Recent Developments on Circuit Satisfiability Algorithms

Transcription

1 Recent Developments on Circuit Satisfiability Algorithms Suguru TAMAKI Kyoto University Fine-Grained Complexity and Algorithm Design Reunion, December 14, 2016, Simons Institute, Berkeley, CA

2 After the Fine-Grained Complexity and Algorithm Our Problem Design Program (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

3 Golovnev-Kulikov-Smal-T, ``Circuit Size Lower Bounds Our Problem and #SAT 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

4 Our Problem Circuit Satisfiability (SAT) Input Boolean Circuit C :{0,1} n {0,1} Output Yes/No x {0,1} Yes x {0,1} No n n, C( x) = 1, C( x) = 0 x + 1? x2 x3 x1? 1??? 4

5 Our Problem Circuit Satisfiability (SAT) Input Boolean Circuit C :{0,1} n {0,1} Output Yes/No x {0,1} Yes x {0,1} No n n, C( x) = 1, C( x) = 0 x x2 x3 x

6 Our Problem Circuit Satisfiability (SAT) Canonical NP-complete problem [Cook-Levin] Solved in time 2 nn poly CC (nn: #variables, CC : size of a circuit CC) CC-SAT input only from a circuit class CC e.g. CC = (kk-)cnf, AC 0, AC 0 [pp], ACC 0, TC 0, NC 1 (Formula), 6

7 Example CNF-SAT CC =(kk-)cnf formulas (conjunctions of clauses) e.g. CC = (xx 1 xx 2 ) ( xx 2 xx 3 xx 4 ) ( xx 3 ) CNF is kk -CNF if each clause has at most kk literals Best known algorithms 3-CNF: nn [Hertli 14, randomized] nn [Makino-T-Yamamoto 13, deterministic] kk-cnf: 2 ( kk )nn [Paturi-Pudlak-Saks-Zane 05, randomized] 2 ( kk )nn [Moser-Scheder 11, deterministic] 7

8 Hardness of Circuit SAT? (Strong) Exponential Time Hypothesis (``quantitatively strengthening of P NP) [Impagliazzo-Paturi-Zane] ETH δδ > 0, 3-CNF-SAT DTIME[2 δδnn ] SETH εε > 0, kk, kk-cnf-sat DTIME[2 (1 εε)nn ] ETH & SETH are widely used to prove the optimality of exact, parametrized, approximation and dynamic algorithms (cf. ``fine-grained complexity theory) 8

9 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: (kk-)cnf AC 0 AC 0 [pp] ACC 0 TC 0 NC 1 CKT 2. If non-trivial algorithms exist for CC-SAT, then improve the running time prove the difficulty of improvement additional properties: deterministic, counting, P-space, 9

10 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

11 Why Algorithms for Circuit SAT? 2. Connections to circuit lower bounds (This Talk) [black box] non-trivial CC-SAT algorithm NEXP CC [Williams 10,11, ] (NEXP: nondeterministic exponential time) [white box] analysis of CC-SAT algorithm average-case lower bounds [Santhanam 10,Seto-T 12,Chen-Kabanets-Kolokolova-Shaltiel- Zuckerman 14, ] 11

12 Boolean Circuit Complexity studies the power of circuit classes defined using several parameters: O( n), O( n 2 ),...,poly( n),..., n polylogn,...,2 #gates: 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, n ε,... 12

13 Circuit Classes (kk-)cnf AC 0 AC 0 [pp] ACC 0 TC 0 NC 1 CKT (kk-)cnf: conjunction of disjunctions of (at most kk) literals AC 0 : constant-depth, unbounded-fan-in, AND/OR/NOT AC 0 [pp]: AC 0 + mod pp gates (pp: prime power) ACC 0 : AC 0 + mod mm gates (mm: integer 2) TC 0 : constant-depth, unbounded-fan-in, nn linear threshold (THR) gates: sgn( ii=1 ww ii xx ii θθ) NC 1 : fan-in 2, fan-out 1, AND/OR/NOT(/XOR) CKT: fan-in 2, AND/OR/NOT(/XOR) CC 1 CC 2 : composition of CC 1 and CC 2 e.g. CNF=AND OR (Note: assume #gates = poly(nn) unless otherwise specified) 13

