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Not Every Domain of a Plain Decompressor
Contains the Domain of a Prefix-Free One

Mikhail Andreev, Ilya Razenshteyn and Alexander Shen


                   July 23, 2010
Kolmogorov complexity
Kolmogorov complexity




      A measure of information in a given string.
Kolmogorov complexity




      A measure of information in a given string.
      One can easily define a set of Martin-L¨f random
                                               o
      sequences (the algorithmic law of large numbers, the
      algorithmic law of iterated logarithm, etc).
Kolmogorov complexity




      A measure of information in a given string.
      One can easily define a set of Martin-L¨f random
                                               o
      sequences (the algorithmic law of large numbers, the
      algorithmic law of iterated logarithm, etc).
      Analysis of running time of algorithms (Moser-Tardos,
      constructive LLL).
Kolmogorov complexity




      A measure of information in a given string.
      One can easily define a set of Martin-L¨f random
                                               o
      sequences (the algorithmic law of large numbers, the
      algorithmic law of iterated logarithm, etc).
      Analysis of running time of algorithms (Moser-Tardos,
      constructive LLL).
      And so on...
Decompressors
Decompressors

      A decompressor is a computable partial function
      D : {0, 1}∗ → {0, 1}∗ .
Decompressors

      A decompressor is a computable partial function
      D : {0, 1}∗ → {0, 1}∗ .
      Kolmogorov complexity of a string x ∈ {0, 1}∗ with respect
      to a decompressor D: CD (x) := min{|y | | D(y ) = x}.
Decompressors

      A decompressor is a computable partial function
      D : {0, 1}∗ → {0, 1}∗ .
      Kolmogorov complexity of a string x ∈ {0, 1}∗ with respect
      to a decompressor D: CD (x) := min{|y | | D(y ) = x}.
      There exists an optimal decompressor U such that CU is
      minimal up to O(1). C (x) := CU (x) is a plain complexity of
      x.
Decompressors

      A decompressor is a computable partial function
      D : {0, 1}∗ → {0, 1}∗ .
      Kolmogorov complexity of a string x ∈ {0, 1}∗ with respect
      to a decompressor D: CD (x) := min{|y | | D(y ) = x}.
      There exists an optimal decompressor U such that CU is
      minimal up to O(1). C (x) := CU (x) is a plain complexity of
      x.
      A decompressor is called prefix-free if its domain is
      prefix-free.
Decompressors

      A decompressor is a computable partial function
      D : {0, 1}∗ → {0, 1}∗ .
      Kolmogorov complexity of a string x ∈ {0, 1}∗ with respect
      to a decompressor D: CD (x) := min{|y | | D(y ) = x}.
      There exists an optimal decompressor U such that CU is
      minimal up to O(1). C (x) := CU (x) is a plain complexity of
      x.
      A decompressor is called prefix-free if its domain is
      prefix-free.
      There exists an optimal prefix-free decompressor V .
      K (x) := CV (x) is a prefix complexity of x.
Decompressors

      A decompressor is a computable partial function
      D : {0, 1}∗ → {0, 1}∗ .
      Kolmogorov complexity of a string x ∈ {0, 1}∗ with respect
      to a decompressor D: CD (x) := min{|y | | D(y ) = x}.
      There exists an optimal decompressor U such that CU is
      minimal up to O(1). C (x) := CU (x) is a plain complexity of
      x.
      A decompressor is called prefix-free if its domain is
      prefix-free.
      There exists an optimal prefix-free decompressor V .
      K (x) := CV (x) is a prefix complexity of x.
      The difference K (x) − C (x) can be as large as log |x|.
The main question
The main question

      In DLT 2008 paper Calude, Nies, Staiger and Stephan
      characterized (supersets of) domains of optimal (prefix-free)
      decompressors.
The main question

      In DLT 2008 paper Calude, Nies, Staiger and Stephan
      characterized (supersets of) domains of optimal (prefix-free)
      decompressors.
      One of their results: a recursively enumerable set W is a
      superset of the domain of an optimal decompressor iff there is
      a constant c such that
      |A ∩ {0, 1}n | + . . . + |A ∩ {0, 1}n+c | ≥ 2n for every n.
The main question

