
≡
read-(k − 1)
×
× read-(k − 1)
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡ |
read-(k − 1)
+ ×
× {z
6∈ read-k
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡ |
read-(k − 1)
+ ×
× {z
6 read-k ∈ ∈ read-k
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡ |
read-(k − 1)
+ ×
× {z
6 read-k ∈ ∈ read-k ∈ read-k1 + read-k2
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡ |
read-(k − 1)
+ ×
× {z
6 read-k ∈ ∈ read-k ∈ read-k1 + read-k2 where k1 + k2 ≤ k
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡
read-(k − 1)
|
+ ×
× {z
6 read-k ∈ ∈ read-k ∈ read-k1 + read-k2 where k1 + k2 ≤ k and | | ≥ Ω nk
}
Introduction
Read-(k + 1) ≤
P2
P2
-Read-k
Conclusion
-Read-k ≤ Read-k
Shattering Read-k Formulae
×
; ∂ ∂x
> 12 n variables
+
≡
read-(k − 1)
|
+ ×
× {z
6 read-k ∈ ∈ read-k ∈ read-k1 + read-k2 where k1 + k2 ≤ k and | | ≥ Ω nk
At most k iterations are required to successfully shatter the formula.
}
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Shattering Lemma Lemma (Shattering Lemma) For any read-k formula F on n variables, there exist disjoint sets of variables P and V , with |P | = O (k 2 R(k )) and |V | = such that
∂F ∂P
n (kR(k ))O(k )
can be written as
Conclusion
Read-(k + 1) ≤
Introduction
P2
P2
-Read-k
-Read-k ≤ Read-k
Shattering Lemma Lemma (Shattering Lemma) For any read-k formula F on n variables, there exist disjoint sets of variables P and V , with |P | = O (k 2 R(k )) and |V | = such that
∂F ∂P
can be written as
+ × ···
n (kR(k ))O(k )
× ···
···
k ′ ≤ k branches × |V | ≤ R(k ′ ) variables in V ···
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion
Conclusion Theorem (Weakened Main) O(k)
+O(k log n) time deterministic algorithm for There is a s O(1) · n k identity testing size-s multilinear read-k formulae.
Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Our argument has two main parts: 1
2
(Fragmenting) – Reduce testing read-(k + 1) to testing P2 -read-k . P (Shattering) – Reduce testing 2 -read-k to testing read-k .
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Algorithms for Restricted Case Deterministic algorithms for bounded-depth formulae: Depth-1 Depth-2 [several] Constant-Top-Fanin Depth-3 [KS07,SS11] Multilinear Constant-Top-Fanin Depth-4 [KMSV10,SV11] Deterministic algorithms for bounded-read formulae: Read-Once Pk -Read-Once [SV08,SV09] Multilinear Read-k [this] Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae.
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Extensions
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Extensions 1 Sparse substituted – quasi-poly-time
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Extensions 1 Sparse substituted – quasi-poly-time Includes depth-four multilinear formulae [KMSV10]
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Extensions 1 Sparse substituted – quasi-poly-time Includes depth-four multilinear formulae [KMSV10] 2
Blackbox – quasi-poly-time
Conclusion
Read-(k + 1) ≤
Introduction
P2
-Read-k
P2
-Read-k ≤ Read-k
Conclusion Theorem (Main) There is a polynomial-time deterministic algorithm for identity testing multilinear constant-read formulae. Extensions 1 Sparse substituted – quasi-poly-time Includes depth-four multilinear formulae [KMSV10] 2
Blackbox – quasi-poly-time Constant-depth formulae – poly-time
Conclusion
Introduction
Read-(k + 1) ≤
P2
-Read-k
Questions?
Thanks!
P2
-Read-k ≤ Read-k
Conclusion
