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 Shuichi Hirahara (The University of Tokyo) Igor C. Oliveira (University of Oxford) Rahul Santhanam (University of Oxford) CCC 2018 @ San Diego 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) Output Input • Truth table 𝑈 ∈ 0,1 2 𝑢 of a Is there a circuit of size ≤ 𝑡 function 𝑔: 0,1 𝑢 → 0,1 that computes 𝑔 ? • Size parameter 𝑡 ∈ ℕ Example: Output: “YES” 𝑡 = 5 𝒚 𝟐 𝒚 𝟑 𝒚 𝟐 ⊕ 𝒚 𝟑 0 0 0 0 1 1 𝑔 = 1 0 1 1 1 0

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

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

Kabanets-Cai Obstacle: Why so difficult? ➢ Suppose that we want to construct a reduction from SAT to MCSP . ↦ (𝑔, 𝑡) 𝜒 ∈ SAT CircSize 𝑔 ≤ 𝑡 ↦ (𝑔, 𝑡) CircSize 𝑔 > 𝑡 𝜒 ∉ SAT Need to prove a circuit lower bound! ➢ Natural reduction techniques would imply E ⊈ SIZE 𝑜 𝑃 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 Output Input • Truth table 𝑈 ∈ 0,1 2 𝑢 of a Is there a 𝒟 -circuit of size ≤ 𝑡 function 𝑔: 0,1 𝑢 → 0,1 that computes 𝑔 ? • Size parameter 𝑡 ∈ ℕ Theorem [Masek (1978 or 79, unpublished)] DNF−MCSP is NP-hard.

DNF - MCSP Output Input • Truth table 𝑈 ∈ 0,1 2 𝑢 of a Is there a DNF formula of size function 𝑔: 0,1 𝑢 → 0,1 ≤ 𝑡 that computes 𝑔 ? • Size parameter 𝑡 ∈ ℕ Depth: 2 ∨ Size: 3 Example of DNFs: ∧ ∧ ∧ ≡ ¬𝑦 1 ∧ 𝑦 2 ∨ 𝑦 2 ∧ ¬𝑦 3 ∨ ¬𝑦 2 ¬𝑦 1 𝑦 2 𝑦 2 ¬𝑦 3 ¬𝑦 2 The size of DNF ≔ #(clauses)

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 More expressive MCSP Hardness based on cryptography ⋯ e.g.) Blum integer 0 - MCSP for large 𝑒 factorization [AHMPS08] AC 𝑒 ⋯ No hardness result Depth3- MCSP Known to be NP-hard DNF - MCSP [Masek (1979)] Remark: The complexity is not necessarily monotone increasing or decreasing.

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

Our Results ➢ The first NP-hardness result for 𝒟 -MCSP for a class 𝒟 ⊃ DNF Theorem (Main Result) DNF ∘ XOR −MCSP is NP-hard under polynomial-time many-one reductions. ➢ Our proof techniques extend to: • DNF ∘ MOD 𝑛 - 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 circuits ( 2 Ω 𝑜 circuit lower bound is known) [Cohen & Shinkar (2016)] ∨ Example Depth 3 ∧ ∧ ∧ Size 3 1 st layer: an OR gate 2 nd layer: AND gates ⊕ ⊕ ⊕ ⊕ 3 rd layer: XOR gates 𝑦 2 ¬𝑦 1 ¬𝑦 3 ¬𝑦 2 The size of DNF ∘ XOR circuits ≔ The number of AND gates ➢ 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) ➢ Our proof techniques extend to the number of all the gates in a DNF ∘ XOR formula.

DNF ∘ XOR circuits ( 2 Ω 𝑜 circuit lower bound is known) [Cohen & Shinkar (2016)] ∨ Example Depth 3 ∧ ∧ ∧ Size 3 1 st layer: an OR gate 2 nd layer: AND gates ⊕ ⊕ ⊕ ⊕ 3 rd layer: XOR gates 𝑦 2 ¬𝑦 1 ¬𝑦 3 ¬𝑦 2 1 ⊕ 𝑦 1 ⊕ 𝑦 2 ⊕ 1 ⊕ 𝑦 3 = 1 outputs 1. ⟺ The subcircuit 𝑦 2 ⊕ 1 ⊕ 𝑦 3 = 1 ← Some linear equations over GF 2 ⟺ 𝑦 1 , 𝑦 2 , 𝑦 3 ∈ 𝐵 (for some affine subspace 𝐵 ⊆ GF 2 𝑜 )

DNF ∘ XOR circuits ( 2 Ω 𝑜 circuit lower bound is known) [Cohen & Shinkar (2016)] 𝑔: 0,1 𝑜 → 0,1 ∨ Example Depth 3 𝐵 3 𝐵 1 𝐵 2 ∧ ∧ ∧ Size 3 1 st layer: an OR gate 2 nd layer: AND gates ⊕ ⊕ ⊕ ⊕ 3 rd layer: XOR gates 𝑦 2 ¬𝑦 1 ¬𝑦 3 ¬𝑦 2 𝑔 −1 1 = 𝐵 1 ∪ 𝐵 2 ∪ 𝐵 3