      In DLT 2008 paper Calude, Nies, Staiger and Stephan
      characterized (supersets of) domains of optimal (prefix-free)
      decompressors.
      One of their results: a recursively enumerable set W is a
      superset of the domain of an optimal decompressor iff there is
      a constant c such that
      |A ∩ {0, 1}n | + . . . + |A ∩ {0, 1}n+c | ≥ 2n for every n.
      The main question: does the domain of every optimal
      decompressor contain the domain of some optimal
      prefix-free decompressor?
The main question

      In DLT 2008 paper Calude, Nies, Staiger and Stephan
      characterized (supersets of) domains of optimal (prefix-free)
      decompressors.
      One of their results: a recursively enumerable set W is a
      superset of the domain of an optimal decompressor iff there is
      a constant c such that
      |A ∩ {0, 1}n | + . . . + |A ∩ {0, 1}n+c | ≥ 2n for every n.
      The main question: does the domain of every optimal
      decompressor contain the domain of some optimal
      prefix-free decompressor?
      We show that the answer is ’NO’.
The main question

      In DLT 2008 paper Calude, Nies, Staiger and Stephan
      characterized (supersets of) domains of optimal (prefix-free)
      decompressors.
      One of their results: a recursively enumerable set W is a
      superset of the domain of an optimal decompressor iff there is
      a constant c such that
      |A ∩ {0, 1}n | + . . . + |A ∩ {0, 1}n+c | ≥ 2n for every n.
      The main question: does the domain of every optimal
      decompressor contain the domain of some optimal
      prefix-free decompressor?
      We show that the answer is ’NO’.
      We build an optimal decompressor U such that for every
      optimal prefix-free decompressor V holds dom V ⊆ dom U.
The construction
The construction
      We show that there exists a set A such that:
        1. A contains the domain of one specific optimal decompressor,
        2. A does not contain the domain of any optimal prefix free
           decompressor.
The construction
      We show that there exists a set A such that:
        1. A contains the domain of one specific optimal decompressor,
        2. A does not contain the domain of any optimal prefix free
           decompressor.
      The first part is easy. It is sufficient to require from A the
      following two properties:
        1. A is recursive,
        2. A is sufficiently dense.
The construction
      We show that there exists a set A such that:
        1. A contains the domain of one specific optimal decompressor,
        2. A does not contain the domain of any optimal prefix free
           decompressor.
      The first part is easy. It is sufficient to require from A the
      following two properties:
        1. A is recursive,
        2. A is sufficiently dense.
      By sufficiently dense we mean that for every n
      |A ∩ {0, 1}n | ≥ ε · 2n .
The construction
      We show that there exists a set A such that:
        1. A contains the domain of one specific optimal decompressor,
        2. A does not contain the domain of any optimal prefix free
           decompressor.
      The first part is easy. It is sufficient to require from A the
      following two properties:
        1. A is recursive,
        2. A is sufficiently dense.
      By sufficiently dense we mean that for every n
      |A ∩ {0, 1}n | ≥ ε · 2n .
      An easy case of a result from [CNSS]. Consider an arbitrary
      optimal decompressor D. A is recursive, so there exists a
      computable injective mapping i : {0, 1}∗ → {0, 1}∗ such that:
        1. i(x) ∈ A for every x,
        2. |i(x)| ≤ |x| + c, where c is a fixed constant (one can actually
           take c := log 1/ε ).
The construction
      We show that there exists a set A such that:
        1. A contains the domain of one specific optimal decompressor,
        2. A does not contain the domain of any optimal prefix free
           decompressor.
      The first part is easy. It is sufficient to require from A the
      following two properties:
        1. A is recursive,
        2. A is sufficiently dense.
      By sufficiently dense we mean that for every n
      |A ∩ {0, 1}n | ≥ ε · 2n .
      An easy case of a result from [CNSS]. Consider an arbitrary
      optimal decompressor D. A is recursive, so there exists a
      computable injective mapping i : {0, 1}∗ → {0, 1}∗ such that:
        1. i(x) ∈ A for every x,
        2. |i(x)| ≤ |x| + c, where c is a fixed constant (one can actually
           take c := log 1/ε ).
      D1 (i(x)) := D(x).
The construction
The construction
      It remains to build a recursive set A with the following
