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# Csr2011 june15 11_30_sokolov

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### Csr2011 june15 11_30_sokolov

1. 1. The complexity of inversion of explicit Goldreich’s function by DPLL algorithms Dmitry Itsykson, Dmitry Sokolov Steklov Institute of Mathematics at St. Petersburg, Academic University CSR 2011, Saint-Petersburg June 15 Sokolov D. 1/9 | Inversion of Goldreich’s function
2. 2. Goldreich’s function f : f0; 1gn 3 f0; 1gn Sokolov D. 2/9 | Inversion of Goldreich’s function
3. 3. Goldreich’s function f : f0; 1gn 3 f0; 1gn x1 x2 x3 . . . . . . xn X Y Sokolov D. 2/9 | Inversion of Goldreich’s function
4. 4. Goldreich’s function f : f0; 1gn 3 f0; 1gn x1 x2 P (xi1 ; xi2 ; : : : ; xid ) x3 . . . . . . xn X Y Sokolov D. 2/9 | Inversion of Goldreich’s function
5. 5. Goldreich’s function f : f0; 1gn 3 f0; 1gn x1 x2 P (xi1 ; xi2 ; : : : ; xid ) x3 G (X ; Y ; E ) is a bipartite . . graph; . . . . xn X Y Sokolov D. 2/9 | Inversion of Goldreich’s function
6. 6. Goldreich’s function f : f0; 1gn 3 f0; 1gn x1 x2 P (xi1 ; xi2 ; : : : ; xid ) x3 G (X ; Y ; E ) is a bipartite . . graph; . . . . Vy P Y deg (y ) = d xn X Y Sokolov D. 2/9 | Inversion of Goldreich’s function
7. 7. Goldreich’s function f : f0; 1gn 3 f0; 1gn x1 x2 P (xi1 ; xi2 ; : : : ; xid ) x3 G (X ; Y ; E ) is a bipartite . . graph; . . . . Vy P Y deg (y ) = d xn d is a constant. X Y Sokolov D. 2/9 | Inversion of Goldreich’s function
8. 8. Goldreich’s function f : f0; 1gn 3 f0; 1gn x1 x2 P (xi1 ; xi2 ; : : : ; xid ) x3 G (X ; Y ; E ) is a bipartite . . graph; . . . . Vy P Y deg (y ) = d xn d is a constant. X Y Goldreich’s conjecture: P is a random predicate; G is an expander; then function f is a one-way. Sokolov D. 2/9 | Inversion of Goldreich’s function
9. 9. Goldreich’s function f : f0; 1gn 3 f0; 1gn x1 x2 P (xi1 ; xi2 ; : : : ; xid ) x3 G (X ; Y ; E ) is a bipartite . . graph; . . . . Vy P Y deg (y ) = d xn d is a constant. X Y Goldreich’s conjecture: P is a random predicate; G is an expander; then function f is a one-way. f is computed by constant depth circuit; Sokolov D. 2/9 | Inversion of Goldreich’s function
10. 10. Goldreich’s function f : f0; 1gn 3 f0; 1gn x1 x2 P (xi1 ; xi2 ; : : : ; xid ) x3 G (X ; Y ; E ) is a bipartite . . graph; . . . . Vy P Y deg (y ) = d xn d is a constant. X Y Goldreich’s conjecture: P is a random predicate; G is an expander; then function f is a one-way. f is computed by constant depth circuit; [Applebaum, Ishai, Kushilevitz 2006] If one-way functions exist then there is a one-way function that can be computed by constant depth circuit. Sokolov D. 2/9 | Inversion of Goldreich’s function
11. 11. DPLL algorithms xi := c xi := 1 c H HH xj := c1 xj := 1   c1 xk := c2 xk := 1   c2 . . . . . . . . . . . . Sokolov D. 3/9 | Inversion of Goldreich’s function
12. 12. DPLL algorithms xi := c xi := 1 c H HH xj := c1 xj := 1   c1 xk := c2 xk := 1   c2 . . . . . . . . . . . . Sokolov D. 3/9 | Inversion of Goldreich’s function
13. 13. DPLL algorithms xi := c xi := 1 c H HH xj := c1 xj := 1   c1 xk := c2 xk := 1   c2 . . . . . . . . . . . . Sokolov D. 3/9 | Inversion of Goldreich’s function
14. 14. DPLL algorithms xi := c xi := 1 c H HH xj := c1 xj := 1   c1 xk := c2 xk := 1   c2 . . . . . . . . . . . . Sokolov D. 3/9 | Inversion of Goldreich’s function
15. 15. DPLL algorithms xi := c xi := 1 c H HH xj := c1 xj := 1   c1 xk := c2 xk := 1   c2 . . . . . . . . . . . . Sokolov D. 3/9 | Inversion of Goldreich’s function
