[TI] Compact: Improve section on Kolmogorov complexity proofs

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

semester3/ti-compact/parts/02_finite-automata.tex

@@ -119,7 +119,8 @@ That is a contradiction, which concludes our proof
 To show that a language needs \textit{at least} $n$ states, use Lemma 3.3 and $n$ words. We thus again do a proof by contradiction:
 \begin{enumerate}
     \item Assume that there exists FA with $|Q| < n$. We now choose $n$ words (as short as possible), as we would for non-regularity proofs using Lemma 3.3 (i.e. find some prefixes).
-        It is usually beneficial to choose prefixes with $|w|$ small (consider just one letter, $\lambda$, then two and more letter words)
+        It is usually beneficial to choose prefixes with $|w|$ small (consider just one letter, $\lambda$, then two and more letter words).
+        An ``easy'' way to find the prefixes is to construct a finite automaton and then picking a prefix from each class
     \item Construct a table for the suffixes using the $n$ chosen words such that one of the words at entry $x_{ij}$ is in the language and the other is not. ($n \times n$ matrix, see below in example)
     \item Conclude that we have reached a contradiction as every field $x_{ij}$ contains a suffix such that one of the two words is in the language and the other one is not.
 \end{enumerate}

semester3/ti-compact/parts/03_turing-machines.tex

@@ -36,6 +36,7 @@ The same ideas as with NFA apply here. The transition function also maps into th
     \delta : (Q - \{ \qacc, \qrej \}) \times \Gamma \rightarrow \cP(Q \times \Gamma \times \{ L, R, N \})
 \end{align*}
 
+\drmvspace
 Again, when constructing a normal TM from a NTM (which is not required at the Midterm, or any other exam for that matter in this course),
 we again apply BFS to the NTM's calculation tree.