NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD Circuits - - PowerPoint PPT Presentation

NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD Circuits

CCC 2018 @ San Diego

Shuichi Hirahara (The University of Tokyo) Igor C. Oliveira (University of Oxford) Rahul Santhanam (University of Oxford)

June 22, 2018

Talk Outline

• 1. MCSP and Its background
• 2. 𝒟-MCSP for a circuit class 𝒟
• 3. Our Results
• 4. Proof Sketch

Talk Outline

• 1. MCSP and Its background
• 2. 𝒟-MCSP for a circuit class 𝒟
• 3. Our Results
• 4. Proof Sketch

Minimum Circuit Size Problem (MCSP)

Input

𝒚𝟐 𝒚𝟑 𝒚𝟐 ⊕ 𝒚𝟑 1 1 1 1 1 1

• Truth table 𝑈 ∈ 0,1 2𝑢 of a

function 𝑔: 0,1 𝑢 → 0,1

Output

Is there a circuit of size ≤ 𝑡 that computes 𝑔?

Example: 𝑡 = 5 𝑔 = Output: “YES”

• Size parameter 𝑡 ∈ ℕ

Brief History of MCSP

1970s Levin delayed publishing his work because he wanted to say something about MCSP. 2000 Kabanets and Cai revived interest, based on natural proofs. [Razborov & Rudich (1997)] 1950s Recognized as an important problem in the Soviet Union [Trakhtenbrot’s survey] 1979 Masek proved NP-completeness of DNF-MCSP.

Since then many papers and results appeared; however, the complexity of MCSP remains elusive.

Current Knowledge about MCSP

✓No strong evidence against NP-completeness

• Weak evidence: [Hirahara-Santhanam (CCC’17)] [Allender-Hirahara 17]…

✓No strong evidence for NP-completeness

• Some new evidence: [Impagliazzo-Kabanets-Volkvovich (CCC’18)] & This work

➢ Lower bound: ∃pseudorandom function generators ⟹ MCSP ∉ 𝐐 ➢ No consensus about the exact complexity of MCSP ➢ Big Open Question: Is MCSP NP-hard? ➢ Upper bound: MCSP ∈ 𝐎𝐐

Kabanets-Cai Obstacle: Why so difficult?

𝜒 ∈ SAT

↦

➢ Suppose that we want to construct a reduction from SAT to MCSP. (𝑔, 𝑡) 𝜒 ∉ SAT

↦

(𝑔, 𝑡) CircSize 𝑔 ≤ 𝑡 CircSize 𝑔 > 𝑡 Need to prove a circuit lower bound! ➢ Natural reduction techniques would imply E ⊈ SIZE 𝑜𝑃 1 .

[Kabanets-Cai (2000)]

Talk Outline

• 1. MCSP and Its background
• 2. 𝒟-MCSP for a circuit class 𝒟
• 3. Our Results
• 4. Proof Sketch

𝒟-MCSP for a circuit class 𝒟

Input

• Truth table 𝑈 ∈ 0,1 2𝑢 of a

function 𝑔: 0,1 𝑢 → 0,1

Output

Is there a 𝒟-circuit of size ≤ 𝑡 that computes 𝑔?

• Size parameter 𝑡 ∈ ℕ

DNF−MCSP is NP-hard.

Theorem [Masek (1978 or 79, unpublished)]

DNF-MCSP

Input

• Truth table 𝑈 ∈ 0,1 2𝑢 of a

function 𝑔: 0,1 𝑢 → 0,1

Output

Is there a DNF formula of size ≤ 𝑡 that computes 𝑔?

• Size parameter 𝑡 ∈ ℕ

Example of DNFs:

Depth: 2 ¬𝑦1 ∧ 𝑦2 ∨ 𝑦2 ∧ ¬𝑦3 ∨ ¬𝑦2

∧ ∨ ∧

¬𝑦1

∧

𝑦2 𝑦2 ¬𝑦3 ¬𝑦2

≡

The size of DNF ≔ #(clauses) Size: 3

𝒟-MCSP for 𝒟 ⊃ DNF

➢ Beyond DNFs, no NP-hardness was proved since the work of Masek (1979). ➢ To quote Allender, Hellerstein, McCabe, Pitassi, and Saks (2008):

“Thus an important open question is to resolve the NP-hardness of … function minimization results above for classes that are stronger than DNF.”

