A Compression Algorithm for AC0[P] Circuits Using Certifying Polynomials

Home , AC0, AC (complexity), P (complexity)

A compression algorithm for AC0[p] circuits using Certifying Polynomials

Srikanth Srinivasan Department of Mathematics, IIT Bombay.

September 30, 2015

S. Compression for AC0(p) September 30, 2015 1 / 27 Example 1: Circuit SAT. Input: Boolean Circuit C, Qn: Is fC ≡ 0?

Example 2: PIT. Input: Algebraic circuit C, Qn: Is PC ≡ 0? Example 3: Minimum Circuit Size problem (MCSP). n I Input: f : {0, 1} → {0, 1}, k. I Qn: Does f have a circuit of size at most k?

Meta-algorithmic problems

Meta-algorithmic problems: Problems that take as input or output the description of a computational model.

S. Compression for AC0(p) September 30, 2015 2 / 27 Example 2: PIT. Input: Algebraic circuit C, Qn: Is PC ≡ 0? Example 3: Minimum Circuit Size problem (MCSP). n I Input: f : {0, 1} → {0, 1}, k. I Qn: Does f have a circuit of size at most k?

Meta-algorithmic problems

Meta-algorithmic problems: Problems that take as input or output the description of a computational model.

Example 1: Circuit SAT. Input: Boolean Circuit C, Qn: Is fC ≡ 0?

S. Compression for AC0(p) September 30, 2015 2 / 27 Example 3: Minimum Circuit Size problem (MCSP). n I Input: f : {0, 1} → {0, 1}, k. I Qn: Does f have a circuit of size at most k?

Meta-algorithmic problems

Meta-algorithmic problems: Problems that take as input or output the description of a computational model.

Example 1: Circuit SAT. Input: Boolean Circuit C, Qn: Is fC ≡ 0?

Example 2: PIT. Input: Algebraic circuit C, Qn: Is PC ≡ 0?

S. Compression for AC0(p) September 30, 2015 2 / 27 n I Input: f : {0, 1} → {0, 1}, k. I Qn: Does f have a circuit of size at most k?

Meta-algorithmic problems

Meta-algorithmic problems: Problems that take as input or output the description of a computational model.

Example 1: Circuit SAT. Input: Boolean Circuit C, Qn: Is fC ≡ 0?

Example 2: PIT. Input: Algebraic circuit C, Qn: Is PC ≡ 0? Example 3: Minimum Circuit Size problem (MCSP).

S. Compression for AC0(p) September 30, 2015 2 / 27 Meta-algorithmic problems

Meta-algorithmic problems: Problems that take as input or output the description of a computational model.

Example 1: Circuit SAT. Input: Boolean Circuit C, Qn: Is fC ≡ 0?

Example 2: PIT. Input: Algebraic circuit C, Qn: Is PC ≡ 0? Example 3: Minimum Circuit Size problem (MCSP). n I Input: f : {0, 1} → {0, 1}, k. I Qn: Does f have a circuit of size at most k?

S. Compression for AC0(p) September 30, 2015 2 / 27 Compression problem for C (Chen, Kabanets, Kolokolova, Shaltiel, Zuckerman (2014)) n I Input: f : {0, 1} → {0, 1} with small Boolean circuits from C. I Qn: Construct a non-trivially small general circuit for f. Non-trivially small: size 2n/n.

(Chen et al.) Compression algorithms imply circuit lower bounds.

Compression Algorithm for a circuit class C

C: a class of circuits (AC0, AC0[p], etc.).

S. Compression for AC0(p) September 30, 2015 3 / 27 n I Input: f : {0, 1} → {0, 1} with small Boolean circuits from C. I Qn: Construct a non-trivially small general circuit for f. Non-trivially small: size 2n/n.

(Chen et al.) Compression algorithms imply circuit lower bounds.

