Improving the Space-Bounded Version ofMuchnik’s Conditional Complexity Theorem via          “Naive” Derandomization       ...
Some disclaimers      This talk is in some sense a continuation of the previous one
Some disclaimers      This talk is in some sense a continuation of the previous one      A similar technique is used to ob...
Some disclaimers      This talk is in some sense a continuation of the previous one      A similar technique is used to ob...
Some disclaimers      This talk is in some sense a continuation of the previous one      A similar technique is used to ob...
Kolmogorov complexity      Kolmogorov complexity C (a|b) of a string a conditional to b      is the minimal length of a pr...
Kolmogorov complexity      Kolmogorov complexity C (a|b) of a string a conditional to b      is the minimal length of a pr...
Kolmogorov complexity      Kolmogorov complexity C (a|b) of a string a conditional to b      is the minimal length of a pr...
Muchnik’s theorem      Muchnik’s theorem (TCS’2002): For any a and b of length n      there exists p of length C (a|b) + O...
Muchnik’s theorem      Muchnik’s theorem (TCS’2002): For any a and b of length n      there exists p of length C (a|b) + O...
Muchnik’s theorem      Muchnik’s theorem (TCS’2002): For any a and b of length n      there exists p of length C (a|b) + O...
Muchnik’s theorem      Muchnik’s theorem (TCS’2002): For any a and b of length n      there exists p of length C (a|b) + O...
Muchnik’s theorem      Muchnik’s theorem (TCS’2002): For any a and b of length n      there exists p of length C (a|b) + O...
Proof of Muchnik’s theorem: fingerprints        {0, 1}n , n = |a|                                 {0, 1}k , k = C(a|b)     ...
Proof of Muchnik’s theorem: fingerprints                                   p         {0, 1}n , n = |a|                     ...
Proof of Muchnik’s theorem: fingerprints                                              p         {0, 1}n , n = |a|          ...
Proof of Muchnik’s theorem: fingerprints                  S = {x: C(x|b) ≤ C(a|b)}                                         ...
Proof of Muchnik’s theorem: fingerprints                  S = {x: C(x|b) ≤ C(a|b)}                                         ...
How to find fingerprints      There is an underlying bipartite graph G = (L, R, E ) with      |L| = 2n and |R| = 2k
How to find fingerprints      There is an underlying bipartite graph G = (L, R, E ) with      |L| = 2n and |R| = 2k      Eac...
How to find fingerprints      There is an underlying bipartite graph G = (L, R, E ) with      |L| = 2n and |R| = 2k      Eac...
How to find fingerprints      There is an underlying bipartite graph G = (L, R, E ) with      |L| = 2n and |R| = 2k      Eac...
How to find fingerprints      There is an underlying bipartite graph G = (L, R, E ) with      |L| = 2n and |R| = 2k      Eac...
How to find fingerprints      There is an underlying bipartite graph G = (L, R, E ) with      |L| = 2n and |R| = 2k      Eac...
How to find fingerprints      There is an underlying bipartite graph G = (L, R, E ) with      |L| = 2n and |R| = 2k      Eac...
How to find fingerprints      There is an underlying bipartite graph G = (L, R, E ) with      |L| = 2n and |R| = 2k      Eac...
What graph properties do we need?   Some graph properties leading to Muchnik’s theorem:       (Muchnik) Expander-like prop...
What graph properties do we need?   Some graph properties leading to Muchnik’s theorem:       (Muchnik) Expander-like prop...
What graph properties do we need?   Some graph properties leading to Muchnik’s theorem:       (Muchnik) Expander-like prop...
What graph properties do we need?   Some graph properties leading to Muchnik’s theorem:       (Muchnik) Expander-like prop...
The essential property: low-congesting graph                                     S, |S| < 2k                           {0,...
The essential property: low-congesting graph                                    S, |S| < 2k                               ...
The essential property: low-congesting graph                                    S, |S| < 2k                               ...
