Recent Developments on Circuit Satisfiability Algorithms

Suguru TAMAKI Kyoto University

Fine-Grained Complexity and Algorithm Design Reunion, December 14, 2016, Simons Institute, Berkeley, CA After the Fine-Grained Complexity and Algorithm OurDesign ProgramProblem (Aug. 19 – Dec. 18, 2015), many papers on Circuit SAT algorithms have been published:

 Chan-Williams, ``Deterministic APSP, Orthogonal Vectors, and More: Quickly Derandomizing Razborov-Smolensky,’’ SODA’16  Chen-Santhanam, ``Satisfiability on Mixed Instances,‘’ ITCS’16  Chen-Santhanam-Srinivasan, ``Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits,’’ CCC’16  Williams, ``Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation,’’ CCC’16

2  Golovnev-Kulikov-Smal-T, ``Circuit Size Lower Bounds Ourand #SATProblem Upper Bounds Through a General Framework,’’ MFCS’16  Sakai-Seto-T-Teruyama, ``Bounded Depth Circuits with Weighted Symmetric Gates: Satisfiability, Lower Bounds and Compression,’’ MFCS’16  Alman-Chan-Williams, ``Polynomial Representations of Threshold Functions and Algorithmic Applications,’’ FOCS’16  Lokshtanov-Paturi-T-Williams-Yu, ``Beating Brute Force for Systems of Polynomial Equations over Finite Fields,’’ SODA’17  T, ``A Satisfiability Algorithm for Depth Two Circuits with a Sub-Quadratic Number of Symmetric and Threshold Gates’’  …… 3 Our Problem

Circuit Satisfiability (SAT) 1? Input Boolean Circuit ∧ C :{0,1}n →{0,1} ∨ ≡ Output Yes/No ∃x ∈{0,1}n ,C(x) =1 x1 + ¬ → Yes ? n x ∀x ∈{0,1} ,C(x) = 0 x2 3 x1 → No ? ? ?

4 Our Problem

Circuit Satisfiability (SAT) 1 Input Boolean Circuit ∧ C :{0,1}n →{0,1} ∨ ≡ Output Yes/No ∃x ∈{0,1}n ,C(x) =1 x1 + ¬ → Yes 1 n x ∀x ∈{0,1} ,C(x) = 0 x2 3 x1 → No 0 0 1

5 Our Problem

Circuit Satisfiability (SAT)

Canonical NP-complete problem [Cook-Levin]

∎ Solved in time 2 × poly ( : #variables,𝑛𝑛 : size of a circuit ) ∎ 𝐶𝐶 -SAT𝑛𝑛 input only from𝐶𝐶 a circuit class 𝐶𝐶 e.g. = ( -)CNF, AC0, AC0[ ], ACC0, TC0, NC1 (Formula),… ∎ 𝒞𝒞 𝒞𝒞 𝒞𝒞 𝑘𝑘 𝑝𝑝

6 Example

CNF-SAT =( -)CNF formulas (conjunctions of clauses)

e.g. = ( ) (¬ ¬ ) (¬ ) 𝒞𝒞 𝑘𝑘 CNF is -CNF1 if2 each clause2 has3 at4 most 3literals 𝐶𝐶 𝑥𝑥 ∨ 𝑥𝑥 ∧ 𝑥𝑥 ∨ 𝑥𝑥 ∨ 𝑥𝑥 ∧ 𝑥𝑥 ∧ ⋯ Best known𝑘𝑘 algorithms 𝑘𝑘 3-CNF: 1.308 [Hertli’14, randomized] 1.331𝑛𝑛 [Makino-T-Yamamoto’13, deterministic]

. 𝑛𝑛 ( ) -CNF: 2 [Paturi-Pudlak-Saks-Zane’05, randomized] 1 64 . (1− 𝑘𝑘 )𝑛𝑛 2 [Moser-Scheder’11, deterministic] 𝑘𝑘 1 44 1− 𝑘𝑘 𝑛𝑛 7 Hardness of Circuit SAT?

(Strong) Exponential Time Hypothesis (``quantitatively strengthening’’ of P NP) [Impagliazzo-Paturi-Zane] ≠ ETH > 0, 3-CNF-SAT DTIME[2 ] SETH > 0, , -CNF-SAT DTIME𝛿𝛿[𝑛𝑛2( ) ] ∃𝛿𝛿 ∉ 1−𝜀𝜀 𝑛𝑛 ETH & ∀SETH𝜀𝜀 are∃𝑘𝑘 widely𝑘𝑘 used to∉ prove the optimality of exact, parametrized, approximation and dynamic algorithms (cf. ``fine-grained complexity’’ theory)

8 General Research Goals

1. Design non-trivial algorithms for a stronger circuit class

Non-trivial: super-polynomially (or exponentially) faster than 2

𝑛𝑛 Stronger circuit class: ( -)CNF AC0 AC0[ ] ACC0 TC0 NC1 … CKT

𝑘𝑘 ⊂ ⊂ 𝑝𝑝 ⊂ ⊆ ⊆ ⊆ ⊆ 2. If non-trivial algorithms exist for -SAT, then improve the running time 𝒞𝒞 prove the difficulty of improvement ∎ additional properties: deterministic, counting, P-space,… ∎

∎ 9 Why Algorithms for Circuit SAT?