16. 16. DPLL algorithms xi := c xi := 1 c H HH xj := c1 xj := 1   c1 xk := c2 xk := 1   c2 . . . . . . . . . . . . Heuristic A chooses a variable for splitting. Sokolov D. 3/9 | Inversion of Goldreich’s function
17. 17. DPLL algorithms xi := c xi := 1 c H HH xj := c1 xj := 1   c1 xk := c2 xk := 1   c2 . . . . . . . . . . . . Heuristic A chooses a variable for splitting. Heuristic B chooses ﬁrst value. Sokolov D. 3/9 | Inversion of Goldreich’s function
18. 18. DPLL algorithms xi := c xi := 1 c H HH xj := c1 xj := 1   c1 xk := c2 xk := 1   c2 . . . . . . . . . . . . Heuristic A chooses a variable for splitting. Heuristic B chooses ﬁrst value. Simpliﬁcation rules: unit clause elimination; pure literal rule. Sokolov D. 3/9 | Inversion of Goldreich’s function
19. 19. Lower bounds for DPLL algorithms Sokolov D. 4/9 | Inversion of Goldreich’s function
20. 20. Lower bounds for DPLL algorithms Unsatisﬁable formulas Exponential lower bounds for resolution refutations of unsatisﬁable formulas translate to backtracking algorithms. [Tseitin, 1968] ... [Pudlak, Implagliazzo, 2000]. Sokolov D. 4/9 | Inversion of Goldreich’s function
21. 21. Lower bounds for DPLL algorithms Unsatisﬁable formulas Exponential lower bounds for resolution refutations of unsatisﬁable formulas translate to backtracking algorithms. [Tseitin, 1968] ... [Pudlak, Implagliazzo, 2000]. Satisﬁable formulas If P = NP then there are no superpolynomial lower bounds for backtracking algorithms since heuristic B may choose correct value. Sokolov D. 4/9 | Inversion of Goldreich’s function
22. 22. Lower bounds for DPLL algorithms Unsatisﬁable formulas Exponential lower bounds for resolution refutations of unsatisﬁable formulas translate to backtracking algorithms. [Tseitin, 1968] ... [Pudlak, Implagliazzo, 2000]. Satisﬁable formulas If P = NP then there are no superpolynomial lower bounds for backtracking algorithms since heuristic B may choose correct value. Inverting of functions corresponds to satisﬁable formulas. Sokolov D. 4/9 | Inversion of Goldreich’s function
23. 23. Lower bounds for DPLL algorithms Unsatisﬁable formulas Exponential lower bounds for resolution refutations of unsatisﬁable formulas translate to backtracking algorithms. [Tseitin, 1968] ... [Pudlak, Implagliazzo, 2000]. Satisﬁable formulas If P = NP then there are no superpolynomial lower bounds for backtracking algorithms since heuristic B may choose correct value. Inverting of functions corresponds to satisﬁable formulas. [Nikolenko 2002], [Achilioptas, Beame, Molloy 2003-2004] exponential lower bound for speciﬁc backtracking algorithms. [Alekhnovich, Hirsch, Itsykson 2005] Exponential lower bound for myopic and drunken algorithms. Sokolov D. 4/9 | Inversion of Goldreich’s function
24. 24. Lower bounds for DPLL algorithms Unsatisﬁable formulas Exponential lower bounds for resolution refutations of unsatisﬁable formulas translate to backtracking algorithms. [Tseitin, 1968] ... [Pudlak, Implagliazzo, 2000]. Satisﬁable formulas If P = NP then there are no superpolynomial lower bounds for backtracking algorithms since heuristic B may choose correct value. Inverting of functions corresponds to satisﬁable formulas. [Nikolenko 2002], [Achilioptas, Beame, Molloy 2003-2004] exponential lower bound for speciﬁc backtracking algorithms. [Alekhnovich, Hirsch, Itsykson 2005] Exponential lower bound for myopic and drunken algorithms. Exponential lower bound for inversion of Goldreich’s function by myopic [J. Cook et al. 2009] and drunken [Itsykson 2010] algorithms. Sokolov D. 4/9 | Inversion of Goldreich’s function