The essential property: low-congesting graph                                    S, |S| < 2k                               ...
The essential property: low-congesting graph      α-clot for S is the set of all vertices having more than αD      neighbo...
How to get a low-congesting graph      A random graph with certain parameters is an extractor with      positive probabili...
How to get a low-congesting graph      A random graph with certain parameters is an extractor with      positive probabili...
How to get a low-congesting graph      A random graph with certain parameters is an extractor with      positive probabili...
How to get a low-congesting graph      A random graph with certain parameters is an extractor with      positive probabili...
How to get a low-congesting graph      A random graph with certain parameters is an extractor with      positive probabili...
How to get a low-congesting graph      A random graph with certain parameters is an extractor with      positive probabili...
Why a pseudorandom graph fits      We use the Nisan pseudorandom generator with polynomial      seed length and exponential...
Why a pseudorandom graph fits      We use the Nisan pseudorandom generator with polynomial      seed length and exponential...
Why a pseudorandom graph fits      We use the Nisan pseudorandom generator with polynomial      seed length and exponential...
Why a pseudorandom graph fits      We use the Nisan pseudorandom generator with polynomial      seed length and exponential...
Why a pseudorandom graph fits      We use the Nisan pseudorandom generator with polynomial      seed length and exponential...
Why a pseudorandom graph fits      We use the Nisan pseudorandom generator with polynomial      seed length and exponential...
Why a pseudorandom graph fits      We use the Nisan pseudorandom generator with polynomial      seed length and exponential...
The circuit                                            S  h1 (x1 ) h2 (x1 )   ···   hD (x1 )   h1 (x2 )        ···     hD ...
How to get Muchnik’s theorem      Firstly, we search for a good seed for the generator and fix it.      This search needs o...
How to get Muchnik’s theorem      Firstly, we search for a good seed for the generator and fix it.      This search needs o...
How to get Muchnik’s theorem                 S = {x: C s (x|b) ≤ k}                                             p      {0,...
How to get Muchnik’s theorem      Firstly, we search for a good seed for the generator and fix it.      This search needs o...
How to get Muchnik’s theorem      Firstly, we search for a good seed for the generator and fix it.      This search needs o...
The final formulation   For any a and b of length n and for any s there exists p of length   C s (a|b) + O(log log s + log ...
Summary of the technology      Take some theorem about Kolmogorov complexity relying on      the existence of some combina...
Summary of the technology      Take some theorem about Kolmogorov complexity relying on      the existence of some combina...
Summary of the technology      Take some theorem about Kolmogorov complexity relying on      the existence of some combina...
Summary of the technology      Take some theorem about Kolmogorov complexity relying on      the existence of some combina...
Summary of the technology      Take some theorem about Kolmogorov complexity relying on      the existence of some combina...
Thank you!   mailto:musatych@gmail.comhttp://musatych.livejournal.com
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Daniil Musatov. Improving the Space-Bounded Version of Muchnik's Conditional Complexity Theorem via ``Naive'' Derandomization

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Csr2011 june14 15_45_musatov

  1. 1. Improving the Space-Bounded Version ofMuchnik’s Conditional Complexity Theorem via “Naive” Derandomization Daniil Musatov1 1 Lomonosov Moscow State University, St. Petersburg, June 14th, 2011
  2. 2. Some disclaimers This talk is in some sense a continuation of the previous one
  3. 3. Some disclaimers This talk is in some sense a continuation of the previous one A similar technique is used to obtain a different result
  4. 4. Some disclaimers This talk is in some sense a continuation of the previous one A similar technique is used to obtain a different result Why “naive”?
  5. 5. Some disclaimers This talk is in some sense a continuation of the previous one A similar technique is used to obtain a different result Why “naive”? Because we simply replace a random object by a pseudo-random one and it still does the job.