1. Useful

SAT can encode many combinatorial problems efficiently

∎ (sometimes) inspire algorithms for other problems e.g. All-Pairs Shortest Paths (APSP) [Williams’14,…] ∎

10 Why Algorithms for Circuit SAT?

2. Connections to circuit lower bounds (This Talk)

[black box] non-trivial -SAT algorithm ⇒ NEXP [Williams’10,11,…] (NEXP: nondeterministic exponential time) ∎ 𝒞𝒞 ⊈ 𝒞𝒞 [white box] analysis of -SAT algorithm ⇒ average-case lower bounds ∎ 𝒞𝒞 [Santhanam’10,Seto-T’12,Chen-Kabanets-Kolokolova-Shaltiel- Zuckerman’14,…]

11 Boolean Circuit Complexity studies the power of circuit classes defined using several parameters:

2 polylogn nε #gates: O(n),O(n ),...,poly(n),...,n ,...,2 ,... Gate type: AND, OR, NOT, XOR, MOD, MAJORITY, THRESHOLD… Structure: ・fanin: 2, 3, …,const, … , unbounded ・fanout: 1, 2, …, const, …, unbounded ・depth: 2, 3, …, const, …, log n, …, polylog n, … ・graph class: tree (=formula, fan-out=1), planar, …

12 Circuit Classes

( -)CNF AC0 AC0[ ] ACC0 TC0 NC1 … CKT

𝑘𝑘 ( -)CNF⊂: conjunction⊂ 𝑝𝑝 ⊂of disjunctions⊆ ⊆ of (at⊆ most⊆ ) literals AC0: constant-depth, unbounded-fan-in, AND/OR/NOT ∎ 𝑘𝑘 𝑘𝑘 AC0[ ]: AC0 + mod gates ( : prime power) ∎ ACC0: AC0 + mod gates ( : integer 2) ∎ 𝑝𝑝 𝑝𝑝 𝑝𝑝 TC0: constant-depth, unbounded-fan-in, ∎ 𝑚𝑚 𝑚𝑚 ≥ linear threshold (THR) gates: sgn( ) ∎ NC1: fan-in 2, fan-out 1, AND/OR/NOT(/XOR)𝑛𝑛 ∑𝑖𝑖=1 𝑤𝑤𝑖𝑖𝑥𝑥𝑖𝑖 − 𝜃𝜃 CKT: fan-in 2, AND/OR/NOT(/XOR) ∎ : composition of and e.g. CNF=AND OR ∎ 1 2 1 2 (Note: assume #gates = poly( ) unless otherwise ∎ 𝒞𝒞 ∘𝒞𝒞 𝒞𝒞 𝒞𝒞 ∘ specified) 13 𝑛𝑛 Circuit Lower Bounds

Typical question:

Can a family of Boolean circuits {Cn} from a circuit class compute a problem X? (Cn is an n-variate circuit for each n) 𝒞𝒞 Important case (NP CKT?): ・ =CKT: circuits with poly(n) number of gates ⊆ ・X=NP-complete problem, e.g., SAT, clique,… 𝒞𝒞

If the answer is no, then P NP (by P CKT) (Approach to P vs. NP [Sipser’s Program]) ≠ ⊊

14 Research Frontier (-2015)

… AC0[ ] ACC0 ACC0 THR TC0 NC1 … CKT

⊂ 𝑝𝑝 ⊂ ⊆ ∘ ⊆ ⊆ ⊆ ⊆ Non-trivial -SAT algorithm: ACC0 THR with a super-poly #gates [Williams’14] 𝒞𝒞 depth 2 TC0 with a linear #wires [Impagliazzo-Paturi- Schneider’13,…]∎ ∘ ∎ AC0: constant-depth, unbounded-fan-in, AND/OR/NOT ACC0: AC0 + mod gates ( : integer 2) ∎ TC0: constant-depth, unbounded-fan-in, ∎ 𝑚𝑚 𝑚𝑚 ≥ linear threshold (THR) gates: sgn( ) ∎ 𝑛𝑛 1 2: composition of 1 2 ∑𝑖𝑖=1 𝑤𝑤𝑖𝑖𝑥𝑥𝑖𝑖 − 𝜃𝜃 ∎ 𝒞𝒞 ∘𝒞𝒞 𝒞𝒞 ∘𝒞𝒞 15 Research Frontier (2016)