      properties:
        1. A is sufficiently dense,
        2. A does not contain the domain of any optimal prefix-free
           decompressor.
The construction
      It remains to build a recursive set A with the following
      properties:
        1. A is sufficiently dense,
        2. A does not contain the domain of any optimal prefix-free
           decompressor.
      A will be, in some sense, a universal set. For every n consider
      strings of length n in the set A. They determine some subset
      in Cantor space, which is open-closed.
      What is needed for A: every open-closed subset of Cantor
      space of measure at least 1/3 is represented at some level and
      even at many subsequent levels
The construction
      It remains to build a recursive set A with the following
      properties:
        1. A is sufficiently dense,
        2. A does not contain the domain of any optimal prefix-free
           decompressor.
      A will be, in some sense, a universal set. For every n consider
      strings of length n in the set A. They determine some subset
      in Cantor space, which is open-closed.
      What is needed for A: every open-closed subset of Cantor
      space of measure at least 1/3 is represented at some level and
      even at many subsequent levels
      Infinite binary tree ↔ {0, 1}∗ ↔ cylinders in {0, 1}ω
The construction
      It remains to build a recursive set A with the following
      properties:
        1. A is sufficiently dense,
        2. A does not contain the domain of any optimal prefix-free
           decompressor.
      A will be, in some sense, a universal set. For every n consider
      strings of length n in the set A. They determine some subset
      in Cantor space, which is open-closed.
      What is needed for A: every open-closed subset of Cantor
      space of measure at least 1/3 is represented at some level and
      even at many subsequent levels
      Infinite binary tree ↔ {0, 1}∗ ↔ cylinders in {0, 1}ω
      Ωx = {α ∈ {0, 1}ω | α begins with x}
The construction
      It remains to build a recursive set A with the following
      properties:
        1. A is sufficiently dense,
        2. A does not contain the domain of any optimal prefix-free
           decompressor.
      A will be, in some sense, a universal set. For every n consider
      strings of length n in the set A. They determine some subset
      in Cantor space, which is open-closed.
      What is needed for A: every open-closed subset of Cantor
      space of measure at least 1/3 is represented at some level and
      even at many subsequent levels
      Infinite binary tree ↔ {0, 1}∗ ↔ cylinders in {0, 1}ω
      Ωx = {α ∈ {0, 1}ω | α begins with x}
      P ⊆ {0, 1}ω is basic if P = n Ωxi .
                                      i=1
The construction
      It remains to build a recursive set A with the following
      properties:
        1. A is sufficiently dense,
        2. A does not contain the domain of any optimal prefix-free
           decompressor.
      A will be, in some sense, a universal set. For every n consider
      strings of length n in the set A. They determine some subset
      in Cantor space, which is open-closed.
      What is needed for A: every open-closed subset of Cantor
      space of measure at least 1/3 is represented at some level and
      even at many subsequent levels
      Infinite binary tree ↔ {0, 1}∗ ↔ cylinders in {0, 1}ω
      Ωx = {α ∈ {0, 1}ω | α begins with x}
      P ⊆ {0, 1}ω is basic if P = n Ωxi .
                                      i=1
      A ⊆ {0, 1}∗ represents P at level n if P = x∈A∩{0,1}n Ωx .
The construction
      It remains to build a recursive set A with the following
      properties:
        1. A is sufficiently dense,
        2. A does not contain the domain of any optimal prefix-free
           decompressor.
      A will be, in some sense, a universal set. For every n consider
      strings of length n in the set A. They determine some subset
      in Cantor space, which is open-closed.
      What is needed for A: every open-closed subset of Cantor
      space of measure at least 1/3 is represented at some level and
      even at many subsequent levels
      Infinite binary tree ↔ {0, 1}∗ ↔ cylinders in {0, 1}ω
      Ωx = {α ∈ {0, 1}ω | α begins with x}
      P ⊆ {0, 1}ω is basic if P = n Ωxi .
                                      i=1
      A ⊆ {0, 1}∗ represents P at level n if P = x∈A∩{0,1}n Ωx .
      Construction of A: for every basic set P of measure at least
      1/3 there are infinitely many n such that A represents P at
      levels n, n + 1, . . . , 2n.
Two games
Two games