25. 25. Drunken and myopic algorithms Sokolov D. 5/9 | Inversion of Goldreich’s function
26. 26. Drunken and myopic algorithms Deﬁnition Drunken algorithms: A: any; B: random 50 : 50. Sokolov D. 5/9 | Inversion of Goldreich’s function
27. 27. Drunken and myopic algorithms Deﬁnition Drunken algorithms: A: any; B: random 50 : 50. Deﬁnition Myopic heuristic: Sokolov D. 5/9 | Inversion of Goldreich’s function
28. 28. Drunken and myopic algorithms Deﬁnition Drunken algorithms: A: any; B: random 50 : 50. Deﬁnition Myopic heuristic: sees structure of the formula; Sokolov D. 5/9 | Inversion of Goldreich’s function
29. 29. Drunken and myopic algorithms Deﬁnition Drunken algorithms: A: any; B: random 50 : 50. Deﬁnition Myopic heuristic: sees structure of the formula; doesn’t see negation signs; Sokolov D. 5/9 | Inversion of Goldreich’s function
30. 30. Drunken and myopic algorithms Deﬁnition Drunken algorithms: A: any; B: random 50 : 50. Deﬁnition Myopic heuristic: sees structure of the formula; doesn’t see negation signs; requests negations in K = n1  clause. (x1  x3  x5) (x2  x3) (x2  x4  x5 ) (x1  x4  x6 ) Sokolov D. 5/9 | Inversion of Goldreich’s function
31. 31. Drunken and myopic algorithms Deﬁnition Drunken algorithms: A: any; B: random 50 : 50. Deﬁnition Myopic heuristic: sees structure of the formula; doesn’t see negation signs; requests negations in K = n1  clause. (x1  x3  x5) (x2  x3) (x2  x4  x5 ) (x1  x4  x6 ) Sokolov D. 5/9 | Inversion of Goldreich’s function
32. 32. Drunken and myopic algorithms Deﬁnition Drunken algorithms: A: any; B: random 50 : 50. Deﬁnition Myopic heuristic: sees structure of the formula; doesn’t see negation signs; requests negations in K = n1  clause. (x1  x3  x5) (x2  x3) (x2  x4  x5 ) (x1  x4  x6 ) Sokolov D. 5/9 | Inversion of Goldreich’s function
33. 33. Drunken and myopic algorithms Deﬁnition Drunken algorithms: A: any; B: random 50 : 50. Deﬁnition Myopic heuristic: sees structure of the formula; doesn’t see negation signs; requests negations in K = n1  clause. (x1  x3  x5) (x1  x3  x5) (x2  x3) A (x2  Xx3) (x2  x4  x5 ) (x2  x4  x5 ) (x1  x4  x6 ) (x1  Xx4  x6 ) Sokolov D. 5/9 | Inversion of Goldreich’s function
34. 34. Myopic algorithms Sokolov D. 6/9 | Inversion of Goldreich’s function
35. 35. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. Sokolov D. 6/9 | Inversion of Goldreich’s function
36. 36. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. [Alekhnovich, Hirsch, Itsykson 2005] P is a linear predicate. G is a random graph. Sokolov D. 6/9 | Inversion of Goldreich’s function
37. 37. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. [Alekhnovich, Hirsch, Itsykson 2005] P is a linear predicate. G is a random graph. [J. Cook, Etesami, Miller, Trevisan 2009] Sokolov D. 6/9 | Inversion of Goldreich’s function
38. 38. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. [Alekhnovich, Hirsch, Itsykson 2005] P is a linear predicate. G is a random graph. [J. Cook, Etesami, Miller, Trevisan 2009] P = x1 + x2 + ¡ ¡ ¡ + xd  2 + xd  1 xd . In fact: P = x1 + x2 + ¡ ¡ ¡ + xd  k + Q (xd  k +1 ; : : : ; xd ). Sokolov D. 6/9 | Inversion of Goldreich’s function
39. 39. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. [Alekhnovich, Hirsch, Itsykson 2005] P is a linear predicate. G is a random graph. [J. Cook, Etesami, Miller, Trevisan 2009] P = x1 + x2 + ¡ ¡ ¡ + xd  2 + xd  1 xd . In fact: P = x1 + x2 + ¡ ¡ ¡ + xd  k + Q (xd  k +1 ; : : : ; xd ). Disadvantages: Sokolov D. 6/9 | Inversion of Goldreich’s function