Compression Algorithm for a circuit class C

C: a class of circuits (AC0, AC0[p], etc.). Compression problem for C (Chen, Kabanets, Kolokolova, Shaltiel, Zuckerman (2014))

S. Compression for AC0(p) September 30, 2015 3 / 27 I Qn: Construct a non-trivially small general circuit for f. Non-trivially small: size 2n/n.

(Chen et al.) Compression algorithms imply circuit lower bounds.

Compression Algorithm for a circuit class C

C: a class of circuits (AC0, AC0[p], etc.). Compression problem for C (Chen, Kabanets, Kolokolova, Shaltiel, Zuckerman (2014)) n I Input: f : {0, 1} → {0, 1} with small Boolean circuits from C.

S. Compression for AC0(p) September 30, 2015 3 / 27 Non-trivially small: size 2n/n.

(Chen et al.) Compression algorithms imply circuit lower bounds.

Compression Algorithm for a circuit class C

C: a class of circuits (AC0, AC0[p], etc.). Compression problem for C (Chen, Kabanets, Kolokolova, Shaltiel, Zuckerman (2014)) n I Input: f : {0, 1} → {0, 1} with small Boolean circuits from C. I Qn: Construct a non-trivially small general circuit for f.

S. Compression for AC0(p) September 30, 2015 3 / 27 (Chen et al.) Compression algorithms imply circuit lower bounds.

Compression Algorithm for a circuit class C

S. Compression for AC0(p) September 30, 2015 3 / 27 Compression Algorithm for a circuit class C

(Chen et al.) Compression algorithms imply circuit lower bounds.

S. Compression for AC0(p) September 30, 2015 3 / 27 Compression problem for DNFs

Input: f : {0, 1}n → {0, 1} with DNFs of size s. Qn: Construct a non-trivially small DNF for f.

f(x)

≤ s ∨

∧ ∧ ··· ∧

x1 x5 ··· x3 x7 ··· x5 x65 ··· x2

S. Compression for AC0(p) September 30, 2015 4 / 27 Finding optimal sized DNF for f: Covering f with subcubes in f −1(1). Number of subcubes: 3n. (Lov´asz1975) O(n)-approximation in time 2O(n).

Compression problem for DNFs

DNF of size s: a union of s subcubes. f −1(0)

f −1(1)

S. Compression for AC0(p) September 30, 2015 5 / 27 Finding optimal sized DNF for f: Covering f with subcubes in f −1(1). Number of subcubes: 3n. (Lov´asz1975) O(n)-approximation in time 2O(n).

Compression problem for DNFs

DNF of size s: a union of s subcubes. f −1(0)

f −1(1)

S. Compression for AC0(p) September 30, 2015 5 / 27 Number of subcubes: 3n. (Lov´asz1975) O(n)-approximation in time 2O(n).

Compression problem for DNFs

DNF of size s: a union of s subcubes. f −1(0) Finding optimal sized DNF for f: Covering f with subcubes in f −1(1). f −1(1)

S. Compression for AC0(p) September 30, 2015 5 / 27 (Lov´asz1975) O(n)-approximation in time 2O(n).

Compression problem for DNFs

DNF of size s: a union of s subcubes. f −1(0) Finding optimal sized DNF for f: Covering f with subcubes in f −1(1). Number of subcubes: 3n. f −1(1)

S. Compression for AC0(p) September 30, 2015 5 / 27 Compression problem for DNFs

DNF of size s: a union of s subcubes. f −1(0) Finding optimal sized DNF for f: Covering f with subcubes in f −1(1). Number of subcubes: 3n. f −1(1) (Lov´asz1975) O(n)-approximation in time 2O(n).

S. Compression for AC0(p) September 30, 2015 5 / 27 Can obtain a DNF of size 2n−Θ(t).

Compression problem for DNFs

Input: f : {0, 1}n → {0, 1} with DNFs of size s. Qn: Construct a non-trivially small DNF for f.

Say s = 2n−t = 2n/superpoly(n).

S. Compression for AC0(p) September 30, 2015 6 / 27 Compression problem for DNFs

Input: f : {0, 1}n → {0, 1} with DNFs of size s. Qn: Construct a non-trivially small DNF for f.