     Lemma: if ni is a computable sequence such that
         −ni ≤ 1, then K (i) ≤ n + O(1).
       i2                       i
Two games



     Lemma: if ni is a computable sequence such that
         −ni ≤ 1, then K (i) ≤ n + O(1).
       i2                       i
     ’Memory allocation’ game:
         Alice: ni such that i 2−ni ≤ 1 (one by one).
         Bob: xi such that |xi | = ni and {xi } is prefix-free set (responds
         on-line).
         Bob wins if he is able to allocate desired strings.
     Theorem: Bob wins.
Two games
Two games


     The modified game:
         Bob: ε > 0.
         Alice: makes at most 2/3 strings of each length forbidden.
         Alice: ni such that i 2−ni ≤ ε (one by one).
         Bob: xi such that |xi | = ni and {xi } is prefix-free set.
         Moreover, xi must not be forbidden (responds on-line).
         Bob wins if he is able to allocate desired strings.
     Theorem: Alice wins.
Two games


     The modified game:
         Bob: ε > 0.
         Alice: makes at most 2/3 strings of each length forbidden.
         Alice: ni such that i 2−ni ≤ ε (one by one).
         Bob: xi such that |xi | = ni and {xi } is prefix-free set.
         Moreover, xi must not be forbidden (responds on-line).
         Bob wins if he is able to allocate desired strings.
     Theorem: Alice wins.
     Technically, we must consider more complicated game,
     because complexity is defined up to a constant.
Two games


     The modified game:
         Bob: ε > 0.
         Alice: makes at most 2/3 strings of each length forbidden.
         Alice: ni such that i 2−ni ≤ ε (one by one).
         Bob: xi such that |xi | = ni and {xi } is prefix-free set.
         Moreover, xi must not be forbidden (responds on-line).
         Bob wins if he is able to allocate desired strings.
     Theorem: Alice wins.
     Technically, we must consider more complicated game,
     because complexity is defined up to a constant.
     The idea of a proof is the following: Alice wins, and her
     winning set is more or less A.
Thank you for your attention.