40. 40. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. [Alekhnovich, Hirsch, Itsykson 2005] P is a linear predicate. G is a random graph. [J. Cook, Etesami, Miller, Trevisan 2009] P = x1 + x2 + ¡ ¡ ¡ + xd  2 + xd  1 xd . In fact: P = x1 + x2 + ¡ ¡ ¡ + xd  k + Q (xd  k +1 ; : : : ; xd ). Disadvantages: G is a random graph. Sokolov D. 6/9 | Inversion of Goldreich’s function
41. 41. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. [Alekhnovich, Hirsch, Itsykson 2005] P is a linear predicate. G is a random graph. [J. Cook, Etesami, Miller, Trevisan 2009] P = x1 + x2 + ¡ ¡ ¡ + xd  2 + xd  1 xd . In fact: P = x1 + x2 + ¡ ¡ ¡ + xd  k + Q (xd  k +1 ; : : : ; xd ). Disadvantages: G is a random graph. K is a constant. Sokolov D. 6/9 | Inversion of Goldreich’s function
42. 42. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. [Alekhnovich, Hirsch, Itsykson 2005] P is a linear predicate. G is a random graph. [J. Cook, Etesami, Miller, Trevisan 2009] P = x1 + x2 + ¡ ¡ ¡ + xd  2 + xd  1 xd . In fact: P = x1 + x2 + ¡ ¡ ¡ + xd  k + Q (xd  k +1 ; : : : ; xd ). Disadvantages: In our work: G is a random graph. K is a constant. Too complicated proof. Sokolov D. 6/9 | Inversion of Goldreich’s function
43. 43. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. [Alekhnovich, Hirsch, Itsykson 2005] P is a linear predicate. G is a random graph. [J. Cook, Etesami, Miller, Trevisan 2009] P = x1 + x2 + ¡ ¡ ¡ + xd  2 + xd  1 xd . In fact: P = x1 + x2 + ¡ ¡ ¡ + xd  k + Q (xd  k +1 ; : : : ; xd ). Disadvantages: In our work: G is a random graph. G is based on expander. K is a constant. Too complicated proof. Sokolov D. 6/9 | Inversion of Goldreich’s function
44. 44. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. [Alekhnovich, Hirsch, Itsykson 2005] P is a linear predicate. G is a random graph. [J. Cook, Etesami, Miller, Trevisan 2009] P = x1 + x2 + ¡ ¡ ¡ + xd  2 + xd  1 xd . In fact: P = x1 + x2 + ¡ ¡ ¡ + xd  k + Q (xd  k +1 ; : : : ; xd ). Disadvantages: In our work: G is a random graph. G is based on expander. K is a constant. K = n1  . Too complicated proof. Sokolov D. 6/9 | Inversion of Goldreich’s function
45. 45. Myopic algorithms Deﬁnition Myopic algorithm: A; B are myopic heuristics. [Alekhnovich, Hirsch, Itsykson 2005] P is a linear predicate. G is a random graph. [J. Cook, Etesami, Miller, Trevisan 2009] P = x1 + x2 + ¡ ¡ ¡ + xd  2 + xd  1 xd . In fact: P = x1 + x2 + ¡ ¡ ¡ + xd  k + Q (xd  k +1 ; : : : ; xd ). Disadvantages: In our work: G is a random graph. G is based on expander. K is a constant. K = n1  . Too complicated proof. “Simple” proof. Sokolov D. 6/9 | Inversion of Goldreich’s function
46. 46. Our results P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ), Q is an arbitrary, k d =4. Sokolov D. 7/9 | Inversion of Goldreich’s function
47. 47. Our results P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ), Q is an arbitrary, k d =4. Theorem There exists an explicit graph G such that every myopic or drunken (1) DPLL algorithm makes at least 2n steps on “f (x ) = f (a)” for almost all a P f0; 1g n. Sokolov D. 7/9 | Inversion of Goldreich’s function
48. 48. Our results P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ), Q is an arbitrary, k d =4. Theorem There exists an explicit graph G such that every myopic or drunken (1) DPLL algorithm makes at least 2n steps on “f (x ) = f (a)” for almost all a P f0; 1g n. G is based on expander instead of random graph. Sokolov D. 7/9 | Inversion of Goldreich’s function