Say s = 2n−t = 2n/superpoly(n). Can obtain a DNF of size 2n−Θ(t).

S. Compression for AC0(p) September 30, 2015 6 / 27 Can obtain a DNF of size 2n−Θ(t).

Compression algorithms for other classes of circuits (Chen et al. (2014))

Input: f : {0, 1}n → {0, 1} with C-circuits of size s. Qn: Construct a non-trivially small circuit for f.

Say C-circuits of size s ⇒ DNFs of size 2n−t.

S. Compression for AC0(p) September 30, 2015 7 / 27 Compression algorithms for other classes of circuits (Chen et al. (2014))

Input: f : {0, 1}n → {0, 1} with C-circuits of size s. Qn: Construct a non-trivially small circuit for f.

Say C-circuits of size s ⇒ DNFs of size 2n−t. Can obtain a DNF of size 2n−Θ(t).

S. Compression for AC0(p) September 30, 2015 7 / 27 fan-in unbounded ∧ ∨ ∧ depth d = O(1) . . ∧ ··· xi1 xis

Compression algorithms for other classes of circuits (Chen et al. (2014))

Gives non-trivial compression algorithms for: AC0 circuits of size 2no(1) (Impagliazzo-Matthews-Paturi, H˚astad).

S. Compression for AC0(p) September 30, 2015 8 / 27 Compression algorithms for other classes of circuits (Chen et al. (2014))

Gives non-trivial compression algorithms for: AC0 circuits of size 2no(1) (Impagliazzo-Matthews-Paturi, H˚astad).

fan-in unbounded ∧ ∨ ∧ depth d = O(1) . . ∧ ··· xi1 xis

S. Compression for AC0(p) September 30, 2015 8 / 27 DeMorgan formulas of size n1.5. (Subbotovskaya, Santhanam) Further algorithms using memoization. General formulas, branching programs.

Compression algorithms for other classes of circuits (Chen et al. (2014))

Gives non-trivial compression algorithms for: AC0 circuits of size 2no(1) (Impagliazzo-Matthews-Paturi, H˚astad).

Theorem (Chen et al. (2014)) Say f has an AC0 circuit of size s. Then the algorithm outputs a circuit of size 2n/M, where M = exp(n/(C log(s/n))d−1).

S. Compression for AC0(p) September 30, 2015 9 / 27 Further algorithms using memoization. General formulas, branching programs.

Compression algorithms for other classes of circuits (Chen et al. (2014))

Gives non-trivial compression algorithms for: AC0 circuits of size 2no(1) (Impagliazzo-Matthews-Paturi, H˚astad).

Theorem (Chen et al. (2014)) Say f has an AC0 circuit of size s. Then the algorithm outputs a circuit of size 2n/M, where M = exp(n/(C log(s/n))d−1).

DeMorgan formulas of size n1.5. (Subbotovskaya, Santhanam)

S. Compression for AC0(p) September 30, 2015 9 / 27 General formulas, branching programs.

Compression algorithms for other classes of circuits (Chen et al. (2014))

Gives non-trivial compression algorithms for: AC0 circuits of size 2no(1) (Impagliazzo-Matthews-Paturi, H˚astad).

Theorem (Chen et al. (2014)) Say f has an AC0 circuit of size s. Then the algorithm outputs a circuit of size 2n/M, where M = exp(n/(C log(s/n))d−1).

DeMorgan formulas of size n1.5. (Subbotovskaya, Santhanam) Further algorithms using memoization.

S. Compression for AC0(p) September 30, 2015 9 / 27 Compression algorithms for other classes of circuits (Chen et al. (2014))

Gives non-trivial compression algorithms for: AC0 circuits of size 2no(1) (Impagliazzo-Matthews-Paturi, H˚astad).

Theorem (Chen et al. (2014)) Say f has an AC0 circuit of size s. Then the algorithm outputs a circuit of size 2n/M, where M = exp(n/(C log(s/n))d−1).