Non-trivial -SAT algorithm: ACC0 THR with a super-poly #gates [Williams’14] 𝒞𝒞 depth 2 TC0 with a linear #wires [Impagliazzo-Paturi- Schneider’13,…]∎ ∘ ∎ New Results! (Note: #gates #wires) ( ) (1) depth TC0 with #wires / [Chen-Santhanam- ≤ 𝑂𝑂 𝑑𝑑 Srinivasan’16] 1+1 2 (2) depth 𝑑𝑑2 TC0 with #gates ≪ 𝑛𝑛 . [Alman-Chan-Williams’16, T] (3) depth 3 TC0 with #gates 1.99, where the top gate is ≪ 𝑛𝑛 unweighted majority [ACW’16] 1 19 (4) Extensions of (2) to ACC0≪THR𝑛𝑛 THR and (3) to MAJ AC0 THR AC0 THR with corresponding restrictions [ACW’16] ∘ ∘ 16 ∘ ∘ ∘ ∘ Research Frontier (-2015)

… AC0[ ] ACC0 ACC0 THR TC0 NC1 … CKT

Lower⊂ Bounds𝑝𝑝 ⊂ for ⊆: ∘ ⊆ ⊆ ⊆ ⊆ majority, mod AC0[ ] [Razborov’87,Smolensky’87] 𝒞𝒞 NEXP ACC0 THR [Williams’14] ∎ 𝑞𝑞 ∉ 𝑝𝑝 parity depth TC0 with #wires / ∎ ⊈ ∘ 𝑑𝑑 inner product depth TC0 with #gates1+1 3 ∎ ∉ 𝑑𝑑 ≪ 𝑛𝑛 [Impagliazzo-Paturi-Saks’93,Gröger-Turán’91] ∉ 𝑑𝑑 ≪ 𝑛𝑛 AC0[ ]: AC0 + mod gates ( : prime power) ACC0: AC0 + mod gates ( : integer 2) ∎ 𝑝𝑝 𝑝𝑝 𝑝𝑝 TC0: constant-depth, unbounded-fan-in, ∎ 𝑚𝑚 𝑚𝑚 ≥ linear threshold (THR) gates: sgn( ) ∎ 𝑛𝑛 1 2: composition of 1 2 ∑𝑖𝑖=1 𝑤𝑤𝑖𝑖𝑥𝑥𝑖𝑖 − 𝜃𝜃 17 ∎ 𝒞𝒞 ∘𝒞𝒞 𝒞𝒞 ∘𝒞𝒞 Research Frontier (2016)

Lower Bounds for : NEXP ACC0 THR [Williams’14] 𝒞𝒞 parity depth TC0 with #wires / ∎ ⊈ ∘ 𝑑𝑑 inner product depth TC0 with #gates1+1 3 ∎ ∉ 𝑑𝑑 ≪ 𝑛𝑛 [Impagliazzo-Paturi-Saks’93,Gröger-Turán’91] ∉ 𝑑𝑑 ≪ 𝑛𝑛 New Results! [Kane-Williams’16] (1) P s.t. depth 2 TC0 with #wires . or #gates . ∃𝐿𝐿 ∈ 𝐿𝐿 ∉ (2) P s.t.2 49 depth 3 TC0 with1 49 ≪ 𝑛𝑛 ≪ 𝑛𝑛 #wires . or #gates . ∃𝐿𝐿 ∈ 𝐿𝐿 ∉ where the top2 49 gate is unweighted1 49 majority ≪ 𝑛𝑛 ≪ 𝑛𝑛 18 Research Frontier (2016)

Lower Bounds for : NEXP ACC0 THR [Williams’14] 𝒞𝒞 parity depth TC0 with #wires / ∎ ⊈ ∘ 𝑑𝑑 inner product depth TC0 with #gates1+1 3 ∎ ∉ 𝑑𝑑 ≪ 𝑛𝑛 [Impagliazzo-Paturi-Saks’93,Gröger-Turán’91] ∉ 𝑑𝑑 ≪ 𝑛𝑛 New Results! (consequences of non-trivial SAT algorithms) (1) E s.t. depth 2 TC0 with #gates . [Alman- Chan-Williams’16,NP T] 1 99 cf. ∃𝐿𝐿 ∈P, #gates𝐿𝐿 ∉ . [Kane-Williams’16] ≪ 𝑛𝑛 (2) Extensions of (1) to1 49 ACC0 THR THR with corresponding restrictions∃𝐿𝐿 ∈ [ACW’16≪] 𝑛𝑛 ∘ ∘ 19 Algorithm Design Paradigms

Paradigm 1: Restrict and Simplify (Divide and Conquer)

