csr2011 june14 15_45_musatov
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Daniil Musatov. Improving the Space-Bounded Version of Muchnik's Conditional Complexity Theorem via ``Naive'' DerandomizationTRANSCRIPT
Improving the Space-Bounded Version ofMuchnik’s Conditional Complexity Theorem via
“Naive” Derandomization
Daniil Musatov1
1Lomonosov Moscow State University,
St. Petersburg, June 14th, 2011
Some disclaimers
I This talk is in some sense a continuation of the previous one
I A similar technique is used to obtain a different result
I Why “naive”? Because we simply replace a random object bya pseudo-random one and it still does the job.
Some disclaimers
I This talk is in some sense a continuation of the previous one
I A similar technique is used to obtain a different result
I Why “naive”? Because we simply replace a random object bya pseudo-random one and it still does the job.
Some disclaimers
I This talk is in some sense a continuation of the previous one
I A similar technique is used to obtain a different result
I Why “naive”?
Because we simply replace a random object bya pseudo-random one and it still does the job.
Some disclaimers
I This talk is in some sense a continuation of the previous one
I A similar technique is used to obtain a different result
I Why “naive”? Because we simply replace a random object bya pseudo-random one and it still does the job.
Kolmogorov complexity
I Kolmogorov complexity C (a|b) of a string a conditional to bis the minimal length of a program p that produces a given b,i.e., p(b) = a
I Space-bounded version: C s(a|b) is the minimal length of aprogram p such that p(b) = a and the computation of p(b)performs in space s.
Kolmogorov complexity
I Kolmogorov complexity C (a|b) of a string a conditional to bis the minimal length of a program p that produces a given b,i.e., p(b) = a
I Space-bounded version: C s(a|b) is the minimal length of aprogram p such that p(b) = a
and the computation of p(b)performs in space s.
Kolmogorov complexity
I Kolmogorov complexity C (a|b) of a string a conditional to bis the minimal length of a program p that produces a given b,i.e., p(b) = a
I Space-bounded version: C s(a|b) is the minimal length of aprogram p such that p(b) = a and the computation of p(b)performs in space s.
Muchnik’s theorem
I Muchnik’s theorem (TCS’2002): For any a and b of length nthere exists p of length C (a|b) + O(log n) such that p(b) = a
and C (p|a) = O(log n).I Space-bounded version (M., Romashchenko, Shen, CSR’2009,
ToCS’2011): For any a and b of length n and for any s thereexists p of length C s(a|b) + O(log3 n) such that:
I p(b) = a;I the computation of p(b) performs in space O(s) + poly(n)I and C poly(n)(p|a) = O(log3 n)
I In current work we get rid of polylogarithmic terms and makethem again logarithmic
Muchnik’s theorem
I Muchnik’s theorem (TCS’2002): For any a and b of length nthere exists p of length C (a|b) + O(log n) such that p(b) = aand C (p|a) = O(log n).
I Space-bounded version (M., Romashchenko, Shen, CSR’2009,ToCS’2011): For any a and b of length n and for any s thereexists p of length C s(a|b) + O(log3 n) such that:
I p(b) = a;I the computation of p(b) performs in space O(s) + poly(n)I and C poly(n)(p|a) = O(log3 n)
I In current work we get rid of polylogarithmic terms and makethem again logarithmic
Muchnik’s theorem
I Muchnik’s theorem (TCS’2002): For any a and b of length nthere exists p of length C (a|b) + O(log n) such that p(b) = aand C (p|a) = O(log n).
I Space-bounded version (M., Romashchenko, Shen, CSR’2009,ToCS’2011): For any a and b of length n and for any s thereexists p of length C s(a|b) + O(log3 n) such that:
I p(b) = a;I the computation of p(b) performs in space O(s) + poly(n)I and C poly(n)(p|a) = O(log3 n)
I In current work we get rid of polylogarithmic terms and makethem again logarithmic
Muchnik’s theorem
I Muchnik’s theorem (TCS’2002): For any a and b of length nthere exists p of length C (a|b) + O(log n) such that p(b) = aand C (p|a) = O(log n).