DeMorgan formulas of size n1.5. (Subbotovskaya, Santhanam) Further algorithms using memoization. General formulas, branching programs.

S. Compression for AC0(p) September 30, 2015 9 / 27 Natural next question: AC0[2]: AC0 augmented with ⊕ gates.

fan-in unbounded ⊕ ∨ ⊕ depth d = O(1) . . ∧ ··· xi1 xis

More general classes of circuits

Compression algorithms for more powerful classes of circuits?

S. Compression for AC0(p) September 30, 2015 10 / 27 More general classes of circuits

Compression algorithms for more powerful classes of circuits? Natural next question: AC0[2]: AC0 augmented with ⊕ gates.

fan-in unbounded ⊕ ∨ ⊕ depth d = O(1) . . ∧ ··· xi1 xis

S. Compression for AC0(p) September 30, 2015 10 / 27 Theorem (This work) Say f has an AC0[2] circuit of size s. Then we can ﬁnd in poly(2n) time a circuit of size 2n/M, where M = exp(n/(C log s)2(d−1)).

Also works for AC0[p] (p prime).

Compression algorithms for AC0[p] circuits

Theorem (Chen et al. (2014)) Say f has an AC0 circuit of size s. Then we can ﬁnd in poly(2n) time a circuit of size 2n/M, where M = exp(n/(C log(s/n))d−1).

S. Compression for AC0(p) September 30, 2015 11 / 27 Also works for AC0[p] (p prime).

Compression algorithms for AC0[p] circuits

Theorem (Chen et al. (2014)) Say f has an AC0 circuit of size s. Then we can ﬁnd in poly(2n) time a circuit of size 2n/M, where M = exp(n/(C log(s/n))d−1).

Theorem (This work) Say f has an AC0[2] circuit of size s. Then we can ﬁnd in poly(2n) time a circuit of size 2n/M, where M = exp(n/(C log s)2(d−1)).

S. Compression for AC0(p) September 30, 2015 11 / 27 Compression algorithms for AC0[p] circuits

Theorem (Chen et al. (2014)) Say f has an AC0 circuit of size s. Then we can ﬁnd in poly(2n) time a circuit of size 2n/M, where M = exp(n/(C log(s/n))d−1).

Theorem (This work) Say f has an AC0[2] circuit of size s. Then we can ﬁnd in poly(2n) time a circuit of size 2n/M, where M = exp(n/(C log s)2(d−1)).

Also works for AC0[p] (p prime).

S. Compression for AC0(p) September 30, 2015 11 / 27 E.g.: x1x2 + x3 + x1x5. Special kind of depth-2 AC0[2] circuit. P n n Degree D ⇒ size i≤D i = ≤D . Say that P ε-approximates f if Prx∈{0,1}n [P (x) 6= f(x)] ≤ ε.

⊕ ∧ ∧ ··· ∧

x1 x5 ··· x3 x7 ··· x5 x6 ··· x2

Polynomials and polynomial approximations

P (x1, . . . , xn) ∈ F2[x1, . . . , xn]. Multilinear.

S. Compression for AC0(p) September 30, 2015 12 / 27 Special kind of depth-2 AC0[2] circuit. P n n Degree D ⇒ size i≤D i = ≤D . Say that P ε-approximates f if Prx∈{0,1}n [P (x) 6= f(x)] ≤ ε.

⊕ ∧ ∧ ··· ∧

x1 x5 ··· x3 x7 ··· x5 x6 ··· x2

Polynomials and polynomial approximations

P (x1, . . . , xn) ∈ F2[x1, . . . , xn]. Multilinear. E.g.: x1x2 + x3 + x1x5.

S. Compression for AC0(p) September 30, 2015 12 / 27 P n n Degree D ⇒ size i≤D i = ≤D . Say that P ε-approximates f if Prx∈{0,1}n [P (x) 6= f(x)] ≤ ε.