  6. 6. Kolmogorov complexity Kolmogorov complexity C (a|b) of a string a conditional to b is the minimal length of a program p that produces a given b, i.e., p(b) = a
  7. 7. Kolmogorov complexity Kolmogorov complexity C (a|b) of a string a conditional to b is the minimal length of a program p that produces a given b, i.e., p(b) = a Space-bounded version: C s (a|b) is the minimal length of a program p such that p(b) = a
  8. 8. Kolmogorov complexity Kolmogorov complexity C (a|b) of a string a conditional to b is the minimal length of a program p that produces a given b, i.e., p(b) = a Space-bounded version: C s (a|b) is the minimal length of a program p such that p(b) = a and the computation of p(b) performs in space s.
  9. 9. Muchnik’s theorem Muchnik’s theorem (TCS’2002): For any a and b of length n there exists p of length C (a|b) + O(log n) such that p(b) = a
  10. 10. Muchnik’s theorem Muchnik’s theorem (TCS’2002): For any a and b of length n there exists p of length C (a|b) + O(log n) such that p(b) = a and C (p|a) = O(log n).
  11. 11. Muchnik’s theorem Muchnik’s theorem (TCS’2002): For any a and b of length n there exists p of length C (a|b) + O(log n) such that p(b) = a and C (p|a) = O(log n). Space-bounded version (M., Romashchenko, Shen, CSR’2009, ToCS’2011): For any a and b of length n and for any s there exists p of length C s (a|b) + O(log3 n) such that:
  12. 12. Muchnik’s theorem Muchnik’s theorem (TCS’2002): For any a and b of length n there exists p of length C (a|b) + O(log n) such that p(b) = a and C (p|a) = O(log n). Space-bounded version (M., Romashchenko, Shen, CSR’2009, ToCS’2011): For any a and b of length n and for any s there exists p of length C s (a|b) + O(log3 n) such that: p(b) = a; the computation of p(b) performs in space O(s) + poly(n) and C poly(n) (p|a) = O(log3 n)
  13. 13. Muchnik’s theorem Muchnik’s theorem (TCS’2002): For any a and b of length n there exists p of length C (a|b) + O(log n) such that p(b) = a and C (p|a) = O(log n). Space-bounded version (M., Romashchenko, Shen, CSR’2009, ToCS’2011): For any a and b of length n and for any s there exists p of length C s (a|b) + O(log3 n) such that: p(b) = a; the computation of p(b) performs in space O(s) + poly(n) and C poly(n) (p|a) = O(log3 n) In current work we get rid of polylogarithmic terms and make them again logarithmic
  14. 14. Proof of Muchnik’s theorem: fingerprints {0, 1}n , n = |a| {0, 1}k , k = C(a|b) a
  15. 15. Proof of Muchnik’s theorem: fingerprints p {0, 1}n , n = |a| {0, 1}k , k = C(a|b) a p is a fingerprint of a
  16. 16. Proof of Muchnik’s theorem: fingerprints p {0, 1}n , n = |a| {0, 1}k , k = C(a|b) a p is a fingerprint of a For C (p|a) to be small, there should be few fingerprints for each a
  17. 17. Proof of Muchnik’s theorem: fingerprints S = {x: C(x|b) ≤ C(a|b)} p {0, 1}n , n = |a| {0, 1}k , k = C(a|b) a p is a fingerprint of a For C (p|a) to be small, there should be few fingerprints for each a For C (a|p, b) to be small,
  18. 18. Proof of Muchnik’s theorem: fingerprints S = {x: C(x|b) ≤ C(a|b)} p {0, 1}n , n = |a| {0, 1}k , k = C(a|b) a p is a fingerprint of a For C (p|a) to be small, there should be few fingerprints for each a For C (a|p, b) to be small, there should be few preimages of p in S
  19. 19. How to find fingerprints There is an underlying bipartite graph G = (L, R, E ) with |L| = 2n and |R| = 2k
  20. 20. How to find fingerprints There is an underlying bipartite graph G = (L, R, E ) with |L| = 2n and |R| = 2k Each left-part vertex has constant degree D
  21. 21. How to find fingerprints There is an underlying bipartite graph G = (L, R, E ) with |L| = 2n and |R| = 2k Each left-part vertex has constant degree D The fingerprint p is chosen among neighbors of a
