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I have hard time understanding the proof by contradiction for the claim "$L=\{x : K(x) \ge |x| \}$" is undecidable ".

The proof is as follows :

M' = " On input $n$

  1. Enumerate over all $n$-bit strings $x$ in lexicographical order
  2. Simulate M on each $x$, where $M$ is the Turing machine that decides $L$.
  3. Output first $x$ which $M$ accepts. "

Since TM $M'$ produced incompressible using only $O(\log n)$ to specify $n$, we can compress incompressible strings which is a contradiction.

I understood the $M'$ construction. However, I do not understand where exactly is the contradiction happening? According to me, $M'$ outputs $x$ which is in-compressible(ensured by TM $M$) but how does it is also compressible at same time?

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Consider the word $w_n$ that is the output of $M'$ given input $n$.

Note that description of $w_n$ is the description of $M'$, whose length is some constant $c$, plus the description of $n$, whose length is $O(\log n)$ since we can express $n$ in the binary representation. So $K(w_n)\le c + O(\log n)$. If $n$ is large enough, we get $$K(w_n)\lt n = |w_n|.$$

That is a contradiction since, as you have noted, $w_n$ should be incompressible.

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  • $\begingroup$ Why is the description of wn is the description of M' ? Also when you assumed large n, why did K(wn) < n = |wn| hold? $\endgroup$
    – Prithi
    Jul 7 '19 at 6:02
  • $\begingroup$ The description of $w_n$, "the output of $M'$ given input $n$" is the sum of the description of $m'$, the description of $n$ and some string of constant length. $\endgroup$
    – John L.
    Jul 7 '19 at 7:45
  • $\begingroup$ $c+O(\log n)$ is smaller than $n$ if $n$ is large enough since $\log n= o(n)$. $\endgroup$
    – John L.
    Jul 7 '19 at 7:46
  • $\begingroup$ Can we say that string is compressable for smaller number of $n$ (then $O(logn)$ is needed to encode), for large values of $n$, we will need whole output $w_n$ ? Also this supports the intuition because I read somewhere that the K-complexity of majority of the strings is almost themselves. i.e $K(x)=|x|+c$ and large values are far more in number than smaller values. $\endgroup$
    – Prithi
    Jul 7 '19 at 8:08
  • $\begingroup$ Are you asking a new question or are you still on "how to understand that proof"? For what "we will need whole output $w_n$"? $\endgroup$
    – John L.
    Jul 7 '19 at 8:20

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