Polynomials and polynomial approximations

P (x1, . . . , xn) ∈ F2[x1, . . . , xn]. Multilinear. E.g.: x1x2 + x3 + x1x5. Special kind of depth-2 AC0[2] circuit.

⊕ ∧ ∧ ··· ∧

x1 x5 ··· x3 x7 ··· x5 x6 ··· x2

S. Compression for AC0(p) September 30, 2015 12 / 27 Say that P ε-approximates f if Prx∈{0,1}n [P (x) 6= f(x)] ≤ ε.

Polynomials and polynomial approximations

P (x1, . . . , xn) ∈ F2[x1, . . . , xn]. Multilinear. E.g.: x1x2 + x3 + x1x5. Special kind of depth-2 AC0[2] circuit. P n n Degree D ⇒ size i≤D i = ≤D .

⊕ ∧ ∧ ··· ∧

x1 x5 ··· x3 x7 ··· x5 x6 ··· x2

S. Compression for AC0(p) September 30, 2015 12 / 27 Polynomials and polynomial approximations

P (x1, . . . , xn) ∈ F2[x1, . . . , xn]. Multilinear. E.g.: x1x2 + x3 + x1x5. Special kind of depth-2 AC0[2] circuit. P n n Degree D ⇒ size i≤D i = ≤D . Say that P ε-approximates f if Prx∈{0,1}n [P (x) 6= f(x)] ≤ ε.

⊕ ∧ ∧ ··· ∧

x1 x5 ··· x3 x7 ··· x5 x6 ··· x2

S. Compression for AC0(p) September 30, 2015 12 / 27 f −1(0)

Circuit has size nO(1) ⇒ degree of f −1(1) polynomial is O(log n)d−1 log(1/ε).

Razborov approximations to AC0[2] circuits

(Razborov 1987): Can ε-approximate small AC0[2] circuits by low-degree polynomials.

S. Compression for AC0(p) September 30, 2015 13 / 27 Razborov approximations to AC0[2] circuits

(Razborov 1987): Can ε-approximate f −1(0) small AC0[2] circuits by low-degree polynomials. Circuit has size nO(1) ⇒ degree of f −1(1) polynomial is O(log n)d−1 log(1/ε).

S. Compression for AC0(p) September 30, 2015 13 / 27 −1 Find low-degree ε-approximation P . f (0) Size(P ) = exp((log n)O(1) log(1/ε)). “Fix” P at all the error points with a f −1(1) circuit of size ε2n. Overall size: Size(P ) + ε2n. Bottleneck: How to ﬁnd P ?

Natural approach to compression problem

Given f : {0, 1}n → {0, 1} with AC0[2] circuits of size s = nO(1).

S. Compression for AC0(p) September 30, 2015 14 / 27 Size(P ) = exp((log n)O(1) log(1/ε)). “Fix” P at all the error points with a circuit of size ε2n. Overall size: Size(P ) + ε2n. Bottleneck: How to ﬁnd P ?

Natural approach to compression problem

Given f : {0, 1}n → {0, 1} with AC0[2] circuits of size s = nO(1). −1 Find low-degree ε-approximation P . f (0)

f −1(1)

S. Compression for AC0(p) September 30, 2015 14 / 27 “Fix” P at all the error points with a circuit of size ε2n. Overall size: Size(P ) + ε2n. Bottleneck: How to ﬁnd P ?

Natural approach to compression problem

Given f : {0, 1}n → {0, 1} with AC0[2] circuits of size s = nO(1). −1 Find low-degree ε-approximation P . f (0) Size(P ) = exp((log n)O(1) log(1/ε)).

f −1(1)

S. Compression for AC0(p) September 30, 2015 14 / 27 Overall size: Size(P ) + ε2n. Bottleneck: How to ﬁnd P ?

Natural approach to compression problem

Given f : {0, 1}n → {0, 1} with AC0[2] circuits of size s = nO(1). −1 Find low-degree ε-approximation P . f (0) Size(P ) = exp((log n)O(1) log(1/ε)). “Fix” P at all the error points with a f −1(1) circuit of size ε2n.