Step 1 Construct a partition (or covering) = {0,1} based on the input

𝑛𝑛 𝐷𝐷Step1 ∪ 𝐷𝐷2 2 For∪ ⋯ each , simplify | 𝒞𝒞 and check the satisfiability of it 𝐷𝐷𝑖𝑖 𝒞𝒞 𝐷𝐷𝑖𝑖 (recursively or by another algorithm)

Typically is a subcube, affine subspace, solution space of polynomial equations etc. 𝐷𝐷𝑖𝑖

20 Algorithm Design Paradigms

Paradigm 2: Polynomial Method (Low Rank Decomposition)

Step 1 Take a ``big OR’’ of the input : ( , ) , 𝒞𝒞 ′ 𝑛𝑛′ Step𝐶𝐶 𝑦𝑦 2≔ Represent∨𝑎𝑎∈ 0 1 𝐶𝐶 𝑦𝑦as𝑎𝑎 , = ( ( ) ( )) ′ ′/ ′ 𝑟𝑟 for some , : 0,1 and : 𝑖𝑖=1 𝑖𝑖0,1 𝑖𝑖 𝐶𝐶 𝐶𝐶′ 𝑦𝑦 𝑦𝑦푦푦 ℎ ∑ 𝑓𝑓 𝑦𝑦푦 𝑔𝑔 𝑦𝑦푦푦 𝑛𝑛−𝑛𝑛 2 Step 3 Define𝑓𝑓𝑖𝑖 𝑔𝑔𝑖𝑖 2 / × and→ 𝑅𝑅 × 2ℎ 𝑅𝑅 →/ matrices ′ ′ 𝑛𝑛−𝑛𝑛 2 𝑛𝑛−𝑛𝑛 2 , , ( ) , 𝑟𝑟 𝑟𝑟 Note that ( ′ ) ′′ = ( ) ( ) 𝐴𝐴𝑦𝑦� 𝑖𝑖 ≔ 𝑓𝑓𝑖𝑖 𝑦𝑦 𝐵𝐵𝑖𝑖 𝑦𝑦, ≔ 𝑔𝑔𝑖𝑖 𝑦𝑦푦푦 i.e. the truth table of 𝑟𝑟is obtained via rectangular matrix 𝐴𝐴𝐵𝐵 𝑦𝑦� 𝑦𝑦𝑦 ∑𝑖𝑖=1 𝑓𝑓𝑖𝑖 𝑦𝑦푦 𝑔𝑔𝑖𝑖 𝑦𝑦푦푦 multiplication (with applied′ later) 𝐶𝐶 21 ℎ A Glance at Some Recent -SAT algorithms

𝒞𝒞

22 Some Recent -SAT algorithms

Attempts to handle TC0 (probably ACC0 THR) depth 2 TC0 with a sub𝒞𝒞-quadratic #gates 𝒞𝒞 ⊈ ∘ AC0 with a limited #symmetric gates: ( ) ∎ 𝑛𝑛 ∎ 𝑔𝑔 ∑𝑖𝑖=1 𝑥𝑥𝑖𝑖 Faster algorithms within AC0[ ] ( ACC0 THR) Systems of degree Polynomial Equations over GF( ) 𝑝𝑝 𝒞𝒞 ⊆ ∘ AND XOR AND XOR ∎ 𝑘𝑘 𝑞𝑞 AC0[ ] ∎ ∘ ∘ ∘ ∎ 𝑝𝑝

23 Depth 2 TC0 with a sub-quadratic #gates

24 Motivation

Think about interesting TC0 ACC0 THR i.e. = depth 2 TC0 with #gates = ( ) 𝒞𝒞 ⊂ ∖ ∘ (Note: ACC0 THR is unknown) 𝒞𝒞 𝜔𝜔 𝑛𝑛 Open question𝒞𝒞 ⊈ until∘ 2015 E s.t. depth-2 TC0 with #gates = ( )

NP ∃ 𝐿𝐿(E∈ : languages𝐿𝐿 ∉ computable by 𝑂𝑂 𝑛𝑛 ( ) 2 NP -time Turing machines with NP oracle) 𝑂𝑂 𝑛𝑛 Possible Attack Non-trivial deterministic algorithms for E [Ben-Sasson-Viola’14] NP 𝒞𝒞 ⇒ ⊈𝒞𝒞 25 Recent -SAT algorithms (1)

