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Csr2011 june15 11_30_sokolov

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  • 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. Goldreich’s function f : f0; 1gn 3 f0; 1gn Sokolov D. 2/9 | Inversion of Goldreich’s function
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Lower bounds for DPLL algorithms Sokolov D. 4/9 | Inversion of Goldreich’s function
  • 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. 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. 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. 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. 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. Drunken and myopic algorithms Sokolov D. 5/9 | Inversion of Goldreich’s function
  • 26. Drunken and myopic algorithms Definition Drunken algorithms: A: any; B: random 50 : 50. Sokolov D. 5/9 | Inversion of Goldreich’s function
  • 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. 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. 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. 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. 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. 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. 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. Myopic algorithms Sokolov D. 6/9 | Inversion of Goldreich’s function
  • 35. Myopic algorithms Definition Myopic algorithm: A; B are myopic heuristics. Sokolov D. 6/9 | Inversion of Goldreich’s function
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Graph construction P (x1 ; : : : ; xd ) = x1 ¨ x2 ¨ : : : ¨ xd  k ¨ Q (xd  k +1; : : : ; xd ) Sokolov D. 8/9 | Inversion of Goldreich’s function
  • 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. 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. 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. 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. 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. 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. Plan of the proof Sokolov D. 9/9 | Inversion of Goldreich’s function
  • 59. Plan of the proof Lower bounds for unsatisfiable formulas. Sokolov D. 9/9 | Inversion of Goldreich’s function
  • 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. 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. 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