S. Compression for AC0(p) September 30, 2015 14 / 27 Bottleneck: How to ﬁnd P ?

Natural approach to compression problem

S. Compression for AC0(p) September 30, 2015 14 / 27 Natural approach to compression problem

S. Compression for AC0(p) September 30, 2015 14 / 27 R 6= 0. Also studied as Algebraic Immunity, Weak degree. Notion of one-sided approximation. Any (*) function f has a certifying polynomial of degree at most n/2. Gives a “local” circuit for f of size n n−1 ≤n/2 = 2 .

Certifying polynomials (ABFR ’94, Green ’95)

R ∈ F2[x1, . . . , xn] is a Certifying polynomial for f if R(x) = 1 ⇒ f(x) = 1.

f −1(0)

f −1(1)

S. Compression for AC0(p) September 30, 2015 15 / 27 Also studied as Algebraic Immunity, Weak degree. Notion of one-sided approximation. Any (*) function f has a certifying polynomial of degree at most n/2. Gives a “local” circuit for f of size n n−1 ≤n/2 = 2 .

Certifying polynomials (ABFR ’94, Green ’95)

R ∈ F2[x1, . . . , xn] is a Certifying polynomial for f if R(x) = 1 ⇒ f(x) = 1. R 6= 0. f −1(0)

f −1(1)

S. Compression for AC0(p) September 30, 2015 15 / 27 Notion of one-sided approximation. Any (*) function f has a certifying polynomial of degree at most n/2. Gives a “local” circuit for f of size n n−1 ≤n/2 = 2 .

Certifying polynomials (ABFR ’94, Green ’95)

R ∈ F2[x1, . . . , xn] is a Certifying polynomial for f if R(x) = 1 ⇒ f(x) = 1. R 6= 0. −1 Also studied as Algebraic Immunity, Weak f (0) degree.

f −1(1)

S. Compression for AC0(p) September 30, 2015 15 / 27 Any (*) function f has a certifying polynomial of degree at most n/2. Gives a “local” circuit for f of size n n−1 ≤n/2 = 2 .

Certifying polynomials (ABFR ’94, Green ’95)

R ∈ F2[x1, . . . , xn] is a Certifying polynomial for f if R(x) = 1 ⇒ f(x) = 1. R 6= 0. −1 Also studied as Algebraic Immunity, Weak f (0) degree. Notion of one-sided approximation. f −1(1)

S. Compression for AC0(p) September 30, 2015 15 / 27 Gives a “local” circuit for f of size n n−1 ≤n/2 = 2 .

Certifying polynomials (ABFR ’94, Green ’95)

R ∈ F2[x1, . . . , xn] is a Certifying polynomial for f if R(x) = 1 ⇒ f(x) = 1. R 6= 0. −1 Also studied as Algebraic Immunity, Weak f (0) degree. Notion of one-sided approximation. f −1(1) Any (*) function f has a certifying polynomial of degree at most n/2.

S. Compression for AC0(p) September 30, 2015 15 / 27 Certifying polynomials (ABFR ’94, Green ’95)

S. Compression for AC0(p) September 30, 2015 15 / 27 Thm (Kopparty-S. ’12): f has AC0[2] circuit of size poly(n) ⇒ certifying polynomials of degree D = n − n . 2 (log n)O(1) Gives a circuit of size 2n/ exp(n/(log n)O(1)).

Certifying polynomials

I Any function f has a certifying polynomial of degree at most n/2. n n−1 I Gives a “local” circuit for f of size ≤n/2 = 2 .

S. Compression for AC0(p) September 30, 2015 16 / 27 Gives a circuit of size 2n/ exp(n/(log n)O(1)).

Certifying polynomials

I Any function f has a certifying polynomial of degree at most n/2. n n−1 I Gives a “local” circuit for f of size ≤n/2 = 2 . Thm (Kopparty-S. ’12): f has AC0[2] circuit of size poly(n) ⇒ certifying polynomials of degree D = n − n . 2 (log n)O(1)

S. Compression for AC0(p) September 30, 2015 16 / 27 Certifying polynomials

S. Compression for AC0(p) September 30, 2015 16 / 27 P Q Finding R = |S|≤D αS i∈S xi. Need R(x) = 0 for all x ∈ f −1(0).

Has solution set VD.