  22. 22. How to find fingerprints There is an underlying bipartite graph G = (L, R, E ) with |L| = 2n and |R| = 2k Each left-part vertex has constant degree D The fingerprint p is chosen among neighbors of a This guarantees that C (p|a, G ) = O(log D)
  23. 23. How to find fingerprints There is an underlying bipartite graph G = (L, R, E ) with |L| = 2n and |R| = 2k Each left-part vertex has constant degree D The fingerprint p is chosen among neighbors of a This guarantees that C (p|a, G ) = O(log D) The statement that C (a|p, b, G ) is small for some p follows from some enumerable graph property (see later)
  24. 24. How to find fingerprints There is an underlying bipartite graph G = (L, R, E ) with |L| = 2n and |R| = 2k Each left-part vertex has constant degree D The fingerprint p is chosen among neighbors of a This guarantees that C (p|a, G ) = O(log D) The statement that C (a|p, b, G ) is small for some p follows from some enumerable graph property (see later) It is proven by the probabilistic method that this property is non-empty
  25. 25. How to find fingerprints There is an underlying bipartite graph G = (L, R, E ) with |L| = 2n and |R| = 2k Each left-part vertex has constant degree D The fingerprint p is chosen among neighbors of a This guarantees that C (p|a, G ) = O(log D) The statement that C (a|p, b, G ) is small for some p follows from some enumerable graph property (see later) It is proven by the probabilistic method that this property is non-empty Hence, the first graph in the enumeration has small complexity
  26. 26. How to find fingerprints There is an underlying bipartite graph G = (L, R, E ) with |L| = 2n and |R| = 2k Each left-part vertex has constant degree D The fingerprint p is chosen among neighbors of a This guarantees that C (p|a, G ) = O(log D) The statement that C (a|p, b, G ) is small for some p follows from some enumerable graph property (see later) It is proven by the probabilistic method that this property is non-empty Hence, the first graph in the enumeration has small complexity Hence, C (p|a) and C (a|p, b) are also small
  27. 27. What graph properties do we need? Some graph properties leading to Muchnik’s theorem: (Muchnik) Expander-like property
  28. 28. What graph properties do we need? Some graph properties leading to Muchnik’s theorem: (Muchnik) Expander-like property (MRS) Possibility of online matching
  29. 29. What graph properties do we need? Some graph properties leading to Muchnik’s theorem: (Muchnik) Expander-like property (MRS) Possibility of online matching (MRS) Extractor
  30. 30. What graph properties do we need? Some graph properties leading to Muchnik’s theorem: (Muchnik) Expander-like property (MRS) Possibility of online matching (MRS) Extractor (This paper) “Low-congesting”
  31. 31. The essential property: low-congesting graph S, |S| < 2k {0, 1}n each vertex has D neighbors {0, 1}k
  32. 32. The essential property: low-congesting graph S, |S| < 2k α-clot {0, 1} n each vertex has D neighbors {0, 1}k α-clot for S is the set of all vertices having more than αD neighbors in S.
  33. 33. The essential property: low-congesting graph S, |S| < 2k α-clot {0, 1} n each vertex has D neighbors {0, 1}k α-congested vertex α-clot for S is the set of all vertices having more than αD neighbors in S. A vertex is α-congested if all its neighbors lie in α-clot.
  34. 34. The essential property: low-congesting graph S, |S| < 2k α-clot {0, 1} n each vertex has D neighbors {0, 1}k α-congested vertex α-clot for S is the set of all vertices having more than αD neighbors in S. A vertex is α-congested if all its neighbors lie in α-clot. The set S is (α, β)-low-congested if it contains less than βK α-congested vertices.