49. 49. Our results P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ), Q is an arbitrary, k d =4. Theorem There exists an explicit graph G such that every myopic or drunken (1) DPLL algorithm makes at least 2n steps on “f (x ) = f (a)” for almost all a P f0; 1g n. G is based on expander instead of random graph. For drunken algorithms the proof follows [Itsykson, CSR-2010] Sokolov D. 7/9 | Inversion of Goldreich’s function
50. 50. Our results P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ), Q is an arbitrary, k d =4. Theorem There exists an explicit graph G such that every myopic or drunken (1) DPLL algorithm makes at least 2n steps on “f (x ) = f (a)” for almost all a P f0; 1g n. G is based on expander instead of random graph. For drunken algorithms the proof follows [Itsykson, CSR-2010] For myopic we simplify previous proof and K = n1  Sokolov D. 7/9 | Inversion of Goldreich’s function
51. 51. Graph construction P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ) Sokolov D. 8/9 | Inversion of Goldreich’s function
52. 52. Graph construction P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ) G G is an expander; Sokolov D. 8/9 | Inversion of Goldreich’s function
53. 53. Graph construction P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ) + G G is an expander; Sokolov D. 8/9 | Inversion of Goldreich’s function
54. 54. Graph construction P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ) + G T G is an expander; G + T has full rank. Vy P Y T; deg (y ) = 1; Sokolov D. 8/9 | Inversion of Goldreich’s function
55. 55. Graph construction P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ) + + G T G is an expander; G + T has full rank. Vy P Y T; deg (y ) = 1; G +T is an expander. Sokolov D. 8/9 | Inversion of Goldreich’s function
56. 56. Graph construction P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ) + + n G T R G is an expander; G + T has full rank. Vy P Y T; deg (y ) = 1; G +T is an expander. R contains nonlinear edges. jfx j x P X ; deg (x ) T= 0gj n . G +T +R is an expander. Sokolov D. 8/9 | Inversion of Goldreich’s function
57. 57. Graph construction P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ) + + n G T R G is an expander; G + T has full rank. Vy P Y T; deg (y ) = 1; G +T is an expander. R contains nonlinear edges. jfx j x P X ; deg (x ) T= 0gj n . G + T + R is an expander. Size of preimages no more than 2n . We can invert fG +T +R ;P in time poly (n)2n , but this is still much! Sokolov D. 8/9 | Inversion of Goldreich’s function
58. 58. Plan of the proof Sokolov D. 9/9 | Inversion of Goldreich’s function
59. 59. Plan of the proof Lower bounds for unsatisﬁable formulas. Sokolov D. 9/9 | Inversion of Goldreich’s function
60. 60. Plan of the proof Lower bounds for unsatisﬁable formulas. G is an expander. P is almost linear. Lower bounds for resolution proofs Sokolov D. 9/9 | Inversion of Goldreich’s function
61. 61. Plan of the proof Lower bounds for unsatisﬁable formulas. G is an expander. P is almost linear. Lower bounds for resolution proofs With probability 1   2 (n) after several steps current formula becomes unsatisﬁable. Sokolov D. 9/9 | Inversion of Goldreich’s function
62. 62. Plan of the proof Lower bounds for unsatisﬁable formulas. G is an expander. P is almost linear. Lower bounds for resolution proofs With probability 1   2 (n) after several steps current formula becomes unsatisﬁable. G is an expander. f is almost bijection. Myopic algorithm can’t recognize diﬀerent absolute terms. Sokolov D. 9/9 | Inversion of Goldreich’s function