Theorem [Alman-Chan-Williams’16, T]: Let = depth𝒞𝒞 2 TC0 with #gates = There is a deterministic algorithm for # -SAT 𝒞𝒞 𝑚𝑚 that runs in time poly( , ) 2 ( , ), 𝒞𝒞 , = / / 𝑛𝑛−𝜇𝜇 ,𝑛𝑛 𝑚𝑚 > 0 where 𝑛𝑛 𝑚𝑚 𝑐𝑐 1 2+𝑜𝑜 1 𝜇𝜇 𝑛𝑛 𝑚𝑚 Ω 𝑛𝑛 𝑚𝑚 ∃𝑐𝑐 Corollary: E s.t. holds with = ( . ) NP 1 99 ∃𝐿𝐿 ∈ 𝐿𝐿 ∉ 𝒞𝒞 𝑚𝑚 𝑂𝑂 𝑛𝑛 Note: These results are incomparable with [Chen-Santhanam-Srinivasan’16,Kane-Williams’16] 26 Sketch of the Proof by [T] based on the Polynomial Method in Circuit Complexity follow the framework for ACC0 THR [Williams’14] probabilistic polynomial for THR [Srinivasan’13] ∎ ∘ PRG for space-bounded computation [Nisan’92] ∎ some transformation techniques due to ∎ [Maciel-Therien’98], [Beigel’92] ∎ fast evaluation algorithm for SYM SYM [Williams’14]

∎ ∘ Some details later

27 AC0 with a limited #symmetric gates

28 Motivation

Think about some interesting TC0 ACC0 THR i.e. = ``AC0 with ( ) weighted symmetric gates’’ 𝒞𝒞 ⊂ ∖ ∘ (Note: ACC0 THR is unknown) 𝒞𝒞 𝑡𝑡 𝑛𝑛 Definition𝒞𝒞 ⊈: ∘ : {0,1} 0,1 is symmetric (SYM) if : 𝑛𝑛 0,1 , = ( ) ∎ 𝑓𝑓 → : {0,1} 0,1 is weighted𝑛𝑛 symmetric ∃𝑔𝑔 ℤ → 𝑓𝑓 𝑔𝑔 ∑𝑖𝑖=1 𝑥𝑥𝑖𝑖 if : 𝑛𝑛 0,1 , , = ( )

∎ 𝑓𝑓 → 𝑛𝑛 AND,∃𝑔𝑔 OR,ℤ → parity, ∃mod𝑤𝑤𝑖𝑖 ∈ ℤ, majority𝑓𝑓 𝑔𝑔 ∑ are𝑖𝑖=1 𝑤𝑤symmetric𝑖𝑖𝑥𝑥𝑖𝑖 THR (sgn( )) is weighted symmetric 𝑛𝑛 𝑚𝑚 𝑖𝑖=1 𝑖𝑖 𝑖𝑖 ∑ 𝑤𝑤 𝑥𝑥 − 𝜃𝜃 29 Motivation

= ``AC0 with ( ) weighted symmetric gates’’

𝒞𝒞Interesting? 𝑡𝑡 𝑛𝑛

contains Max SAT when depth 2, = 1 (THR OR) non-trivial algorithm for weighted Max 3-SAT is open ∎ 𝑡𝑡 𝑛𝑛 ∘ (cf. 2 . time algorithm for Max 2-SAT [Williams’04])

0 791𝑛𝑛 Lower bounds: generalized inner product (GIP) AC0 with ∎ #symmetric gates = ( ) or #THRs = / ( ) ∉ […,Lovett-Srinivasan'11] 1−𝑜𝑜 1 1 2−𝑜𝑜 1 𝑛𝑛 𝑛𝑛

30 Recent -SAT algorithms (2)

Theorem [Sakai-Seto-T-Teruyama]: Let = ``AC0𝒞𝒞 with ( ) weighted symmetric gates’’ ( ) where = ( ), | | = 2 𝒞𝒞 𝑡𝑡 𝑛𝑛 𝑜𝑜 1 There is a non-trivial𝑜𝑜 1 deterministic𝑛𝑛 algorithm for # -SAT 𝑡𝑡 𝑛𝑛 𝑛𝑛 𝑤𝑤𝑖𝑖 (Note: we assume evaluation of symmetric gate is easy) 𝒞𝒞 Special Case: / ( ) Max SAT can be solved in deterministic time 2 1 𝑂𝑂 𝑘𝑘 when #clauses = ( ) 𝑛𝑛−𝑛𝑛 (Note: Max -SAT ⇒ #𝑘𝑘clauses = ( )) 𝑂𝑂 𝑛𝑛 𝑘𝑘 𝑘𝑘 𝑂𝑂 𝑛𝑛 31 By-products

Average-case lower bounds P s.t. Pr = + 1 1 if = ``AC0 with ( ) weighted symmetric𝜔𝜔 1 gates’’ ∃𝐿𝐿 ∈ 𝐿𝐿𝑛𝑛 𝑥𝑥 𝐶𝐶 𝑥𝑥 ≤ 2 𝑛𝑛 ( ) where = ( ), | | = 2 𝐶𝐶 ∈ 𝒞𝒞 𝑡𝑡 𝑛𝑛 𝑜𝑜 1 𝑜𝑜 1 𝑛𝑛 𝑡𝑡 𝑛𝑛 𝑛𝑛 𝑤𝑤𝑖𝑖 Worst-case lower bounds P s.t. MAJ , where is as above