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CiE 2010 talk

  • 1. Not Every Domain of a Plain Decompressor Contains the Domain of a Prefix-Free One Mikhail Andreev, Ilya Razenshteyn and Alexander Shen July 23, 2010
  • 3. Kolmogorov complexity A measure of information in a given string.
  • 4. Kolmogorov complexity A measure of information in a given string. One can easily define a set of Martin-L¨f random o sequences (the algorithmic law of large numbers, the algorithmic law of iterated logarithm, etc).
  • 5. Kolmogorov complexity A measure of information in a given string. One can easily define a set of Martin-L¨f random o sequences (the algorithmic law of large numbers, the algorithmic law of iterated logarithm, etc). Analysis of running time of algorithms (Moser-Tardos, constructive LLL).
  • 6. Kolmogorov complexity A measure of information in a given string. One can easily define a set of Martin-L¨f random o sequences (the algorithmic law of large numbers, the algorithmic law of iterated logarithm, etc). Analysis of running time of algorithms (Moser-Tardos, constructive LLL). And so on...
  • 8. Decompressors A decompressor is a computable partial function D : {0, 1}∗ → {0, 1}∗ .
  • 9. Decompressors A decompressor is a computable partial function D : {0, 1}∗ → {0, 1}∗ . Kolmogorov complexity of a string x ∈ {0, 1}∗ with respect to a decompressor D: CD (x) := min{|y | | D(y ) = x}.
  • 10. Decompressors A decompressor is a computable partial function D : {0, 1}∗ → {0, 1}∗ . Kolmogorov complexity of a string x ∈ {0, 1}∗ with respect to a decompressor D: CD (x) := min{|y | | D(y ) = x}. There exists an optimal decompressor U such that CU is minimal up to O(1). C (x) := CU (x) is a plain complexity of x.
  • 11. Decompressors A decompressor is a computable partial function D : {0, 1}∗ → {0, 1}∗ . Kolmogorov complexity of a string x ∈ {0, 1}∗ with respect to a decompressor D: CD (x) := min{|y | | D(y ) = x}. There exists an optimal decompressor U such that CU is minimal up to O(1). C (x) := CU (x) is a plain complexity of x. A decompressor is called prefix-free if its domain is prefix-free.
  • 12. Decompressors A decompressor is a computable partial function D : {0, 1}∗ → {0, 1}∗ . Kolmogorov complexity of a string x ∈ {0, 1}∗ with respect to a decompressor D: CD (x) := min{|y | | D(y ) = x}. There exists an optimal decompressor U such that CU is minimal up to O(1). C (x) := CU (x) is a plain complexity of x. A decompressor is called prefix-free if its domain is prefix-free. There exists an optimal prefix-free decompressor V . K (x) := CV (x) is a prefix complexity of x.
  • 13. Decompressors A decompressor is a computable partial function D : {0, 1}∗ → {0, 1}∗ . Kolmogorov complexity of a string x ∈ {0, 1}∗ with respect to a decompressor D: CD (x) := min{|y | | D(y ) = x}. There exists an optimal decompressor U such that CU is minimal up to O(1). C (x) := CU (x) is a plain complexity of x. A decompressor is called prefix-free if its domain is prefix-free. There exists an optimal prefix-free decompressor V . K (x) := CV (x) is a prefix complexity of x. The difference K (x) − C (x) can be as large as log |x|.
  • 15. The main question In DLT 2008 paper Calude, Nies, Staiger and Stephan characterized (supersets of) domains of optimal (prefix-free) decompressors.
  • 16. The main question In DLT 2008 paper Calude, Nies, Staiger and Stephan characterized (supersets of) domains of optimal (prefix-free) decompressors. One of their results: a recursively enumerable set W is a superset of the domain of an optimal decompressor iff there is a constant c such that |A ∩ {0, 1}n | + . . . + |A ∩ {0, 1}n+c | ≥ 2n for every n.
  • 17. The main question In DLT 2008 paper Calude, Nies, Staiger and Stephan characterized (supersets of) domains of optimal (prefix-free) decompressors. One of their results: a recursively enumerable set W is a superset of the domain of an optimal decompressor iff there is a constant c such that |A ∩ {0, 1}n | + . . . + |A ∩ {0, 1}n+c | ≥ 2n for every n. The main question: does the domain of every optimal decompressor contain the domain of some optimal prefix-free decompressor?
  • 18. The main question In DLT 2008 paper Calude, Nies, Staiger and Stephan characterized (supersets of) domains of optimal (prefix-free) decompressors. One of their results: a recursively enumerable set W is a superset of the domain of an optimal decompressor iff there is a constant c such that |A ∩ {0, 1}n | + . . . + |A ∩ {0, 1}n+c | ≥ 2n for every n. The main question: does the domain of every optimal decompressor contain the domain of some optimal prefix-free decompressor? We show that the answer is ’NO’.