Known results about 𝒟-MCSP

DNF-MCSP AC𝑒

0-MCSP for large 𝑒

MCSP

Hardness based on cryptography Known to be NP-hard

[Masek (1979)]

More expressive

Depth3-MCSP ⋯ ⋯

No hardness result Remark: The complexity is not necessarily monotone increasing or decreasing. e.g.) Blum integer factorization [AHMPS08]

Talk Outline

• 1. MCSP and Its background
• 2. 𝒟-MCSP for a circuit class 𝒟
• 3. Our Results
• 4. Proof Sketch

➢ The first NP-hardness result for 𝒟-MCSP for a class 𝒟 ⊃ DNF

Our Results

➢ Our proof techniques extend to:

• DNF ∘ MOD𝑛 -MCSP′ is NP-hard for any 𝑛 ≥ 2,

but the input is a truth table of an 𝑛-valued function 𝑔: ℤ/𝑛ℤ 𝑢 → 0,1 .

DNF ∘ XOR −MCSP is NP-hard under polynomial-time many-one reductions.

Theorem (Main Result)

DNF ∘ XOR circuits

Depth 3

∧ ∨ ∧

¬𝑦1

∧

𝑦2 ¬𝑦3 ¬𝑦2 The size of DNF ∘ XOR circuits ≔ The number of AND gates

⊕ ⊕ ⊕ ⊕ Example

1st layer: an OR gate 2nd layer: AND gates 3rd layer: XOR gates Size 3

➢ This is a convenient circuit size measure as advocated by Cohen & Shinkar (2016). 1. Nice combinatorial meaning 2. W.l.o.g., # XOR gates ≤ 𝑜 ⋅ #(AND gates) (2Ω 𝑜 circuit lower bound is known)

[Cohen & Shinkar (2016)]

➢ Our proof techniques extend to the number of all the gates in a DNF ∘ XOR formula.

DNF ∘ XOR circuits

Depth 3

∧ ∨ ∧

¬𝑦1

∧

𝑦2 ¬𝑦3 ¬𝑦2

⊕ ⊕ ⊕ ⊕ Example

1st layer: an OR gate 2nd layer: AND gates 3rd layer: XOR gates Size 3

1 ⊕ 𝑦1 ⊕ 𝑦2 ⊕ 1 ⊕ 𝑦3 = 1 𝑦2 ⊕ 1 ⊕ 𝑦3 = 1 The subcircuit

• utputs 1. ⟺

⟺

𝑦1, 𝑦2, 𝑦3 ∈ 𝐵 (for some affine subspace 𝐵 ⊆ GF 2 𝑜) ← Some linear equations over GF 2 (2Ω 𝑜 circuit lower bound is known)

[Cohen & Shinkar (2016)]

DNF ∘ XOR circuits

Depth 3

∧ ∨ ∧

¬𝑦1

∧

𝑦2 ¬𝑦3 ¬𝑦2

⊕ ⊕ ⊕ ⊕ Example

1st layer: an OR gate 2nd layer: AND gates 3rd layer: XOR gates Size 3 𝑔: 0,1 𝑜 → 0,1 𝐵1 𝐵2 𝐵3

𝑔−1 1 = 𝐵1 ∪ 𝐵2 ∪ 𝐵3

(2Ω 𝑜 circuit lower bound is known)

[Cohen & Shinkar (2016)]

The Important Observation

The minimum DNF ∘ XOR circuit size for computing 𝑔 The minimum number 𝑛 of affine subspaces needed to cover 𝑔−1 1 : that is, ∃𝐵1, … , 𝐵𝑛: affine subspaces of 0,1 𝑜

=

𝐵𝑗 ⊆ 𝑔−1 1 𝐵1 ∪ ⋯ ∪ 𝐵𝑛 = 𝑔−1 1 and

Talk Outline

• 1. MCSP and Its background
• 2. 𝒟-MCSP for a circuit class 𝒟
• 3. Our Results
• 4. Proof Sketch

➢ Our proof was inspired by a simple proof of Masek’s result given by [Allender, Hellerstein, McCabe, Pitassi, and Saks (2008)]. ➢ We extend and generalize their ideas significantly.