∃𝐿𝐿 ∈ 𝐿𝐿 ∉ ∘𝒞𝒞 𝒞𝒞 Remark (for specialists) These seem to be the first lower bounds for such bypassing communication complexity arguments 𝒞𝒞 32 Core Technical Result

Lemma: Let =(weighted SYM) AND where #ANDs = , | | 𝒞𝒞 ∘ There is a deterministic algorithm for # -SAT 𝑚𝑚 𝑤𝑤𝑖𝑖 ≤ 𝑤𝑤 that runs in time poly( , , log )2 ( , , ) 𝒞𝒞 where , , = ( /log ( )) ( 𝑛𝑛−𝜇𝜇 /𝑛𝑛 𝑚𝑚 𝑤𝑤 ) 𝑛𝑛 𝑚𝑚 𝑤𝑤 Ω log 𝑛𝑛 log 𝑚𝑚 Based on𝜇𝜇 ``Restrict𝑛𝑛 𝑚𝑚 𝑤𝑤 and𝑛𝑛 Simplify’’𝑚𝑚𝑚𝑚 & DP

(Note: Theorem follows from Lemma and transformation AC0 with symmetric gates ⇒ SYM AND using [Beigel-Reingold-Spielman'91,Beigel'92,Beame-Impagliazzo- Srinivasan'12]) ∘ 33 Faster algorithms within ACC0

34 Motivation

( -)CNF AC0 AC0[ ] ACC0

𝑘𝑘: -SAT ⊂in time⊂ T, condition𝑝𝑝 ⊂ (1) -CNF: 2 ( ), = 1/ ( ) 𝒞𝒞 𝒞𝒞 [Paturi-Pudlak𝑛𝑛 1-−Zane’97,…]𝜇𝜇 𝑘𝑘 𝑘𝑘 𝜇𝜇 𝑘𝑘 𝑂𝑂 𝑘𝑘 (2) CNF: 2 ( ), = 1/ (log ), #clauses = [Schuler’05,Calabro𝑛𝑛 1−𝜇𝜇 𝑐𝑐 -Impagliazzo-Paturi’06,…] 𝜇𝜇 𝑐𝑐 𝑂𝑂 𝑐𝑐 𝑐𝑐𝑐𝑐 (3) AC0: 2 ( , ), , = 1/ (log + log ) depth 𝑛𝑛, 1#−gates𝜇𝜇 𝑐𝑐 𝑑𝑑 = [Impagliazzo-Matthews-Paturi’12]𝑑𝑑−1 𝜇𝜇 𝑐𝑐 𝑑𝑑 𝑂𝑂 ( ) 𝑐𝑐 𝑑𝑑 𝑑𝑑 (4) ACC0: 2 , , , = / 𝑑𝑑 𝑐𝑐𝑐𝑐 𝑂𝑂 𝑑𝑑 𝑛𝑛−𝜇𝜇 𝑛𝑛 𝑑𝑑 ( ) 1 2 depth , #gates = 2 [Williams’11] 𝜇𝜇 𝑛𝑛 𝑜𝑜𝑑𝑑1 𝑛𝑛 𝑛𝑛 𝑑𝑑 35 Recent -SAT algorithms (3)

Theorem [Lokshtanov-Paturi-T-Williams-Yu]: : -SAT in time𝒞𝒞 T, condition (1) Systems of degree- Polynomial Equations over GF( ) 𝒞𝒞 𝒞𝒞 (=AND XOR AND -CNF = AND OR if = 2): k 𝑘𝑘 k 𝑞𝑞 ( ), = 1/ ( ) ∘ ∘ ⊃ 𝑘𝑘 ∘ 𝑞𝑞 𝑛𝑛 1−𝜇𝜇 𝑘𝑘 (2)𝑞𝑞 AND XOR𝜇𝜇 AND𝑘𝑘 XOR𝑂𝑂 (𝑘𝑘 CNF = AND OR): 2 ( ), = 1/ (log ), #ANDs = at depth 3 ∘ ∘ ∘ ⊃ ∘ 𝑛𝑛 1−𝜇𝜇 𝑐𝑐 0 0 (3) AC [ ] ( 𝜇𝜇 AC𝑐𝑐 ): 𝑂𝑂 𝑐𝑐 𝑐𝑐𝑐𝑐 2 ( , ), , = 1/ (log ) 𝑝𝑝 ⊃ depth𝑛𝑛 1−𝜇𝜇 𝑑𝑑, 𝑚𝑚#gates = 𝑑𝑑−1 𝜇𝜇 𝑑𝑑 𝑚𝑚 𝑂𝑂 𝑚𝑚 𝑑𝑑 𝑚𝑚 36 Proof Sketch based on the Polynomial Method in Circuit Complexity probabilistic polynomial for AND/OR [Razborov’87,Smolensky’87] ∎ AC0[ ] [Kopparty-Srinivasan’12] fast evaluation algorithm for polynomial 𝑝𝑝 degree reduction for AND XOR AND XOR ∎ extending Schuler’s width reduction for CNF ∎ ∘ ∘ ∘ (``Restrict and Simplify’’, where each partition is a solution space of low-degree polynomial equations)