  35. 35. The essential property: low-congesting graph α-clot for S is the set of all vertices having more than αD neighbors in S. A vertex is α-congested if all its neighbors lie in α-clot. The set S is (α, β)-low-congested if it contains less than βK α-congested vertices. We call a set relevant if it has the form {x|C s (x|b) < k}. We call a graph (α, β)-low-congesting if all relevant sets are (α, β)-low-congested.
  36. 36. How to get a low-congesting graph A random graph with certain parameters is an extractor with positive probability.
  37. 37. How to get a low-congesting graph A random graph with certain parameters is an extractor with positive probability. Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k, )-extractor graph any S is (2, 2 )-low-congested. Hence, in an extractor any relevant set is low-congested and the graph itself is low-congesting.
  38. 38. How to get a low-congesting graph A random graph with certain parameters is an extractor with positive probability. Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k, )-extractor graph any S is (2, 2 )-low-congested. Hence, in an extractor any relevant set is low-congested and the graph itself is low-congesting. But even the description of the graph is exponential in size, so we cannot find it in polynomial space.
  39. 39. How to get a low-congesting graph A random graph with certain parameters is an extractor with positive probability. Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k, )-extractor graph any S is (2, 2 )-low-congested. Hence, in an extractor any relevant set is low-congested and the graph itself is low-congesting. But even the description of the graph is exponential in size, so we cannot find it in polynomial space. Central idea: replace a random graph by a pseudorandom one To make this idea work, we need:
  40. 40. How to get a low-congesting graph A random graph with certain parameters is an extractor with positive probability. Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k, )-extractor graph any S is (2, 2 )-low-congested. Hence, in an extractor any relevant set is low-congested and the graph itself is low-congesting. But even the description of the graph is exponential in size, so we cannot find it in polynomial space. Central idea: replace a random graph by a pseudorandom one To make this idea work, we need: to prove that a pseudorandom graph is low-congesting with positive probability
  41. 41. How to get a low-congesting graph A random graph with certain parameters is an extractor with positive probability. Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k, )-extractor graph any S is (2, 2 )-low-congested. Hence, in an extractor any relevant set is low-congested and the graph itself is low-congesting. But even the description of the graph is exponential in size, so we cannot find it in polynomial space. Central idea: replace a random graph by a pseudorandom one To make this idea work, we need: to prove that a pseudorandom graph is low-congesting with positive probability to show that the seed this graph is generated from may be found in polynomial space
  42. 42. Why a pseudorandom graph fits We use the Nisan pseudorandom generator with polynomial seed length and exponential output.
  43. 43. Why a pseudorandom graph fits We use the Nisan pseudorandom generator with polynomial seed length and exponential output. It is well-known that this generator fools any circuit from AC0
  44. 44. Why a pseudorandom graph fits We use the Nisan pseudorandom generator with polynomial seed length and exponential output. It is well-known that this generator fools any circuit from AC0 The size of a fooled circuit may be exponential since the size of output is exponential.
  45. 45. Why a pseudorandom graph fits We use the Nisan pseudorandom generator with polynomial seed length and exponential output. It is well-known that this generator fools any circuit from AC0 The size of a fooled circuit may be exponential since the size of output is exponential. We cannot check the low-congesting property literally, but using circuits for approximate counting we build a circuit C such that:
  46. 46. Why a pseudorandom graph fits We use the Nisan pseudorandom generator with polynomial seed length and exponential output. It is well-known that this generator fools any circuit from AC0 The size of a fooled circuit may be exponential since the size of output is exponential. We cannot check the low-congesting property literally, but using circuits for approximate counting we build a circuit C such that: If G is (2, 2 )-low-congesting then C (G ) = 1;
  47. 47. Why a pseudorandom graph fits We use the Nisan pseudorandom generator with polynomial seed length and exponential output. It is well-known that this generator fools any circuit from AC0 The size of a fooled circuit may be exponential since the size of output is exponential. We cannot check the low-congesting property literally, but using circuits for approximate counting we build a circuit C such that: If G is (2, 2 )-low-congesting then C (G ) = 1; If C (G ) = 1 then G is (2.01, 2.01 )-low-congesting.
