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 n ∃x ∈{0,1} ,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 n ∃x ∈{0,1} ,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, : size of a circuit ) ∎ -SAT input only from a circuit class e.g. = ( -)CNF, AC0, AC0[ ], ACC0, TC0, NC1 (Formula),… ∎ 6 Example CNF-SAT =( -)CNF formulas (conjunctions of clauses) e.g. = ( ) (¬ ¬ ) (¬ ) CNF is -CNF1 if2 each clause2 has3 at4 most 3literals ∨ ∧ ∨ ∨ ∧ ∧ ⋯ Best known algorithms 3-CNF: 1.308 [Hertli’14, randomized] 1.331 [Makino-T-Yamamoto’13, deterministic] . ( ) -CNF: 2 [Paturi-Pudlak-Saks-Zane’05, randomized] 1 64 . (1− ) 2 [Moser-Scheder’11, deterministic] 1 44 1− 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 DTIME[2( ) ] ∃ ∉ 1− ETH & ∀SETH are∃ widely 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 Stronger circuit class: ( -)CNF AC0 AC0[ ] ACC0 TC0 NC1 … CKT ⊂ ⊂ ⊂ ⊆ ⊆ ⊆ ⊆ 2. If non-trivial algorithms exist for -SAT, then improve the running time 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) ∎ ⊈ [white box] analysis of -SAT algorithm ⇒ average-case lower bounds ∎ [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 ( -)CNF⊂: conjunction⊂ ⊂of disjunctions⊆ ⊆ of (at⊆ most⊆ ) literals AC0: constant-depth, unbounded-fan-in, AND/OR/NOT ∎ AC0[ ]: AC0 + mod gates ( : prime power) ∎ ACC0: AC0 + mod gates ( : integer 2) ∎ TC0: constant-depth, unbounded-fan-in, ∎ ≥ linear threshold (THR) gates: sgn( ) ∎ NC1: fan-in 2, fan-out 1, AND/OR/NOT(/XOR) ∑=1 − 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 ∎ ∘ ∘ specified) 13 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) Important case (NP CKT?): ・ =CKT: circuits with poly(n) number of gates ⊆ ・X=NP-complete problem, e.g., SAT, clique,… 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 ⊂ ⊂ ⊆ ∘ ⊆ ⊆ ⊆ ⊆ Non-trivial -SAT algorithm: ACC0 THR with a super-poly #gates [Williams’14] 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, ∎ ≥ linear threshold (THR) gates: sgn( ) ∎ 1 2: composition of 1 2 ∑=1 − ∎ ∘ ∘ 15 Research Frontier (2016) Non-trivial -SAT algorithm: ACC0 THR with a super-poly #gates [Williams’14] depth 2 TC0 with a linear #wires [Impagliazzo-Paturi- Schneider’13,…]∎ ∘ ∎ New Results! (Note: #gates #wires) ( ) (1) depth TC0 with #wires / [Chen-Santhanam- ≤ Srinivasan’16] 1+1 2 (2) depth 2 TC0 with #gates ≪ . [Alman-Chan-Williams’16, T] (3) depth 3 TC0 with #gates 1.99, where the top gate is ≪ unweighted majority [ACW’16] 1 19 (4) Extensions of (2) to ACC0≪THR 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⊂ Bounds ⊂ for ⊆: ∘ ⊆ ⊆ ⊆ ⊆ majority, mod AC0[ ] [Razborov’87,Smolensky’87] NEXP ACC0 THR [Williams’14] ∎ ∉ 0 / parity depth TC with #wires ∎ ⊈ ∘ inner product depth TC0 with #gates1+1 3 ∎ ∉ ≪ [Impagliazzo-Paturi-Saks’93,Gröger-Turán’91] ∉ ≪ AC0[ ]: AC0 + mod gates ( : prime power) ACC0: AC0 + mod gates ( : integer 2) ∎ TC0: constant-depth, unbounded-fan-in, ∎ ≥ linear threshold (THR) gates: sgn( ) ∎ 1 2: composition of 1 2 ∑=1 − 17 ∎ ∘ ∘ Research Frontier (2016) Lower Bounds for : NEXP ACC0 THR [Williams’14] 0 / parity depth TC with #wires ∎ ⊈ ∘ inner product depth TC0 with #gates1+1 3 ∎ ∉ ≪ [Impagliazzo-Paturi-Saks’93,Gröger-Turán’91] ∉ ≪ New Results! [Kane-Williams’16] (1) P s.t. depth 2 TC0 with #wires . or #gates . ∃ ∈ ∉ (2) P s.t.2 49 depth 3 TC0 with1 49 ≪ ≪ #wires . or #gates . ∃ ∈ ∉ where the top2 49 gate is unweighted1 49 majority ≪ ≪ 18 Research Frontier (2016) Lower Bounds for : NEXP ACC0 THR [Williams’14] 0 / parity depth TC with #wires ∎ ⊈ ∘ inner product depth TC0 with #gates1+1 3 ∎ ∉ ≪ [Impagliazzo-Paturi-Saks’93,Gröger-Turán’91] ∉ ≪ 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. ∃ ∈P, #gates ∉ . [Kane-Williams’16] ≪ (2) Extensions of (1) to1 49 ACC0 THR THR with corresponding restrictions∃ ∈ [ACW’16≪] ∘ ∘ 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 Step1 ∪ 2 2 For∪ ⋯ each , simplify | and check the satisfiability of it (recursively or by another algorithm) Typically is a subcube, affine subspace, solution space of polynomial equations etc. 20 Algorithm Design Paradigms Paradigm 2: Polynomial Method (Low Rank Decomposition) Step 1 Take a ``big OR’’ of the input : ( , ) , ′ ′ Step 2≔ Represent∨∈ 0 1 as , = ( ( ) ( )) ′ ′/ ′ for some , : 0,1 and : =1 0,1 ′ 푦푦 ℎ ∑ 푦 푦푦 − 2 Step 3 Define 2 / × and→ × 2ℎ →/ matrices ′ ′ − 2 − 2 , , ( ) , Note that ( ′ ) ′′ = ( ) ( ) ′ ≔ , ≔ 푦푦 i.e. the truth table of is obtained via rectangular matrix ′ ∑=1 푦 푦푦 multiplication (with applied′ later) 21 ℎ A Glance at Some Recent -SAT algorithms 22 Some Recent -SAT algorithms Attempts to handle TC0 (probably ACC0 THR) depth 2 TC0 with a sub-quadratic #gates ⊈ ∘ AC0 with a limited #symmetric gates: ( ) ∎ ∎ ∑=1 Faster algorithms within AC0[ ] ( ACC0 THR) Systems of degree Polynomial Equations over GF( ) ⊆ ∘ AND XOR AND XOR ∎ AC0[ ] ∎ ∘ ∘ ∘ ∎ 23 Depth 2 TC0 with a sub-quadratic #gates 24 Motivation Think about interesting TC0 ACC0 THR i.e. = depth 2 TC0 with #gates = ( ) ⊂ ∖ ∘ (Note: ACC0 THR is unknown) Open question ⊈ until∘ 2015 0 E s.t. depth-2 TC with #gates = ( ) NP ∃ (E∈ : languages ∉ computable by ( ) 2 NP -time Turing machines with NP oracle) Possible Attack Non-trivial deterministic algorithms for E [Ben-Sasson-Viola’14] NP ⇒ ⊈ 25 Recent -SAT algorithms (1) Theorem [Alman-Chan-Williams’16, T]: Let = depth 2 TC0 with #gates = There is a deterministic algorithm for # -SAT that runs in time poly( , ) 2 ( , ), , = / / − , > 0 where 1 2+ 1 Ω ∃ Corollary: E s.t. holds with = ( . ) NP 1 99 ∃ ∈ ∉ 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