37 Algorithms via Polynomial Method

38 Polynomial Method

Example [Razborov’87,Smolensky’87]: 1. AC0[ ] can be well approximated by a low-degree GF( ) polynomial 𝑝𝑝 2. majority, mod cannot be well approximated 𝑝𝑝 by a low-degree GF( ) polynomial 𝑞𝑞 0 1+2 ⇒ majority, mod 𝑝𝑝 AC [ ]

𝑞𝑞 ∉ 𝑝𝑝 item 1 is useful in algorithm design (``sparse’’ suffices instead of ``low-degree’’ in many cases)

39 Polynomial Method

Definition: Let : {0,1} {0,1}

𝑛𝑛 A distribution𝑓𝑓 → over polynomials is an -error probabilistic polynomial for 𝑃𝑃 if , Pr ( ) 𝜖𝜖 𝑓𝑓 deg∀(𝑥𝑥 ) 𝑝𝑝∼𝑃𝑃 if𝑝𝑝 Pr𝑥𝑥 ≠ 𝑓𝑓deg𝑥𝑥 ≤ 𝜖𝜖 = 1

𝑃𝑃 ≤ 𝑑𝑑 𝑝𝑝∼𝑃𝑃 𝑝𝑝 ≤ 𝑑𝑑 −error probabilistic -circuit is defined analogously

𝜖𝜖 𝒞𝒞

40 Algorithm for -SAT

Input: : {0,1} 0,1 ,

𝑛𝑛 𝒞𝒞 Step 1.𝐶𝐶 Define →: 0,1 𝐶𝐶 ∈𝒞𝒞 0,1 , OR as ′ (𝑛𝑛−, 𝑛𝑛) (Note: | | 2 | |) 𝐶𝐶,′ → 𝐶𝐶′ ∈ ∘𝒞𝒞 ′ 𝑛𝑛′ ′ 𝑛𝑛� Step𝐶𝐶 𝑦𝑦2. Construct≔ ∨𝑎𝑎∈ 0 1 (1/3𝐶𝐶 )𝑦𝑦-error𝑎𝑎 probabilistic𝐶𝐶 ≈ polynomial𝐶𝐶 for in time ( , , | |) 𝑝𝑝 Step 3. ′Evaluate ( ) for′ all 0,1 𝐶𝐶 𝑇𝑇 𝑛𝑛 𝑛𝑛 𝐶𝐶 ′ in time ( , , | |) 𝑛𝑛−𝑛𝑛 𝑝𝑝 𝑦𝑦 𝑦𝑦 ∈ Step 4. repeat 2-′3 ( ) times to reduce error probability 𝑇𝑇′ 𝑛𝑛 𝑛𝑛 𝐶𝐶 Output: truth table𝑂𝑂 𝑛𝑛 of such that , Pr ( ) 2′ 𝑉𝑉 𝐶𝐶 Running Time: ( ( ( , ,−|2𝑛𝑛|)+ ( , , | |))) ∀𝑦𝑦 𝑉𝑉 𝑦𝑦 ≠ 𝐶𝐶� 𝑦𝑦 ≤ 41 ′ ′ 𝑂𝑂 𝑛𝑛 𝑇𝑇 𝑛𝑛 𝑛𝑛 𝐶𝐶 𝑇𝑇′ 𝑛𝑛 𝑛𝑛 𝐶𝐶

Ingredients for THR THR (1/3)

Lemma [Razborov’87,Smolensky’87]: There exists -error probabilistic polynomial∘ for AND/OR of degree log(1/ ) and it is efficiently samplable 𝜖𝜖 Lemma [Srinivasan’13]𝜖𝜖 : There exists -error probabilistic polynomial for THR of degree polylog( / ) and it is efficiently samplable 𝜖𝜖 𝑛𝑛 𝑛𝑛 𝜖𝜖

42 Ingredients for THR THR (2/3)

Lemma: XOR AND XOR AND XOR AND∘

Lemma∘ [Maciel∘ -Therien’98]∘ ⊂: ∘ THR AC0[2] SYM

Lemma⊂ [Beigel’92]∘ : AND SYM weighted SYM

∘ ⊂

43 Ingredients for THR THR (3/3)

Lemma [Williams’14]: Let : {0,1} 0,1 , SYM (weighted∘ SYM) where top gate has𝑛𝑛 fan-in 𝐶𝐶 → 𝐶𝐶 ∈ ∘ bottom gates have maximum weight 𝑢𝑢 such that 2 . 𝑤𝑤 Then, truth table 0of1𝑛𝑛 can be generated in time poly 2 𝑢𝑢𝑢𝑢 ≤ 𝑛𝑛 𝐶𝐶 𝑛𝑛