14 Circuit Lower Bounds Typical question: Can a family of Boolean circuits {C n } from a circuit class CC compute a problem X? (C n is an n-variate circuit for each n) Important case (NP CKT?): CC=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

15 Research Frontier (-2015) AC 0 [pp] ACC 0 ACC 0 THR TC 0 NC 1 CKT Non-trivial CC-SAT algorithm: ACC 0 THR with a super-poly #gates [Williams 14] depth 2 TC 0 with a linear #wires [Impagliazzo-Paturi- Schneider 13, ] AC 0 : constant-depth, unbounded-fan-in, AND/OR/NOT ACC 0 : AC 0 + mod mm gates (mm: integer 2) TC 0 : constant-depth, unbounded-fan-in, nn linear threshold (THR) gates: sgn( ww ii xx ii θθ) CC 1 CC 2 : composition of CC 1 CC 2 ii=1 15

16 Research Frontier (2016) Non-trivial CC-SAT algorithm: ACC 0 THR with a super-poly #gates [Williams 14] depth 2 TC 0 with a linear #wires [Impagliazzo-Paturi- Schneider 13, ] New Results! (Note: #gates #wires) (1) depth dd TC 0 with #wires nn 1+1/2OO(dd) [Chen-Santhanam- Srinivasan 16] (2) depth 2 TC 0 with #gates nn 1.99 [Alman-Chan-Williams 16, T] (3) depth 3 TC 0 with #gates nn 1.19, where the top gate is unweighted majority [ACW 16] (4) Extensions of (2) to ACC 0 THR THR and (3) to MAJ AC 0 THR AC 0 THR with corresponding restrictions [ACW 16] 16

17 Research Frontier (-2015) AC 0 [pp] ACC 0 ACC 0 THR TC 0 NC 1 CKT Lower Bounds for CC: majority, mod qq AC 0 [pp] [Razborov 87,Smolensky 87] NEXP ACC 0 THR [Williams 14] parity depth dd TC 0 with #wires nn 1+1/3dd inner product depth dd TC 0 with #gates nn [Impagliazzo-Paturi-Saks 93,Gröger-Turán 91] AC 0 [pp]: AC 0 + mod pp gates (pp: prime power) ACC 0 : AC 0 + mod mm gates (mm: integer 2) TC 0 : constant-depth, unbounded-fan-in, nn linear threshold (THR) gates: sgn( ww ii xx ii θθ) CC 1 CC 2 : composition of CC 1 CC 2 17 ii=1

18 Research Frontier (2016) Lower Bounds for CC: NEXP ACC 0 THR [Williams 14] parity depth dd TC 0 with #wires nn 1+1/3dd inner product depth dd TC 0 with #gates nn [Impagliazzo-Paturi-Saks 93,Gröger-Turán 91] New Results! [Kane-Williams 16] (1) LL P s.t. LL depth 2 TC 0 with #wires nn 2.49 or #gates nn 1.49 (2) LL P s.t. LL depth 3 TC 0 with #wires nn 2.49 or #gates nn 1.49 where the top gate is unweighted majority 18

19 Research Frontier (2016) Lower Bounds for CC: NEXP ACC 0 THR [Williams 14] parity depth dd TC 0 with #wires nn 1+1/3dd inner product depth dd TC 0 with #gates nn [Impagliazzo-Paturi-Saks 93,Gröger-Turán 91] New Results! (consequences of non-trivial SAT algorithms) (1) LL E NP s.t. LL depth 2 TC 0 with #gates nn 1.99 [Alman- Chan-Williams 16, T] cf. LL P, #gates nn 1.49 [Kane-Williams 16] (2) Extensions of (1) to ACC 0 THR THR with corresponding restrictions [ACW 16] 19

20 Algorithm Design Paradigms Paradigm 1: Restrict and Simplify (Divide and Conquer) Step 1 Construct a partition (or covering) DD 1 DD 2 = {0,1} nn based on the input CC Step 2 For each DD ii, simplify CC DDii and check the satisfiability of it (recursively or by another algorithm) Typically DD ii is a subcube, affine subspace, solution space of polynomial equations etc. 20