  • 19. The main question In DLT 2008 paper Calude, Nies, Staiger and Stephan characterized (supersets of) domains of optimal (prefix-free) decompressors. One of their results: a recursively enumerable set W is a superset of the domain of an optimal decompressor iff there is a constant c such that |A ∩ {0, 1}n | + . . . + |A ∩ {0, 1}n+c | ≥ 2n for every n. The main question: does the domain of every optimal decompressor contain the domain of some optimal prefix-free decompressor? We show that the answer is ’NO’. We build an optimal decompressor U such that for every optimal prefix-free decompressor V holds dom V ⊆ dom U.
  • 21. The construction We show that there exists a set A such that: 1. A contains the domain of one specific optimal decompressor, 2. A does not contain the domain of any optimal prefix free decompressor.
  • 22. The construction We show that there exists a set A such that: 1. A contains the domain of one specific optimal decompressor, 2. A does not contain the domain of any optimal prefix free decompressor. The first part is easy. It is sufficient to require from A the following two properties: 1. A is recursive, 2. A is sufficiently dense.
  • 23. The construction We show that there exists a set A such that: 1. A contains the domain of one specific optimal decompressor, 2. A does not contain the domain of any optimal prefix free decompressor. The first part is easy. It is sufficient to require from A the following two properties: 1. A is recursive, 2. A is sufficiently dense. By sufficiently dense we mean that for every n |A ∩ {0, 1}n | ≥ ε · 2n .
  • 24. The construction We show that there exists a set A such that: 1. A contains the domain of one specific optimal decompressor, 2. A does not contain the domain of any optimal prefix free decompressor. The first part is easy. It is sufficient to require from A the following two properties: 1. A is recursive, 2. A is sufficiently dense. By sufficiently dense we mean that for every n |A ∩ {0, 1}n | ≥ ε · 2n . An easy case of a result from [CNSS]. Consider an arbitrary optimal decompressor D. A is recursive, so there exists a computable injective mapping i : {0, 1}∗ → {0, 1}∗ such that: 1. i(x) ∈ A for every x, 2. |i(x)| ≤ |x| + c, where c is a fixed constant (one can actually take c := log 1/ε ).
  • 25. The construction We show that there exists a set A such that: 1. A contains the domain of one specific optimal decompressor, 2. A does not contain the domain of any optimal prefix free decompressor. The first part is easy. It is sufficient to require from A the following two properties: 1. A is recursive, 2. A is sufficiently dense. By sufficiently dense we mean that for every n |A ∩ {0, 1}n | ≥ ε · 2n . An easy case of a result from [CNSS]. Consider an arbitrary optimal decompressor D. A is recursive, so there exists a computable injective mapping i : {0, 1}∗ → {0, 1}∗ such that: 1. i(x) ∈ A for every x, 2. |i(x)| ≤ |x| + c, where c is a fixed constant (one can actually take c := log 1/ε ). D1 (i(x)) := D(x).
  • 27. The construction It remains to build a recursive set A with the following properties: 1. A is sufficiently dense, 2. A does not contain the domain of any optimal prefix-free decompressor.
  • 28. The construction It remains to build a recursive set A with the following properties: 1. A is sufficiently dense, 2. A does not contain the domain of any optimal prefix-free decompressor. A will be, in some sense, a universal set. For every n consider strings of length n in the set A. They determine some subset in Cantor space, which is open-closed. What is needed for A: every open-closed subset of Cantor space of measure at least 1/3 is represented at some level and even at many subsequent levels
  • 29. The construction It remains to build a recursive set A with the following properties: 1. A is sufficiently dense, 2. A does not contain the domain of any optimal prefix-free decompressor. A will be, in some sense, a universal set. For every n consider strings of length n in the set A. They determine some subset in Cantor space, which is open-closed. What is needed for A: every open-closed subset of Cantor space of measure at least 1/3 is represented at some level and even at many subsequent levels Infinite binary tree ↔ {0, 1}∗ ↔ cylinders in {0, 1}ω