Need non-zero element of VD.

Finding Certifying polynomials

0 I Thm (Kopparty-S. ’12): f has AC [2] circuit of size poly(n) ⇒ n n certifying polynomials of degree D = 2 − (log n)O(1) . n O(1) I Gives a circuit of size 2 / exp(n/(log n) ).

S. Compression for AC0(p) September 30, 2015 17 / 27 Need R(x) = 0 for all x ∈ f −1(0).

Has solution set VD.

Need non-zero element of VD.

Finding Certifying polynomials

0 I Thm (Kopparty-S. ’12): f has AC [2] circuit of size poly(n) ⇒ n n certifying polynomials of degree D = 2 − (log n)O(1) . n O(1) I Gives a circuit of size 2 / exp(n/(log n) ).

P Q Finding R = |S|≤D αS i∈S xi.

S. Compression for AC0(p) September 30, 2015 17 / 27 Has solution set VD.

Need non-zero element of VD.

Finding Certifying polynomials

0 I Thm (Kopparty-S. ’12): f has AC [2] circuit of size poly(n) ⇒ n n certifying polynomials of degree D = 2 − (log n)O(1) . n O(1) I Gives a circuit of size 2 / exp(n/(log n) ).

P Q Finding R = |S|≤D αS i∈S xi. Need R(x) = 0 for all x ∈ f −1(0).

S. Compression for AC0(p) September 30, 2015 17 / 27 Need non-zero element of VD.

Finding Certifying polynomials

0 I Thm (Kopparty-S. ’12): f has AC [2] circuit of size poly(n) ⇒ n n certifying polynomials of degree D = 2 − (log n)O(1) . n O(1) I Gives a circuit of size 2 / exp(n/(log n) ).

P Q Finding R = |S|≤D αS i∈S xi. Need R(x) = 0 for all x ∈ f −1(0).

Has solution set VD.

S. Compression for AC0(p) September 30, 2015 17 / 27 Finding Certifying polynomials

0 I Thm (Kopparty-S. ’12): f has AC [2] circuit of size poly(n) ⇒ n n certifying polynomials of degree D = 2 − (log n)O(1) . n O(1) I Gives a circuit of size 2 / exp(n/(log n) ).

P Q Finding R = |S|≤D αS i∈S xi. Need R(x) = 0 for all x ∈ f −1(0).

Has solution set VD.

Need non-zero element of VD.

S. Compression for AC0(p) September 30, 2015 17 / 27 Using VD to compress

f −1(0) −1 Each R ∈ VD covers a subset of f (1).

Select a few R1,...,Rm ∈ VD such that W −1 i Ri = f. f (1)

S. Compression for AC0(p) September 30, 2015 18 / 27 I Need to say that F is small.

Each R ∈ VD might cover only small subset of f −1(1) \ F .

Problems with the approach

There are points in f −1(1) that are never covered by R ∈ VD. f −1(0) −1 I F = {x ∈ f (1) | R ∈ VD ⇒ R(x) = 0}. f −1(1)

S. Compression for AC0(p) September 30, 2015 19 / 27 Each R ∈ VD might cover only small subset of f −1(1) \ F .

Problems with the approach

There are points in f −1(1) that are never covered by R ∈ VD. f −1(0) −1 I F = {x ∈ f (1) | R ∈ VD ⇒ R(x) = 0}. I Need to say that F is small. f −1(1)

S. Compression for AC0(p) September 30, 2015 19 / 27 Problems with the approach

There are points in f −1(1) that are never covered by R ∈ VD. f −1(0) −1 I F = {x ∈ f (1) | R ∈ VD ⇒ R(x) = 0}. I Need to say that F is small. f −1(1) Each R ∈ VD might cover only small subset of f −1(1) \ F .