21 Algorithm Design Paradigms Paradigm 2: Polynomial Method (Low Rank Decomposition) Step 1 Take a ``big OR of the input CC: CC yy aa 0,1 nn CC(yy, aa) Step 2 Represent CC as CC yy, yyyy = h( ii=1 ff ii (yyy)gg ii (yyyy)) for some ff ii, gg ii : 0,1 nn nn /2 RR and h: RR 0,1 Step 3 Define 2 nn nn /2 rr and rr 2 nn nn /2 matrices AA yyy,ii ff ii yy, BB ii,yy gg ii (yyyy) rr Note that (AABB) yyy,yyyy = ii=1 ff ii (yyy)gg ii (yyyy) i.e. the truth table of CC is obtained via rectangular matrix multiplication (with h applied later) 21 rr

22 A Glance at Some Recent CC-SAT algorithms 22

23 Some Recent CC-SAT algorithms Attempts to handle TC 0 (probably CC ACC 0 THR) depth 2 TC 0 with a sub-quadratic #gates AC 0 nn with a limited #symmetric gates: gg( ii=1 xx ii ) Faster algorithms within AC 0 [pp] (CC ACC 0 THR) Systems of degree kk Polynomial Equations over GF(qq) AND XOR AND XOR AC 0 [pp] 23

24 Depth 2 TC 0 with a sub-quadratic #gates 24

25 Motivation Think about interesting CC TC 0 ACC 0 THR i.e. CC= depth 2 TC 0 with #gates = ωω(nn) (Note: CC ACC 0 THR is unknown) Open question until 2015 LL E NP s.t. LL depth-2 TC 0 with #gates = OO(nn) (E NP : languages computable by 2 OO(nn) -time Turing machines with NP oracle) Possible Attack Non-trivial deterministic algorithms for CC E NP CC [Ben-Sasson-Viola 14] 25

26 Recent CC-SAT algorithms (1) Theorem [Alman-Chan-Williams 16, T]: Let CC= depth 2 TC 0 with #gates = mm There is a deterministic algorithm for #CC-SAT that runs in time poly(nn, mm) 2 nn μμ(nn,mm), where μμ nn, mm = Ω nn/mm 1/2+oo 1 cc, cc > 0 Corollary: LL E NP s.t. LL CC holds with mm = OO(nn 1.99 ) Note: These results are incomparable with [Chen-Santhanam-Srinivasan 16,Kane-Williams 16] 26

27 Sketch of the Proof by [T] based on the Polynomial Method in Circuit Complexity follow the framework for ACC 0 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

28 AC 0 with a limited #symmetric gates 28

29 Motivation Think about some interesting CC TC 0 ACC 0 THR i.e. CC= ``AC 0 with tt(nn) weighted symmetric gates (Note: CC ACC 0 THR is unknown) Definition: ff: {0,1} nn 0,1 is symmetric (SYM) nn if gg: Z 0,1, ff = gg( ii=1 xx ii ) ff: {0,1} nn 0,1 is weighted symmetric nn if gg: Z 0,1, ww ii Z, ff = gg( ww ii xx ii ii=1 ) AND, OR, parity, mod mm, majority are symmetric nn THR (sgn( ww ii xx ii θθ)) is weighted symmetric ii=1 29

30 Motivation CC= ``AC 0 with tt(nn) weighted symmetric gates Interesting? contains Max SAT when depth 2, tt nn = 1 (THR OR) non-trivial algorithm for weighted Max 3-SAT is open (cf nn time algorithm for Max 2-SAT [Williams 04]) Lower bounds: generalized inner product (GIP) AC 0 with #symmetric gates = nn 1 oo(1) or #THRs = nn 1/2 oo(1) [,Lovett-Srinivasan'11] 30

31 Recent CC-SAT algorithms (2) Theorem [Sakai-Seto-T-Teruyama]: Let CC= ``AC 0 with tt(nn) weighted symmetric gates where tt nn = nn oo(1), ww ii = 2 nnoo(1) There is a non-trivial deterministic algorithm for #CC-SAT (Note: we assume evaluation of symmetric gate is easy) Special Case: Max SAT can be solved in deterministic time 2 nn nn1/oo(kk) when #clauses = OO(nn kk ) (Note: Max kk-sat #clauses = OO(nn kk )) 31

