diff --git a/semester3/ti-compact/parts/02_finite-automata.tex b/semester3/ti-compact/parts/02_finite-automata.tex index a62b96f..24d5c45 100644 --- a/semester3/ti-compact/parts/02_finite-automata.tex +++ b/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} diff --git a/semester3/ti-compact/ti-compact.pdf b/semester3/ti-compact/ti-compact.pdf index c1912f1..07bea4b 100644 Binary files a/semester3/ti-compact/ti-compact.pdf and b/semester3/ti-compact/ti-compact.pdf differ