I Space-bounded version (M., Romashchenko, Shen, CSR’2009,ToCS’2011): For any a and b of length n and for any s thereexists p of length C s(a|b) + O(log3 n) such that:
I p(b) = a;I the computation of p(b) performs in space O(s) + poly(n)I and C poly(n)(p|a) = O(log3 n)
I In current work we get rid of polylogarithmic terms and makethem again logarithmic
Muchnik’s theorem
I Muchnik’s theorem (TCS’2002): For any a and b of length nthere exists p of length C (a|b) + O(log n) such that p(b) = aand C (p|a) = O(log n).
I Space-bounded version (M., Romashchenko, Shen, CSR’2009,ToCS’2011): For any a and b of length n and for any s thereexists p of length C s(a|b) + O(log3 n) such that:
I p(b) = a;I the computation of p(b) performs in space O(s) + poly(n)I and C poly(n)(p|a) = O(log3 n)
I In current work we get rid of polylogarithmic terms and makethem again logarithmic
Proof of Muchnik’s theorem: fingerprints
{0, 1}n, n = |a|
{0, 1}k, k = C(a|b)
a
Proof of Muchnik’s theorem: fingerprints
{0, 1}n, n = |a|
{0, 1}k, k = C(a|b)
p
a
p is a fingerprint of a
For C (p|a) to be small, there should be few fingerprints for each a
Proof of Muchnik’s theorem: fingerprints
{0, 1}n, n = |a|
{0, 1}k, k = C(a|b)
p
a
p is a fingerprint of aFor C (p|a) to be small, there should be few fingerprints for each a
Proof of Muchnik’s theorem: fingerprints
{0, 1}n, n = |a|
{0, 1}k, k = C(a|b)
S = {x:C(x|b) ≤ C(a|b)}
p
a
p is a fingerprint of aFor C (p|a) to be small, there should be few fingerprints for each aFor C (a|p, b) to be small,
there should be few preimages of p in S
Proof of Muchnik’s theorem: fingerprints
{0, 1}n, n = |a|
{0, 1}k, k = C(a|b)
S = {x:C(x|b) ≤ C(a|b)}
p
a
p is a fingerprint of aFor C (p|a) to be small, there should be few fingerprints for each aFor C (a|p, b) to be small, there should be few preimages of p in S
How to find fingerprints
I There is an underlying bipartite graph G = (L,R,E ) with|L| = 2n and |R| = 2k
I Each left-part vertex has constant degree DI The fingerprint p is chosen among neighbors of a
I This guarantees that C (p|a,G ) = O(log D)
I The statement that C (a|p, b,G ) is small for some p followsfrom some enumerable graph property (see later)
I It is proven by the probabilistic method that this property isnon-empty
I Hence, the first graph in the enumeration has small complexity
I Hence, C (p|a) and C (a|p, b) are also small
How to find fingerprints
I There is an underlying bipartite graph G = (L,R,E ) with|L| = 2n and |R| = 2k
I Each left-part vertex has constant degree D
I The fingerprint p is chosen among neighbors of aI This guarantees that C (p|a,G ) = O(log D)
I The statement that C (a|p, b,G ) is small for some p followsfrom some enumerable graph property (see later)
I It is proven by the probabilistic method that this property isnon-empty
I Hence, the first graph in the enumeration has small complexity
I Hence, C (p|a) and C (a|p, b) are also small
How to find fingerprints
I There is an underlying bipartite graph G = (L,R,E ) with|L| = 2n and |R| = 2k
I Each left-part vertex has constant degree DI The fingerprint p is chosen among neighbors of a
I This guarantees that C (p|a,G ) = O(log D)
I The statement that C (a|p, b,G ) is small for some p followsfrom some enumerable graph property (see later)
I It is proven by the probabilistic method that this property isnon-empty
I Hence, the first graph in the enumeration has small complexity
I Hence, C (p|a) and C (a|p, b) are also small
How to find fingerprints
I There is an underlying bipartite graph G = (L,R,E ) with|L| = 2n and |R| = 2k
I Each left-part vertex has constant degree DI The fingerprint p is chosen among neighbors of a
I This guarantees that C (p|a,G ) = O(log D)
I The statement that C (a|p, b,G ) is small for some p followsfrom some enumerable graph property (see later)