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, ∃𝐵 1 , … , 𝐵 𝑛 : affine subspaces of 0,1 𝑜 𝐵 𝑗 ⊆ 𝑔 −1 1 𝐵 1 ∪ ⋯ ∪ 𝐵 𝑛 = 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 Theorem (Main Result) 𝑞 NP ≤ 𝑛 DNF ∘ XOR −MCSP Step 1. 2-factor approx. of DNF ∘ XOR - MCSP ZPP ≤ 𝑛 𝑠 -Bounded Set Cover for partial functions (NP-hard [Trevisan 2001] ) DNF ∘ XOR - MCSP Step 2. ZPP DNF ∘ XOR - MCSP ≤ 𝑛 for partial functions Derandomization using 𝜗 -biased generators Step 3. [Naor & Naor (1993)]

Proof Outline Theorem (Main Result) 𝑞 NP ≤ 𝑛 DNF ∘ XOR −MCSP Step 1. 2-factor approx. of DNF ∘ XOR - MCSP ZPP ≤ 𝑛 𝑠 -Bounded Set Cover for partial functions (NP-hard [Trevisan 2001] ) DNF ∘ XOR - MCSP Step 2. ZPP DNF ∘ XOR - MCSP ≤ 𝑛 for partial functions Derandomization using 𝜗 -biased generators Step 3. [Naor & Naor (1993)]

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

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

The 𝑠 -Bounded Set Cover Problem A universe 𝑉 and a collection of sets 𝒯 ⊆ 2 𝑉 Input: such that 𝑇 ≤ 𝑠 for every 𝑇 ∈ 𝒯 . Output: The minimum 𝒟 such that 𝒟 ⊆ 𝒯 and ڂ 𝐷∈𝒟 𝐷 = 𝑉 Example: 𝑉 = { … } , 𝒯 = { … } 4-bounded, but not 3-bounded ➢ We set 𝑠 to be large enough [Feige (1998)] [Trevisan (2001)] so that a 2 -factor approx. is NP-hard. Approximation of 1 − 𝑝 1 ln 𝑠 is NP -hard.

Proof Outline Theorem (Main Result) 𝑞 NP ≤ 𝑛 DNF ∘ XOR −MCSP Step 1. 2-factor approx. of DNF ∘ XOR - MCSP ZPP ≤ 𝑛 𝑠 -Bounded Set Cover for partial functions (NP-hard [Trevisan 2001] ) DNF ∘ XOR - MCSP Step 2. ZPP DNF ∘ XOR - MCSP ≤ 𝑛 for partial functions Derandomization using 𝜗 -biased generators Step 3. [Naor & Naor (1993)]

DNF ∘ XOR - MCSP ∗ for partial functions Output Input • Truth table of a partial Is there a circuit of size ≤ 𝑡 function that agrees with 𝑔 on inputs 𝑔: 0,1 𝑢 → 0,1,∗ from 𝑔 −1 0,1 ? • Size parameter 𝑡 ∈ ℕ Example: 𝒚 𝟐 𝒚 𝟑 𝒈 𝒚 𝟐 , 𝒚 𝟑 0 0 ∗ 0 1 1 𝑔 = ∗ 1 0 1 1 0

Claim DNF ∘ XOR - MCSP 2-factor approx. of ZPP ≤ 𝑛 for partial functions 𝑠 -Bounded Set Cover ➢ Given: 𝑉 = 1, … , 𝑂 , 𝒯 = {𝑇 1 , … , 𝑇 𝑛 } Construct 𝑔: 0,1 𝑢 → 0,1,∗ for 𝑢 = 𝑃 log 𝑂 ➢ Goal: DNF ∘ XOR - MCSP Set Cover 𝑤 𝑗 ∼ 0,1 𝑢 (A uniformly random vector) 𝑗 ∈ 𝑉 ↦ span 𝑗∈𝑇 𝑘 𝑤 𝑗 ⊆ 0,1 𝑢 𝑇 𝑘 ∈ 𝒯 ↦ span 𝑗∈𝑇 𝑤 𝑗 ⊆ 0,1 𝑢 Cover 𝒟 ⊆ 𝒯 ↦ ራ 𝑇∈𝒟

Set Cover DNF ∘ XOR - MCSP 𝑔: 0,1 𝑢 → 0,1,∗ 𝑉 = 1,2,3 , 𝒯 = {𝑇 1 , 𝑇 2 } 𝑇 1 = 1,2 1 2 3 𝑇 2 = 2,3 𝑔 𝑤 𝑗 ≔ 1 for any 𝑗 ∈ 𝑉 . ➢ 𝑔 𝑦 ≔ 0 for all 𝑦 ∉ span 𝑤 1 , 𝑤 2 ∪ span 𝑤 2 , 𝑤 3 . ➢ ➢ 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 = 𝑤 1 , 𝑤 2 , 𝑤 3 .