32 By-products Average-case lower bounds LL P s.t. Pr LL nn xx = CC xx nn ωω 1 if CC CC= ``AC 0 with tt(nn) weighted symmetric gates where tt nn = nn oo(1), ww ii = 2 nnoo(1) Worst-case lower bounds LL P s.t. LL MAJ CC, where CC is as above Remark (for specialists) These seem to be the first lower bounds for such CC bypassing communication complexity arguments 32

33 Core Technical Result Lemma: Let CC=(weighted SYM) AND where #ANDs = mm, ww ii ww There is a deterministic algorithm for #CC-SAT that runs in time poly(nn, mm, log ww)2 nn μμ(nn,mm,ww) where μμ nn, mm, ww = (nn/log(mmmm)) Ω(log nn/ log mm) Based on ``Restrict and Simplify & DP (Note: Theorem follows from Lemma and transformation AC 0 with symmetric gates SYM AND using [Beigel-Reingold-Spielman'91,Beigel'92,Beame-Impagliazzo- Srinivasan'12]) 33

34 Faster algorithms within ACC 0 34

35 Motivation (kk-)cnf AC 0 AC 0 [pp] ACC 0 CC: CC-SAT in time T, condition (1) kk-cnf: 2 nn(1 μμ kk ), μμ kk = 1/OO(kk) [Paturi-Pudlak-Zane 97, ] (2) CNF: 2 nn(1 μμ cc ), μμ cc = 1/OO(log cc), #clauses = cccc [Schuler 05,Calabro-Impagliazzo-Paturi 06, ] (3) AC 0 : 2 nn(1 μμ cc,dd ), μμ cc, dd = 1/OO(log cc + dd log dd) dd 1 depth dd, #gates = cccc [Impagliazzo-Matthews-Paturi 12] (4) ACC 0 : 2 nn μμ nn,dd, μμ nn, dd = nn 1/2OO(dd) depth dd, #gates = 2 nnoo(1) [Williams 11] 35

36 Recent CC-SAT algorithms (3) Theorem [Lokshtanov-Paturi-T-Williams-Yu]: CC: CC-SAT in time T, condition (1) Systems of degree-kk Polynomial Equations over GF(qq) (=AND XOR AND k kk-cnf = AND OR k if qq = 2): qq nn(1 μμ kk ), μμ kk = 1/OO(kk) (2) AND XOR AND XOR ( CNF = AND OR): 2 nn(1 μμ cc ), μμ cc = 1/OO(log cc), #ANDs = cccc at depth 3 (3) AC 0 [pp] ( AC 0 ): 2 nn(1 μμ dd,mm ), μμ dd, mm = 1/OO(log mm) dd 1 depth dd, #gates = mm 36

37 Proof Sketch based on the Polynomial Method in Circuit Complexity probabilistic polynomial for AND/OR [Razborov 87,Smolensky 87] AC 0 [pp] [Kopparty-Srinivasan 12] fast evaluation algorithm for polynomial 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

38 Algorithms via Polynomial Method 38

39 Polynomial Method Example [Razborov 87,Smolensky 87]: 1. AC 0 [pp] can be well approximated by a low-degree GF(pp) polynomial 2. majority, mod qq cannot be well approximated by a low-degree GF(pp) polynomial 1+2 majority, mod qq AC 0 [pp] item 1 is useful in algorithm design (``sparse suffices instead of ``low-degree in many cases) 39

40 Polynomial Method Definition: Let ff: {0,1} nn {0,1} A distribution PP over polynomials is an εε-error probabilistic polynomial for ff if xx, Pr pp PP pp xx ff(xx) εε deg(pp) dd if Pr pp PP deg pp dd = 1 εε error probabilistic CC-circuit is defined analogously 40