I It is proven by the probabilistic method that this property isnon-empty
I Hence, the first graph in the enumeration has small complexity
I Hence, C (p|a) and C (a|p, b) are also small
How to find fingerprints
I There is an underlying bipartite graph G = (L,R,E ) with|L| = 2n and |R| = 2k
I Each left-part vertex has constant degree DI The fingerprint p is chosen among neighbors of a
I This guarantees that C (p|a,G ) = O(log D)
I The statement that C (a|p, b,G ) is small for some p followsfrom some enumerable graph property (see later)
I It is proven by the probabilistic method that this property isnon-empty
I Hence, the first graph in the enumeration has small complexity
I Hence, C (p|a) and C (a|p, b) are also small
How to find fingerprints
I There is an underlying bipartite graph G = (L,R,E ) with|L| = 2n and |R| = 2k
I Each left-part vertex has constant degree DI The fingerprint p is chosen among neighbors of a
I This guarantees that C (p|a,G ) = O(log D)
I The statement that C (a|p, b,G ) is small for some p followsfrom some enumerable graph property (see later)
I It is proven by the probabilistic method that this property isnon-empty
I Hence, the first graph in the enumeration has small complexity
I Hence, C (p|a) and C (a|p, b) are also small
How to find fingerprints
I There is an underlying bipartite graph G = (L,R,E ) with|L| = 2n and |R| = 2k
I Each left-part vertex has constant degree DI The fingerprint p is chosen among neighbors of a
I This guarantees that C (p|a,G ) = O(log D)
I The statement that C (a|p, b,G ) is small for some p followsfrom some enumerable graph property (see later)
I It is proven by the probabilistic method that this property isnon-empty
I Hence, the first graph in the enumeration has small complexity
I Hence, C (p|a) and C (a|p, b) are also small
How to find fingerprints
I There is an underlying bipartite graph G = (L,R,E ) with|L| = 2n and |R| = 2k
I Each left-part vertex has constant degree DI The fingerprint p is chosen among neighbors of a
I This guarantees that C (p|a,G ) = O(log D)
I The statement that C (a|p, b,G ) is small for some p followsfrom some enumerable graph property (see later)
I It is proven by the probabilistic method that this property isnon-empty
I Hence, the first graph in the enumeration has small complexity
I Hence, C (p|a) and C (a|p, b) are also small
What graph properties do we need?
Some graph properties leading to Muchnik’s theorem:
I (Muchnik) Expander-like property
I (MRS) Possibility of online matching
I (MRS) Extractor
I (This paper) “Low-congesting”
What graph properties do we need?
Some graph properties leading to Muchnik’s theorem:
I (Muchnik) Expander-like property
I (MRS) Possibility of online matching
I (MRS) Extractor
I (This paper) “Low-congesting”
What graph properties do we need?
Some graph properties leading to Muchnik’s theorem:
I (Muchnik) Expander-like property
I (MRS) Possibility of online matching
I (MRS) Extractor
I (This paper) “Low-congesting”
What graph properties do we need?
Some graph properties leading to Muchnik’s theorem:
I (Muchnik) Expander-like property
I (MRS) Possibility of online matching
I (MRS) Extractor
I (This paper) “Low-congesting”
The essential property: low-congesting graph
{0, 1}n
each vertex has D neighbors {0, 1}k
S, |S| < 2k
The essential property: low-congesting graph
{0, 1}n
each vertex has D neighbors {0, 1}k
S, |S| < 2k
α-clot
I α-clot for S is the set of all vertices having more than αDneighbors in S .
The essential property: low-congesting graph
{0, 1}n
each vertex has D neighbors {0, 1}k
S, |S| < 2k
α-clot
α-congested vertex
I α-clot for S is the set of all vertices having more than αDneighbors in S .
I A vertex is α-congested if all its neighbors lie in α-clot.
I The set S is (α, β)-low-congested if it contains less than βKα-congested vertices.