Proof Outline

2-factor approx. of 𝑠-Bounded Set Cover ≤𝑛

ZPP

DNF ∘ XOR -MCSP for partial functions DNF ∘ XOR -MCSP ≤𝑛

ZPP

[Naor & Naor (1993)] (NP-hard [Trevisan 2001])

Derandomization using 𝜗-biased generators

NP ≤𝑛

𝑞

DNF ∘ XOR −MCSP

Theorem (Main Result) Step 1. Step 2. DNF ∘ XOR -MCSP for partial functions Step 3.

Proof Outline

2-factor approx. of 𝑠-Bounded Set Cover ≤𝑛

ZPP

DNF ∘ XOR -MCSP for partial functions DNF ∘ XOR -MCSP ≤𝑛

ZPP

[Naor & Naor (1993)] (NP-hard [Trevisan 2001])

Derandomization using 𝜗-biased generators

NP ≤𝑛

𝑞

DNF ∘ XOR −MCSP

Theorem (Main Result) Step 1. Step 2. DNF ∘ XOR -MCSP for partial functions Step 3.

The Set Cover Problem

The minimum 𝒟 such that 𝒟 ⊆ 𝒯 and ڂ𝐷∈𝒟 𝐷 = 𝑉 Example: 𝑉 = { … }, 𝒯 = { … } Input: A universe 𝑉 and a collection of sets 𝒯 ⊆ 2𝑉 Output:

The Set Cover Problem

The minimum 𝒟 such that 𝒟 ⊆ 𝒯 and ڂ𝐷∈𝒟 𝐷 = 𝑉 Example: 𝑉 = { … }, 𝒯 = { … } Input: A universe 𝑉 and a collection of sets 𝒯 ⊆ 2𝑉 Output:

A minimum cover 𝒟:

The 𝑠-Bounded Set Cover Problem

The minimum 𝒟 such that 𝒟 ⊆ 𝒯 and ڂ𝐷∈𝒟 𝐷 = 𝑉 Example: 𝑉 = { … }, 𝒯 = { … } Input: A universe 𝑉 and a collection of sets 𝒯 ⊆ 2𝑉 Output: such that 𝑇 ≤ 𝑠 for every 𝑇 ∈ 𝒯.

4-bounded, but not 3-bounded [Feige (1998)] [Trevisan (2001)] Approximation of 1 − 𝑝 1 ln 𝑠 is NP-hard. ➢ We set 𝑠 to be large enough so that a 2-factor approx. is NP-hard.

Proof Outline

2-factor approx. of 𝑠-Bounded Set Cover ≤𝑛

ZPP

DNF ∘ XOR -MCSP for partial functions DNF ∘ XOR -MCSP ≤𝑛

ZPP

[Naor & Naor (1993)] (NP-hard [Trevisan 2001])

Derandomization using 𝜗-biased generators

NP ≤𝑛

𝑞

DNF ∘ XOR −MCSP

Theorem (Main Result) Step 1. Step 2. DNF ∘ XOR -MCSP for partial functions Step 3.

DNF ∘ XOR -MCSP∗ for partial functions

𝒚𝟐 𝒚𝟑 𝒈 𝒚𝟐, 𝒚𝟑 ∗ 1 1 1 ∗ 1 1

Example:

𝑔 =

Input

• Truth table of a partial

function 𝑔: 0,1 𝑢 → 0,1,∗

Output

Is there a circuit of size ≤ 𝑡 that agrees with 𝑔 on inputs from 𝑔−1 0,1 ?

• Size parameter 𝑡 ∈ ℕ

2-factor approx. of 𝑠-Bounded Set Cover ≤𝑛

ZPP

Claim DNF ∘ XOR -MCSP for partial functions ➢ Given: 𝑉 = 1, … , 𝑂 , 𝒯 = {𝑇1, … , 𝑇𝑛} 𝑗 ∈ 𝑉 𝑤𝑗 ∼ 0,1 𝑢

(A uniformly random vector)

