From 08e9a2cac28be880f9d0f1928e5c3cc7618b5450 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 3 Nov 2025 08:16:21 +0100 Subject: [PATCH] [TI] Compact: Add notes on proofs --- .../ti-compact/parts/01_words-alphabets.tex | 4 +++- semester3/ti-compact/ti-compact.pdf | Bin 297506 -> 297885 bytes 2 files changed, 3 insertions(+), 1 deletion(-) diff --git a/semester3/ti-compact/parts/01_words-alphabets.tex b/semester3/ti-compact/parts/01_words-alphabets.tex index 077b0d9..af8c06b 100644 --- a/semester3/ti-compact/parts/01_words-alphabets.tex +++ b/semester3/ti-compact/parts/01_words-alphabets.tex @@ -51,7 +51,9 @@ where the Program doesn't have to compile, i.e. we can describe processes inform \inlinetheorem Kolmogorov-Complexity doesn't depend on programming language. It only differs in constant -\fancydef{Randomness} $x$ random if $K(x) \geq |x|$, thus for $n$, $K(n) \geq \ceil{\log_2(n + 1)} - 1$ +\fancydef{Randomness} $x \in \wordbool$ random if $K(x) \geq |x|$, thus for $n \in \N$, $K(n) \geq \ceil{\log_2(n + 1)} - 1$ \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 $2^n - 1$ words $x$ with $K(x) < n$). diff --git a/semester3/ti-compact/ti-compact.pdf b/semester3/ti-compact/ti-compact.pdf index d0fea01ef74b37f3960f29a2df8f52da6825d66d..da4139f6d6b0fc414412323ce033cbd6ae1b5cb4 100644 GIT binary patch delta 22824 zcmV(&K;gflloFkn60ltfmr*waDu0z#O>>(t5WVv&bQCi|mV`i}$99@FIdrB|-_jfi z$T1B9%11ozukQ*;PMkW29um-MKi+%0tSNd#6n%+4ZlldLSEwLG&U3WgAyJT==13NT 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