The essential property: low-congesting graph
{0, 1}n
each vertex has D neighbors {0, 1}k
S, |S| < 2k
α-clot
α-congested vertex
I α-clot for S is the set of all vertices having more than αDneighbors in S .
I A vertex is α-congested if all its neighbors lie in α-clot.
I The set S is (α, β)-low-congested if it contains less than βKα-congested vertices.
The essential property: low-congesting graph
I α-clot for S is the set of all vertices having more than αDneighbors in S .
I A vertex is α-congested if all its neighbors lie in α-clot.
I The set S is (α, β)-low-congested if it contains less than βKα-congested vertices.
I We call a set relevant if it has the form {x |C s(x |b) < k}.
I We call a graph (α, β)-low-congesting if all relevant sets are(α, β)-low-congested.
How to get a low-congesting graph
I A random graph with certain parameters is an extractor withpositive probability.
I Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k ,ε)-extractor graph any S is (2, 2ε)-low-congested.
I Hence, in an extractor any relevant set is low-congested andthe graph itself is low-congesting.
I But even the description of the graph is exponential in size, sowe cannot find it in polynomial space.
I Central idea: replace a random graph by a pseudorandomone
I To make this idea work, we need:I to prove that a pseudorandom graph is low-congesting with
positive probabilityI to show that the seed this graph is generated from may be
found in polynomial space
How to get a low-congesting graph
I A random graph with certain parameters is an extractor withpositive probability.
I Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k ,ε)-extractor graph any S is (2, 2ε)-low-congested.
I Hence, in an extractor any relevant set is low-congested andthe graph itself is low-congesting.
I But even the description of the graph is exponential in size, sowe cannot find it in polynomial space.
I Central idea: replace a random graph by a pseudorandomone
I To make this idea work, we need:I to prove that a pseudorandom graph is low-congesting with
positive probabilityI to show that the seed this graph is generated from may be
found in polynomial space
How to get a low-congesting graph
I A random graph with certain parameters is an extractor withpositive probability.
I Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k ,ε)-extractor graph any S is (2, 2ε)-low-congested.
I Hence, in an extractor any relevant set is low-congested andthe graph itself is low-congesting.
I But even the description of the graph is exponential in size, sowe cannot find it in polynomial space.
I Central idea: replace a random graph by a pseudorandomone
I To make this idea work, we need:I to prove that a pseudorandom graph is low-congesting with
positive probabilityI to show that the seed this graph is generated from may be
found in polynomial space
How to get a low-congesting graph
I A random graph with certain parameters is an extractor withpositive probability.
I Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k ,ε)-extractor graph any S is (2, 2ε)-low-congested.
I Hence, in an extractor any relevant set is low-congested andthe graph itself is low-congesting.
I But even the description of the graph is exponential in size, sowe cannot find it in polynomial space.
I Central idea: replace a random graph by a pseudorandomone
I To make this idea work, we need:
I to prove that a pseudorandom graph is low-congesting withpositive probability
I to show that the seed this graph is generated from may befound in polynomial space
How to get a low-congesting graph
I A random graph with certain parameters is an extractor withpositive probability.
I Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k ,ε)-extractor graph any S is (2, 2ε)-low-congested.
I Hence, in an extractor any relevant set is low-congested andthe graph itself is low-congesting.
I But even the description of the graph is exponential in size, sowe cannot find it in polynomial space.
I Central idea: replace a random graph by a pseudorandomone
I To make this idea work, we need:I to prove that a pseudorandom graph is low-congesting with
positive probability
I to show that the seed this graph is generated from may befound in polynomial space
How to get a low-congesting graph
I A random graph with certain parameters is an extractor withpositive probability.
I Lemma. (Buhrman, Fortnow, Laplante ’2002) In an (k ,ε)-extractor graph any S is (2, 2ε)-low-congested.
I Hence, in an extractor any relevant set is low-congested andthe graph itself is low-congesting.
I But even the description of the graph is exponential in size, sowe cannot find it in polynomial space.
