From 57d96ac1563b6313a0a47265c255cff66a6d0f04 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sun, 9 Nov 2025 07:26:49 +0100 Subject: [PATCH] [TI] Compact: Improve section on Kolmogorov complexity proofs --- .../ti-compact/parts/01_words-alphabets.tex | 10 ++++++---- semester3/ti-compact/ti-compact.pdf | Bin 331337 -> 332049 bytes 2 files changed, 6 insertions(+), 4 deletions(-) diff --git a/semester3/ti-compact/parts/01_words-alphabets.tex b/semester3/ti-compact/parts/01_words-alphabets.tex index e184f0f..e8e4ad0 100644 --- a/semester3/ti-compact/parts/01_words-alphabets.tex +++ b/semester3/ti-compact/parts/01_words-alphabets.tex @@ -62,11 +62,13 @@ where the Program doesn't have to compile, i.e. we can describe processes inform \stepLabelNumber{theorem} \fancytheorem{Prime number} $\displaystyle \limni \frac{\text{Prime}(n)}{\frac{n}{\ln(n)}}$ -\fhlc{Cyan}{Proofs} Most of the proofs start with defining the number of words of exactly the required length and we can then usually deduce some kind of indirect proof -(using the fact that there are at most $2^k - 1$ words $x$ with $K(x) < k$). +\fhlc{Cyan}{Proofs} Proofs in which we need to show a lower bound for Kolmogorov-Complexity (almost) always work as follows: +Assume for contradiction that there are no words with $K(w) > f$ for all $w \in W$. +We count the number $m$ of words in $W$ and the number $n$ of programs of length $\leq f$ ($f$ being the given, lower bound). +We will have $m - n > 0$, which means, there are more different words than there are Programs with Kolmogorov-Complexity $\leq f$, +which is a contradiction to our assumption. -It is useful to remember the laws of logarithm and the fact that there are $\floor{\frac{n}{k}} + 1$ numbers divisible by $k$ in the set $\{ 0, 1, \ldots, n \}$. -Additionally, the pigeonhole principle can come in very handy +There are $\floor{\frac{n}{k}} + 1$ numbers divisible by $k$ in the set $\{ 0, 1, \ldots, n \}$. \shade{Orange}{Laws of logarithm} \drmvspace diff --git 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