  • 30. The construction It remains to build a recursive set A with the following properties: 1. A is sufficiently dense, 2. A does not contain the domain of any optimal prefix-free decompressor. A will be, in some sense, a universal set. For every n consider strings of length n in the set A. They determine some subset in Cantor space, which is open-closed. What is needed for A: every open-closed subset of Cantor space of measure at least 1/3 is represented at some level and even at many subsequent levels Infinite binary tree ↔ {0, 1}∗ ↔ cylinders in {0, 1}ω Ωx = {α ∈ {0, 1}ω | α begins with x}
  • 31. The construction It remains to build a recursive set A with the following properties: 1. A is sufficiently dense, 2. A does not contain the domain of any optimal prefix-free decompressor. A will be, in some sense, a universal set. For every n consider strings of length n in the set A. They determine some subset in Cantor space, which is open-closed. What is needed for A: every open-closed subset of Cantor space of measure at least 1/3 is represented at some level and even at many subsequent levels Infinite binary tree ↔ {0, 1}∗ ↔ cylinders in {0, 1}ω Ωx = {α ∈ {0, 1}ω | α begins with x} P ⊆ {0, 1}ω is basic if P = n Ωxi . i=1
  • 32. The construction It remains to build a recursive set A with the following properties: 1. A is sufficiently dense, 2. A does not contain the domain of any optimal prefix-free decompressor. A will be, in some sense, a universal set. For every n consider strings of length n in the set A. They determine some subset in Cantor space, which is open-closed. What is needed for A: every open-closed subset of Cantor space of measure at least 1/3 is represented at some level and even at many subsequent levels Infinite binary tree ↔ {0, 1}∗ ↔ cylinders in {0, 1}ω Ωx = {α ∈ {0, 1}ω | α begins with x} P ⊆ {0, 1}ω is basic if P = n Ωxi . i=1 A ⊆ {0, 1}∗ represents P at level n if P = x∈A∩{0,1}n Ωx .
  • 33. The construction It remains to build a recursive set A with the following properties: 1. A is sufficiently dense, 2. A does not contain the domain of any optimal prefix-free decompressor. A will be, in some sense, a universal set. For every n consider strings of length n in the set A. They determine some subset in Cantor space, which is open-closed. What is needed for A: every open-closed subset of Cantor space of measure at least 1/3 is represented at some level and even at many subsequent levels Infinite binary tree ↔ {0, 1}∗ ↔ cylinders in {0, 1}ω Ωx = {α ∈ {0, 1}ω | α begins with x} P ⊆ {0, 1}ω is basic if P = n Ωxi . i=1 A ⊆ {0, 1}∗ represents P at level n if P = x∈A∩{0,1}n Ωx . Construction of A: for every basic set P of measure at least 1/3 there are infinitely many n such that A represents P at levels n, n + 1, . . . , 2n.
  • 35. Two games Lemma: if ni is a computable sequence such that −ni ≤ 1, then K (i) ≤ n + O(1). i2 i
  • 36. Two games Lemma: if ni is a computable sequence such that −ni ≤ 1, then K (i) ≤ n + O(1). i2 i ’Memory allocation’ game: Alice: ni such that i 2−ni ≤ 1 (one by one). Bob: xi such that |xi | = ni and {xi } is prefix-free set (responds on-line). Bob wins if he is able to allocate desired strings. Theorem: Bob wins.
  • 38. Two games The modified game: Bob: ε > 0. Alice: makes at most 2/3 strings of each length forbidden. Alice: ni such that i 2−ni ≤ ε (one by one). Bob: xi such that |xi | = ni and {xi } is prefix-free set. Moreover, xi must not be forbidden (responds on-line). Bob wins if he is able to allocate desired strings. Theorem: Alice wins.
  • 39. Two games The modified game: Bob: ε > 0. Alice: makes at most 2/3 strings of each length forbidden. Alice: ni such that i 2−ni ≤ ε (one by one). Bob: xi such that |xi | = ni and {xi } is prefix-free set. Moreover, xi must not be forbidden (responds on-line). Bob wins if he is able to allocate desired strings. Theorem: Alice wins. Technically, we must consider more complicated game, because complexity is defined up to a constant.
  • 40. Two games The modified game: Bob: ε > 0. Alice: makes at most 2/3 strings of each length forbidden. Alice: ni such that i 2−ni ≤ ε (one by one). Bob: xi such that |xi | = ni and {xi } is prefix-free set. Moreover, xi must not be forbidden (responds on-line). Bob wins if he is able to allocate desired strings. Theorem: Alice wins. Technically, we must consider more complicated game, because complexity is defined up to a constant. The idea of a proof is the following: Alice wins, and her winning set is more or less A.
  • 41. Thank you for your attention.