I Central idea: replace a random graph by a pseudorandomone
I To make this idea work, we need:I to prove that a pseudorandom graph is low-congesting with
positive probabilityI to show that the seed this graph is generated from may be
found in polynomial space
Why a pseudorandom graph fits
I We use the Nisan pseudorandom generator with polynomialseed length and exponential output.
I It is well-known that this generator fools any circuit from AC0
I The size of a fooled circuit may be exponential since the sizeof output is exponential.
I We cannot check the low-congesting property literally, butusing circuits for approximate counting we build a circuit Csuch that:
I If G is (2, 2ε)-low-congesting then C (G ) = 1;I If C (G ) = 1 then G is (2.01, 2.01ε)-low-congesting.
I This circuit accepts a random graph with sufficient probability,hence it does the same with a pseudorandom one.
Why a pseudorandom graph fits
I We use the Nisan pseudorandom generator with polynomialseed length and exponential output.
I It is well-known that this generator fools any circuit from AC0
I The size of a fooled circuit may be exponential since the sizeof output is exponential.
I We cannot check the low-congesting property literally, butusing circuits for approximate counting we build a circuit Csuch that:
I If G is (2, 2ε)-low-congesting then C (G ) = 1;I If C (G ) = 1 then G is (2.01, 2.01ε)-low-congesting.
I This circuit accepts a random graph with sufficient probability,hence it does the same with a pseudorandom one.
Why a pseudorandom graph fits
I We use the Nisan pseudorandom generator with polynomialseed length and exponential output.
I It is well-known that this generator fools any circuit from AC0
I The size of a fooled circuit may be exponential since the sizeof output is exponential.
I We cannot check the low-congesting property literally, butusing circuits for approximate counting we build a circuit Csuch that:
I If G is (2, 2ε)-low-congesting then C (G ) = 1;I If C (G ) = 1 then G is (2.01, 2.01ε)-low-congesting.
I This circuit accepts a random graph with sufficient probability,hence it does the same with a pseudorandom one.
Why a pseudorandom graph fits
I We use the Nisan pseudorandom generator with polynomialseed length and exponential output.
I It is well-known that this generator fools any circuit from AC0
I The size of a fooled circuit may be exponential since the sizeof output is exponential.
I We cannot check the low-congesting property literally, butusing circuits for approximate counting we build a circuit Csuch that:
I If G is (2, 2ε)-low-congesting then C (G ) = 1;I If C (G ) = 1 then G is (2.01, 2.01ε)-low-congesting.
I This circuit accepts a random graph with sufficient probability,hence it does the same with a pseudorandom one.
Why a pseudorandom graph fits
I We use the Nisan pseudorandom generator with polynomialseed length and exponential output.
I It is well-known that this generator fools any circuit from AC0
I The size of a fooled circuit may be exponential since the sizeof output is exponential.
I We cannot check the low-congesting property literally, butusing circuits for approximate counting we build a circuit Csuch that:
I If G is (2, 2ε)-low-congesting then C (G ) = 1;
I If C (G ) = 1 then G is (2.01, 2.01ε)-low-congesting.
I This circuit accepts a random graph with sufficient probability,hence it does the same with a pseudorandom one.
Why a pseudorandom graph fits
I We use the Nisan pseudorandom generator with polynomialseed length and exponential output.
I It is well-known that this generator fools any circuit from AC0
I The size of a fooled circuit may be exponential since the sizeof output is exponential.
I We cannot check the low-congesting property literally, butusing circuits for approximate counting we build a circuit Csuch that:
I If G is (2, 2ε)-low-congesting then C (G ) = 1;I If C (G ) = 1 then G is (2.01, 2.01ε)-low-congesting.
I This circuit accepts a random graph with sufficient probability,hence it does the same with a pseudorandom one.
Why a pseudorandom graph fits
I We use the Nisan pseudorandom generator with polynomialseed length and exponential output.
I It is well-known that this generator fools any circuit from AC0
I The size of a fooled circuit may be exponential since the sizeof output is exponential.
I We cannot check the low-congesting property literally, butusing circuits for approximate counting we build a circuit Csuch that:
I If G is (2, 2ε)-low-congesting then C (G ) = 1;I If C (G ) = 1 then G is (2.01, 2.01ε)-low-congesting.