𝑇

𝑘 ∈ 𝒯

span𝑗∈𝑇𝑘 𝑤𝑗 ⊆ 0,1 𝑢 DNF ∘ XOR -MCSP Set Cover ↦ ↦ 𝒟 ⊆ 𝒯 Cover ↦ ራ

𝑇∈𝒟

span𝑗∈𝑇 𝑤𝑗 ⊆ 0,1 𝑢 ➢ Goal: Construct 𝑔: 0,1 𝑢 → 0,1,∗ for 𝑢 = 𝑃 log 𝑂

1

𝑉 = 1,2,3 , 𝒯 = {𝑇1, 𝑇2}

2 3

𝑇1 = 1,2 𝑇2 = 2,3

DNF ∘ XOR -MCSP Set Cover

➢ 𝑔 𝑤𝑗 ≔ 1 for any 𝑗 ∈ 𝑉. ➢ 𝑔 𝑦 ≔ 0 for all 𝑦 ∉ span 𝑤1, 𝑤2 ∪ span 𝑤2, 𝑤3 . ➢ 𝑔 𝑧 ≔ ∗ for any other vector 𝑧 ∈ 0,1 𝑢.

• The minimum DNF ∘ XOR circuit size for computing 𝑔

The minimum number of affine subspaces 𝐵 ⊆ 𝑔−1 1,∗ needed to cover 𝑔−1 1 = 𝑤1, 𝑤2, 𝑤3 .

=

𝑔: 0,1 𝑢 → 0,1,∗

Intuition: When 𝐵 is Linear

𝐵 ⊆ 𝑔−1 1,∗ = span 𝑤1, 𝑤2 ∪ span(𝑤2, 𝑤3)

⟹

with high probability (if 𝐵 is a linear subspace)

𝐵 ⊆ span 𝑤1, 𝑤2 or 𝐵 ⊆ span(𝑤2, 𝑤3)

⟹

The set of points 𝑗 ∈ 1,2,3 𝑤𝑗 ∈ 𝐵 } covered by 𝐵 is contained in some legal set 𝑇1 or 𝑇2 ∈ 𝒯.

⟹

The minimum number of linear subspaces needed to cover {𝑤1, 𝑤2, 𝑤3}

= The minimum set cover size

Random linear subspaces of small dimension 𝑠

Intuition: When 𝐵 is Affine

𝐵 ⊆ 𝑔−1 1,∗ = span 𝑤1, 𝑤2 ∪ span(𝑤2, 𝑤3)

⟹

with high probability (if 𝐵 is an affine subspace)

𝐵 ⊆ span 𝑤1, 𝑤2 or 𝐵 ⊆ span(𝑤2, 𝑤3) The set of points 𝑗 ∈ 1,2,3 𝑤𝑗 ∈ 𝐵 } covered by 𝐵 is contained in 𝑇𝑏 ∪ 𝑇𝑐 for some two legal sets 𝑇𝑏, 𝑇𝑐 ∈ 𝒯

⟹ The minimum number of affine subspaces needed to cover 𝑤1, 𝑤2, 𝑤3

is a 2-factor approximation of the minimum set cover size.

?

Counterexample: 𝐵 ≔ 𝑤1, 𝑤3 = 𝑤1 ⊕ 0, 𝑤1 ⊕ 𝑤3 ➢ Still, we can prove that:

Fomally:

For 𝑢 ≥ 𝑃 𝑠 log 𝑂 , the following holds with high probability: The minimum set cover size ≤ 2 × The minimum DNF ∘ XOR circuit size

Claim (Hard part)

The minimum DNF ∘ XOR circuit size ≤ The minimum set cover size

Claim (Easy part)

𝑔 𝑦 = ቐ 1 ∗ (𝑦 = 𝑤𝑗 for some 𝑗) (𝑦 ∉ڂ𝑇∈𝒯 span𝑗∈𝑇(𝑤𝑗)) (otherwise)

𝑔: 0,1 𝑢 → 0,1,∗

➢ By a delicate probabilistic argument, it can be shown:

Summary of Step 1

1. Input: 𝑉 = 1, … , 𝑂 , 𝒯 = {𝑇1, … , 𝑇𝑛} 2. Let 𝑢 ≔ Θ log 𝑂 . 3. Pick 𝑤𝑗 ∼ 0,1 𝑢 randomly for each 𝑗 ∈ 𝑉. 4. Verify that 𝑤𝑗

𝑗∈𝑉 satisfies a certain condition.