44 Algorithm for THR THR-SAT

Input: : {0,1} 0,1 , THR THR

𝑛𝑛 ∘ Step 1.𝐶𝐶 Define →: {0,1} 𝐶𝐶 ∈ 0,1 ∘, OR as (𝑛𝑛′, ) (Note: | | 2 | |) 𝐶𝐶,′ → 𝐶𝐶 ∈ ∘𝒞𝒞 ′ 𝑛𝑛′ ′ 𝑛𝑛� Step𝐶𝐶 𝑦𝑦2. Construct≔ ∨𝑎𝑎∈ 0 1 (1/3𝐶𝐶 )𝑦𝑦-error𝑎𝑎 probabilistic𝐶𝐶 ≈ circuit𝐶𝐶 for (XOR AND) (XOR AND) (XOR AND) ′′SYM ′ 𝐶𝐶 𝐶𝐶 Step′′ 3. transform into XOR (weighted SYM) 𝐶𝐶 ∈ ∘ ∘ ∘ ∘ ⋯ ∘ ∘ ∘ Step 4. Evaluate ′′( ) for all′′′ 0,1 𝐶𝐶 𝐶𝐶 ∈ ∘ ′ Step 5. Repeat 2-4′′ ′ ( ) times to reduce𝑛𝑛− error𝑛𝑛 probability 𝐶𝐶 𝑦𝑦 𝑦𝑦 ∈ Output: truth table 𝑂𝑂of 𝑛𝑛

′ (Note: need more work𝐶𝐶 for deterministic algorithms) 45 Recent -SAT algorithms (1)

46 Bottleneck for Improvements

Let : {0,1} {0,1} A distribution𝑛𝑛 over polynomials is 𝑓𝑓 → an -error probabilistic polynomial for 𝑃𝑃 if , Pr ( ) 𝜖𝜖 𝑓𝑓

𝑝𝑝∼𝑃𝑃 Lemma∀𝑥𝑥 [Srinivasan’13]𝑝𝑝 𝑥𝑥 ≠: 𝑓𝑓 𝑥𝑥 ≤ 𝜖𝜖 There exists -error probabilistic polynomial for THR of degree polylog( / ) and it is efficiently samplable 𝜖𝜖

𝑛𝑛 𝑛𝑛 𝜖𝜖 We cannot improve the lemma under the above definition of -error probabilistic polynomial

𝜖𝜖 47 Further Improvement?

All we want to do in the polynomial method is: Step 1 Take a ``big OR’’ of the input : ( , ) , 𝒞𝒞 ′ 𝑛𝑛′ Step𝐶𝐶 𝑦𝑦 2≔ Represent∨𝑎𝑎∈ 0 1 𝐶𝐶 𝑦𝑦as𝑎𝑎 , ( ( ) ( )) ′ ′/ ′ 𝑟𝑟 for some , : 0,1 and : 𝑖𝑖=1 𝑖𝑖0,1 𝑖𝑖 𝐶𝐶 𝐶𝐶′ 𝑦𝑦 𝑦𝑦푦푦 ≔ ℎ ∑ 𝑓𝑓 𝑦𝑦푦 𝑔𝑔 𝑦𝑦푦푦 𝑛𝑛−𝑛𝑛 2 Typically, 𝑓𝑓𝑖𝑖 𝑔𝑔𝑖𝑖 ( ) ( ) is→ a 𝑅𝑅sum ofℎ monomials𝑅𝑅 → and 𝑟𝑟 is a threshold𝑖𝑖=1 𝑖𝑖 or modulo𝑖𝑖 function ∑ 𝑓𝑓 𝑦𝑦푦 𝑔𝑔 𝑦𝑦푦푦 ℎTry different representations, e.g. 1,2 instead of 0,1 , =complex numbers, ( ) ( ) as a trigonometric polynomial,… etc. 𝑟𝑟 𝑅𝑅 ∑𝑖𝑖=1 𝑓𝑓𝑖𝑖 𝑦𝑦푦 𝑔𝑔𝑖𝑖 𝑦𝑦푦푦 48 Conclusion

49 Some Recent -SAT algorithms

50 Open Questions

without polynomial method? (in polynomial space?) other algebraic representations useful? ∎ e.g. Barrington’s theorem, randomized encoding, span program,…∎ other lower bound techniques useful? e.g. communication complexity, proof complexity, ∎mathematical programming,…

lower bound for non-trivial -SAT algorithm? (cf. [Carmosino-Impagliazzo-Kabanets-Kolokolova,CCC'16]) ∎ 𝒞𝒞 ⟺ 𝒞𝒞

Thank you! 51