Computability

1
Computability

• Kolmogorov-Chaitin-Solomonoff. Other topics.
• Homework Prepare presentations.

2
Information

• Shannon definition series of (binary) choices.
  Information in measured in bits.
• Computability definition Let x be a binary
  string. A minimal description of x, d(x) is the
  shortest string,ltM,wgt, where Turing Machine M on
  input w halts with x on the tape. The descriptive
  complexity (aka Kolmogorov or Kolmogorov-Chaitin
  complexity) is K(x) d(x)
• the length of this shortest string.
• Note there may be more than one.

3
Informally

• Suppose we have a string consisting of 100 groups
  of '0110'. This is a description and it seems
  like it would be shorter than writing out the
  whole string.
• The formal description includes the TM (or
  program) that knows what to do with 100 groups of
  something. The full minimal description
  consists of this TM plus an encoding of 100
  groups, 0110.
• Program that takes description and produces
  string.
• This definition requires a TM for even the
  definition that is the whole string.

4
Common example

• Mandelbrot fractals are very intricate patterns
  and yet can be produced (re-produced?) by simple
  computer programs.

5
Claim

• There exists a constant c, such that for all x,
  K(x)lt x c.
• intuitively take a fixed TM that halts
  immediately. Then it halts with the string x on
  it.
• There exists a constant c, such that for all x
  and y, K(xy)lt2K(x) K(y)c

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Incompressible strings

• Definitions a string x is c-compressible if
  K(x)ltx-c.
• If x is not c-compressible, x is incompressible
  by c.
• If x is incompressible by 1, x is incompressible.

7
Incompressible strings exist!

• The number of strings of length n is greater than
  the number of descriptions of length less than n.
  So, some string of length n is not described by
  any description of length less than n.

8
K(x) is not computable!

• and not because the definition is tied to any
  one of several models of computing
• See http//en.wikipedia.org/wiki/Kolmogorov_comple
  xity It is a Halting Problem type of proof.

9
Berry Paradox

• Let n be the smallest positive integer that
  cannot be defined in fewer than twenty English
  words.
• oops!
• Relates also to Godel incompleteness results.

10
Recursion Theorem

• also called fixed point theorem.
• There exists a program that prints itself.
• General strategy create constant that represents
  working part of program and write program that
  prints out constant.
• See Logo example in Shai lectures.
• c example next slidehttp//igoro.com/archive/ho
  w-to-write-a-self-printing-program/

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c example

• May need to force to be in two lines
• class Pstatic void Main()var S"class Pstatic
  void Main()var S101System.Console
• .Write(S,S,'1')"System.Console.Write(S,S,'
  "')

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finish Breaking the code

• Discussion

13
Homework

• Prepare presentations!