41 Algorithm for CC-SAT Input: CC: {0,1} nn 0,1, CC CC Step 1. Define CC : 0,1 nn nn 0,1, CC OR CC as CC yy aa 0,1 nn CC(yy, aa) (Note: CC 2 nnn CC ) Step 2. Construct (1/3)-error probabilistic polynomial pp for CC in time TT(nn, nn, CC ) Step 3. Evaluate pp(yy) for all yy 0,1 nn nn in time TT (nn, nn, CC ) Step 4. repeat 2-3 OO(nn) times to reduce error probability Output: truth table VV of CC such that yy, Pr VV yy CCC(yy) 2 2nn Running Time: OO(nn(TT(nn, nn, CC )+ TT (nn, nn, CC ))) 41

42 Ingredients for THR THR (1/3) Lemma [Razborov 87,Smolensky 87]: There exists εε-error probabilistic polynomial for AND/OR of degree log(1/εε) and it is efficiently samplable Lemma [Srinivasan 13]: There exists εε-error probabilistic polynomial for THR of degree nn polylog(nn/εε) and it is efficiently samplable 42

43 Ingredients for THR THR (2/3) Lemma: XOR AND XOR AND XOR AND Lemma [Maciel-Therien 98]: THR AC 0 [2] SYM Lemma [Beigel 92]: AND SYM weighted SYM 43

44 Ingredients for THR THR (3/3) Lemma [Williams 14]: Let CC: {0,1} nn 0,1, CC SYM (weighted SYM) where top gate has fan-in uu bottom gates have maximum weight ww such that uuuu 2 0.1nn Then, truth table of CC can be generated in time poly nn 2 nn 44

45 Algorithm for THR THR-SAT Input: CC: {0,1} nn 0,1, CC THR THR Step 1. Define CC : {0,1} nn 0,1, CC OR CC as CC yy aa 0,1 nn CC(yy, aa) (Note: CC 2 nnn CC ) Step 2. Construct (1/3)-error probabilistic circuit CC for CC CC (XOR AND) (XOR AND) (XOR AND) SYM Step 3. transform CC into CC XOR (weighted SYM) Step 4. Evaluate CC (yy) for all yy 0,1 nn nn Step 5. Repeat 2-4 OO(nn) times to reduce error probability Output: truth table of CC (Note: need more work for deterministic algorithms) 45

46 Recent CC-SAT algorithms (1) Theorem [Alman-Chan-Williams 16,T]: Let CC= depth 2 TC 0 with #gates = mm There is a deterministic algorithm for #CC-SAT that runs in time poly(nn, mm) 2 nn μμ(nn,mm), where μμ nn, mm = Ω nn/mm 1/2+oo 1 cc, cc > 0 Corollary: E NP CC holds with mm = OO(nn 1.99 ) 46

47 Bottleneck for Improvements Let ff: {0,1} nn {0,1} A distribution PP over polynomials is an εε-error probabilistic polynomial for ff if xx, Pr pp PP pp xx ff(xx) εε Lemma [Srinivasan 13]: There exists εε-error probabilistic polynomial for THR of degree nn polylog(nn/εε) and it is efficiently samplable We cannot improve the lemma under the above definition of εε-error probabilistic polynomial 47

48 Further Improvement? All we want to do in the polynomial method is: Step 1 Take a ``big OR of the input CC: CC yy aa 0,1 nn CC(yy, aa) Step 2 Represent CC as CC yy, yyyy h( ii=1 ff ii (yyy)gg ii (yyyy)) for some ff ii, gg ii : 0,1 nn nn /2 RR and h: RR 0,1 rr Typically, ii=1 ff ii (yyy)gg ii (yyyy) is a sum of monomials and h is a threshold or modulo function Try different representations, e.g. 1,2 instead of 0,1, rr RR=complex numbers, ii=1 ff ii (yyy)gg ii (yyyy) as a trigonometric polynomial, etc. rr 48

49 Conclusion 49

50 Some Recent CC-SAT algorithms Attempts to handle TC 0 (probably CC ACC 0 THR) depth 2 TC 0 with a sub-quadratic #gates AC 0 nn with a limited #symmetric gates: gg( ii=1 xx ii ) Faster algorithms within AC 0 [pp] (CC ACC 0 THR) Systems of degree kk Polynomial Equations over GF(qq) AND XOR AND XOR AC 0 [pp] 50

51 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 CC non-trivial CC-SAT algorithm? (cf. [Carmosino-Impagliazzo-Kabanets-Kolokolova,CCC'16]) Thank you! 51