  48. 48. Why a pseudorandom graph fits We use the Nisan pseudorandom generator with polynomial seed length and exponential output. It is well-known that this generator fools any circuit from AC0 The size of a fooled circuit may be exponential since the size of output is exponential. We cannot check the low-congesting property literally, but using circuits for approximate counting we build a circuit C such that: If G is (2, 2 )-low-congesting then C (G ) = 1; If C (G ) = 1 then G is (2.01, 2.01 )-low-congesting. This circuit accepts a random graph with sufficient probability, hence it does the same with a pseudorandom one.
  49. 49. The circuit S h1 (x1 ) h2 (x1 ) ··· hD (x1 ) h1 (x2 ) ··· hD (x2 ) ··· ≡ ··· ≡ ≡ ··· ≡ ··· ≡ ··· Counting circuit Counting circuit 1, if Σinputs > 2.01D 1, if Σinputs > 2.01D ··· 0, if Σinputs < 2D 0, if Σinputs < 2D ∧ ··· Counting circuit 1, if Σinputs < 2ϵK 0, if Σinputs > 2.01ϵK ∧S
  50. 50. How to get Muchnik’s theorem Firstly, we search for a good seed for the generator and fix it. This search needs only polynomial space.
  51. 51. How to get Muchnik’s theorem Firstly, we search for a good seed for the generator and fix it. This search needs only polynomial space. If a is not congested in {x|C s (x|b) < k} then it has a neighbor p outside the clot. This p satisfies all requirements.
  52. 52. How to get Muchnik’s theorem S = {x: C s (x|b) ≤ k} p {0, 1}n , n = |a| {0, 1}k , k = C s (a|b) a
  53. 53. How to get Muchnik’s theorem Firstly, we search for a good seed for the generator and fix it. This search needs only polynomial space. If a is not congested in {x|C s (x|b) < k} then it has a neighbor p outside the clot. This p satisfies all requirements. If a is congested then we repeat the whole construction replacing the relevant sets by the sets of congested vertices in relevant sets.
  54. 54. How to get Muchnik’s theorem Firstly, we search for a good seed for the generator and fix it. This search needs only polynomial space. If a is not congested in {x|C s (x|b) < k} then it has a neighbor p outside the clot. This p satisfies all requirements. If a is congested then we repeat the whole construction replacing the relevant sets by the sets of congested vertices in relevant sets. There may be several iterations but since the upper bound on the size of a relevant set decreases exponentially there is at most linear number of steps, hence all polynomial bounds remain.
  55. 55. The final formulation For any a and b of length n and for any s there exists p of length C s (a|b) + O(log log s + log n) such that: p(b) = a; the computation of p(b) performs in space O(s) + poly(n) and C O(s)+poly(n) (p|a) = O(log log s + log n)
  56. 56. Summary of the technology Take some theorem about Kolmogorov complexity relying on the existence of some combinatorial object
  57. 57. Summary of the technology Take some theorem about Kolmogorov complexity relying on the existence of some combinatorial object Build a constant-depth circuit recognizing this object
  58. 58. Summary of the technology Take some theorem about Kolmogorov complexity relying on the existence of some combinatorial object Build a constant-depth circuit recognizing this object Replace a random object by a pseudorandom one
  59. 59. Summary of the technology Take some theorem about Kolmogorov complexity relying on the existence of some combinatorial object Build a constant-depth circuit recognizing this object Replace a random object by a pseudorandom one Obtain a space-bounded version of the theorem
  60. 60. Summary of the technology Take some theorem about Kolmogorov complexity relying on the existence of some combinatorial object Build a constant-depth circuit recognizing this object Replace a random object by a pseudorandom one Obtain a space-bounded version of the theorem ??????? PROFIT
  61. 61. Thank you! mailto:musatych@gmail.comhttp://musatych.livejournal.com
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