Computability Kolmogorov-Chaitin-Solomonoff. Other topics. Homework: Prepare presentations.

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,, 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.

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, –Program that takes description and produces string. This definition requires a TM for even the definition that is the whole string.

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

Claim There exists a constant c, such that for all x, K(x)<= |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)<=2K(x) + K(y)+c

Incompressible strings Definitions: a string x is c-compressible if K(x)<=|x|-c. –If x is not c-compressible, x is incompressible by c. –If x is incompressible by 1, x is incompressible.

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.

K(x) is not computable! [and not because the definition is tied to any one of several models of computing] –See lexity It is a Halting Problem type of proof. lexity

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.

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 slide: printing-program/ printing-program/

c# example May need to force to be in two lines class P{static void Main(){var S="class P{{static void Main(){{var S={1}{0}{1};System.Console.Write(S,S,'{1}');}}}}";System.Console.Write( S,S,'"');}}

finish Breaking the code Discussion

Homework Prepare presentations!