I This circuit accepts a random graph with sufficient probability,hence it does the same with a pseudorandom one.
The circuit
h1(x1) h2(x1) · · · hD(x1) h1(x2) · · · hD(x2) · · ·
≡ ≡ ≡ ≡ ≡· · · · · · · · · · · ·
Counting circuit1, if Σinputs > 2.01D0, if Σinputs < 2D
Counting circuit1, if Σinputs > 2.01D0, if Σinputs < 2D
· · ·
∧ · · ·
Counting circuit1, if Σinputs < 2ϵK0, if Σinputs > 2.01ϵK
∧S
S
How to get Muchnik’s theorem
I Firstly, we search for a good seed for the generator and fix it.This search needs only polynomial space.
I If a is not congested in {x |C s(x |b) < k} then it has aneighbor p outside the clot.
I This p satisfies all requirements.
How to get Muchnik’s theorem
I Firstly, we search for a good seed for the generator and fix it.This search needs only polynomial space.
I If a is not congested in {x |C s(x |b) < k} then it has aneighbor p outside the clot.
I This p satisfies all requirements.
How to get Muchnik’s theorem
{0, 1}n, n = |a|
{0, 1}k, k = Cs(a|b)
S = {x:Cs(x|b) ≤ k}
p
a
How to get Muchnik’s theorem
I Firstly, we search for a good seed for the generator and fix it.This search needs only polynomial space.
I If a is not congested in {x |C s(x |b) < k} then it has aneighbor p outside the clot.
I This p satisfies all requirements.
I If a is congested then we repeat the whole constructionreplacing the relevant sets by the sets of congested vertices inrelevant sets.
I There may be several iterations but since the upper bound onthe size of a relevant set decreases exponentially there is atmost linear number of steps, hence all polynomial boundsremain.
How to get Muchnik’s theorem
I Firstly, we search for a good seed for the generator and fix it.This search needs only polynomial space.
I If a is not congested in {x |C s(x |b) < k} then it has aneighbor p outside the clot.
I This p satisfies all requirements.
I If a is congested then we repeat the whole constructionreplacing the relevant sets by the sets of congested vertices inrelevant sets.
I There may be several iterations but since the upper bound onthe size of a relevant set decreases exponentially there is atmost linear number of steps, hence all polynomial boundsremain.
The final formulation
For any a and b of length n and for any s there exists p of lengthC s(a|b) + O(log log s + log n) such that:
I p(b) = a;
I the computation of p(b) performs in space O(s) + poly(n)
I and CO(s)+poly(n)(p|a) = O(log log s + log n)
Summary of the technology
I Take some theorem about Kolmogorov complexity relying onthe existence of some combinatorial object
I Build a constant-depth circuit recognizing this object
I Replace a random object by a pseudorandom one
I Obtain a space-bounded version of the theorem
I ???????
I PROFIT
Summary of the technology
I Take some theorem about Kolmogorov complexity relying onthe existence of some combinatorial object
I Build a constant-depth circuit recognizing this object
I Replace a random object by a pseudorandom one
I Obtain a space-bounded version of the theorem
I ???????
I PROFIT
Summary of the technology
I Take some theorem about Kolmogorov complexity relying onthe existence of some combinatorial object
I Build a constant-depth circuit recognizing this object
I Replace a random object by a pseudorandom one
I Obtain a space-bounded version of the theorem
I ???????
I PROFIT
Summary of the technology
I Take some theorem about Kolmogorov complexity relying onthe existence of some combinatorial object
I Build a constant-depth circuit recognizing this object
I Replace a random object by a pseudorandom one
I Obtain a space-bounded version of the theorem
I ???????
I PROFIT
Summary of the technology
I Take some theorem about Kolmogorov complexity relying onthe existence of some combinatorial object
I Build a constant-depth circuit recognizing this object
I Replace a random object by a pseudorandom one
I Obtain a space-bounded version of the theorem
I ???????
I PROFIT
Thank you!mailto:[email protected]
http://musatych.livejournal.com