5. Define 𝑔: 0,1 𝑢 → {0,1,∗} as follows and output its truth table.

𝑔 𝑦 = ቐ 1 ∗ (𝑦 = 𝑤𝑗 for some 𝑗) (𝑦 ∉ڂ𝑇∈𝒯 span𝑗∈𝑇(𝑤𝑗)) (otherwise)

Proof Outline

2-factor approx. of 𝑠-Bounded Set Cover ≤𝑛

ZPP

DNF ∘ XOR -MCSP for partial functions DNF ∘ XOR -MCSP ≤𝑛

ZPP

[Naor & Naor (1993)] (NP-hard [Trevisan 2001])

Derandomization using 𝜗-biased generators

NP ≤𝑛

𝑞

DNF ∘ XOR −MCSP

Theorem (Main Result) Step 1. Step 2. DNF ∘ XOR -MCSP for partial functions Step 3.

DNF ∘ XOR -MCSP for partial functions ≤𝑛

ZPP

Claim DNF ∘ XOR -MCSP ➢ Given: a partial function 𝑔: 0,1 𝑢 → {0,1,∗} ➢ Output: a total function 𝑕: 0,1 𝑢+𝑡 → 0,1 ➢ For each 𝑦 ∈ 0,1 𝑢, we encode each value 𝑔 𝑦 ∈ {0,1,∗} as a Boolean function 𝑕𝑦 ≔ 𝑕 𝑦,⋅ on a hypercube 0,1 𝑡.

Step 2: Making it a total function

𝑔 𝑦 ∈ 0,1

𝑔 𝑦

𝑔 𝑦 = ∗

1 1 1

𝑕𝑦: 0,1 𝑡 → 0,1

➢ Pick a random linear subspace 𝑀𝑦 and define 𝑕𝑦 as its characteristic function.

For each 𝑦 ∈ 0,1 𝑢:

➢ Define 𝑕 𝑦, 𝑧 ≔ 𝑕𝑦 𝑧 .

𝑕𝑦 0𝑡 ≔ 𝑔(𝑦) 𝑕𝑦 𝑧 ≔ 0 elsewhere

➢Imagine an optimal way of covering 𝑕−1 1 .

• 𝑕−1 1 consists of 𝑔−1 1 × 0 𝑡 and 𝑦 × 𝑀𝑦 for each 𝑦 ∈ 𝑔−1 ∗ .

➢In order to cover 𝑕−1 1 by affine subspaces, random linear subspaces 𝑦 × 𝑀𝑦 should be used for each 𝑦 ∈ 𝑔−1(∗). ➢Then we need to cover 𝑔−1 1 × 0 𝑡, but we may optionally cover 𝑔−1 ∗ × 0 𝑡. (The minimum DNF ∘ XOR circuit size for 𝑕) = Claim (The minimum circuit size for 𝑔) + 𝑔−1 ∗ The following holds with high probability:

Idea:

Proof Outline

2-factor approx. of 𝑠-Bounded Set Cover ≤𝑛

ZPP

DNF ∘ XOR -MCSP for partial functions DNF ∘ XOR -MCSP ≤𝑛

ZPP

[Naor & Naor (1993)] (NP-hard [Trevisan 2001])

Derandomization using 𝜗-biased generators

NP ≤𝑛

𝑞

DNF ∘ XOR −MCSP

Theorem (Main Result) Step 1. Step 2. DNF ∘ XOR -MCSP for partial functions Step 3.

Step 3: Derandomization

Fact (folklore; a nearly optimal PRG for AND ∘ XOR circuits) Any 𝜗-biased generator 𝜗-fools any AND ∘ XOR circuit. ➢ Can be proved by using a simple Fourier analysis. ➢ Our probabilistic arguments work even if randomness is replaced by the output of an 𝜗-biased generator.

• Careful analysis: sub-conditions can be checked by AND ∘ XOR circuits

➢ Extending the fact to AND ∘ MOD𝑛 requires some extra work.

Open Problems

➢NP-hardness of Depth3-AC0-MCSP under quasipolynomial-time deterministic reductions, or randomized polynomial-time reductions?

• The Kabanets-Cai obstacle is not applied to these

reductions.

➢What about 𝒟-MCSP for 𝒟 = MAJ ∘ MAJ, OR ∘ MAJ?