S. Compression for AC0(p) September 30, 2015 19 / 27 Picking R1,...,RO(n) ∈u VD covers f −1(1) with high probability. Can be easily derandomized using Error-Correcting codes.

Handling second issue

A random R ∈ VD covers each x 6∈ F −1 1 f (0) with probability 2 .

f −1(1)

S. Compression for AC0(p) September 30, 2015 20 / 27 Picking R1,...,RO(n) ∈u VD covers f −1(1) with high probability. Can be easily derandomized using Error-Correcting codes.

Handling second issue

A random R ∈ VD covers each x 6∈ F −1 1 f (0) with probability 2 .

f −1(1)

S. Compression for AC0(p) September 30, 2015 20 / 27 Picking R1,...,RO(n) ∈u VD covers f −1(1) with high probability. Can be easily derandomized using Error-Correcting codes.

Handling second issue

A random R ∈ VD covers each x 6∈ F −1 1 f (0) with probability 2 .

f −1(1)

S. Compression for AC0(p) September 30, 2015 20 / 27 Picking R1,...,RO(n) ∈u VD covers f −1(1) with high probability. Can be easily derandomized using Error-Correcting codes.

Handling second issue

A random R ∈ VD covers each x 6∈ F −1 1 f (0) with probability 2 .

f −1(1)

S. Compression for AC0(p) September 30, 2015 20 / 27 Can be easily derandomized using Error-Correcting codes.

Handling second issue

A random R ∈ VD covers each x 6∈ F −1 1 f (0) with probability 2 . Picking R1,...,RO(n) ∈u VD covers −1 f (1) with high probability. f −1(1)

S. Compression for AC0(p) September 30, 2015 20 / 27 Handling second issue

A random R ∈ VD covers each x 6∈ F −1 1 f (0) with probability 2 . Picking R1,...,RO(n) ∈u VD covers −1 f (1) with high probability. f −1(1) Can be easily derandomized using Error-Correcting codes.

S. Compression for AC0(p) September 30, 2015 20 / 27 −1 Obtain m = O(n) polynomials R1,...,Rm ∈ VD covering f (1) \ F . W Output C = i Ri ∨ ϕ, where ϕ is a brute-force DNF accepting F . Size(C) = 2n/ exp(n/(log n)O(1)) + |F |.

Overall approach summary

Argue that for D = n − n , F is small. 2 (log n)O(1)

S. Compression for AC0(p) September 30, 2015 21 / 27 W Output C = i Ri ∨ ϕ, where ϕ is a brute-force DNF accepting F . Size(C) = 2n/ exp(n/(log n)O(1)) + |F |.

Overall approach summary

Argue that for D = n − n , F is small. 2 (log n)O(1) −1 Obtain m = O(n) polynomials R1,...,Rm ∈ VD covering f (1) \ F .

S. Compression for AC0(p) September 30, 2015 21 / 27 Size(C) = 2n/ exp(n/(log n)O(1)) + |F |.

Overall approach summary

Argue that for D = n − n , F is small. 2 (log n)O(1) −1 Obtain m = O(n) polynomials R1,...,Rm ∈ VD covering f (1) \ F . W Output C = i Ri ∨ ϕ, where ϕ is a brute-force DNF accepting F .

S. Compression for AC0(p) September 30, 2015 21 / 27 Overall approach summary

Argue that for D = n − n , F is small. 2 (log n)O(1) −1 Obtain m = O(n) polynomials R1,...,Rm ∈ VD covering f (1) \ F . W Output C = i Ri ∨ ϕ, where ϕ is a brute-force DNF accepting F . Size(C) = 2n/ exp(n/(log n)O(1)) + |F |.

S. Compression for AC0(p) September 30, 2015 21 / 27 Find non-zero Q of degree D2 s.t. Q|E = 0. Need n ≥ ε2n. ≤D2 n p D2 = 2 − Θ( n log(1/ε)).

Constructing certifying polynomials