From 3e26c08ca2911bda890a79d74f5c34eb4b41f96d Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sat, 1 Nov 2025 08:08:57 +0100 Subject: [PATCH] [TI] Compact: Add section on minimum states proofs --- .../ti-compact/parts/02_finite-automata.tex | 53 ++++++++++++++++-- semester3/ti-compact/ti-compact.pdf | Bin 290391 -> 297284 bytes 2 files changed, 49 insertions(+), 4 deletions(-) diff --git a/semester3/ti-compact/parts/02_finite-automata.tex b/semester3/ti-compact/parts/02_finite-automata.tex index 5ffde71..679a9cb 100644 --- a/semester3/ti-compact/parts/02_finite-automata.tex +++ b/semester3/ti-compact/parts/02_finite-automata.tex @@ -51,7 +51,15 @@ Below is an informal guide as to how to do proofs using each of the methods and For all of them start by assuming that $L$ is regular. \fhlc{Cyan}{Lemma 3.3} -\rmvspace +\setLabelNumber{lemma}{3} +\begin{lemma}[]{Unterscheidung von Wörtern} + Sei $A$ ein EA über $\Sigma$ und $x \neq y \in \Sigma^*$ so dass $\hdelta_A (q_0, x) = \hdelta(q_0, y)$. + Dann existiert für jedes $z \in \Sigma^*$ ein $r \in Q$, so dass $xz, yz \in \class[r]$, also gilt insbesondere + \rmvspace + \begin{align*} + xz \in L(A) \Longleftrightarrow yz \in L(A) + \end{align*} +\end{lemma} \begin{enumerate}[noitemsep] \item Pick a FA $A$ over $\Sigma$ and say that $L(A) = L$ \item Pick $|Q| + 1$ words $x$ such that $xy = w \in L$ with $|y| > 0$. @@ -65,7 +73,18 @@ That is a contradiction, which concludes our proof \fhlc{Cyan}{Pumping Lemma} -\rmvspace +\begin{lemma}[]{Pumping-Lemma für reguläre Sprachen} + Sei $L$ regulär. Dann existiert eine Konstante $n_0 \in \N$, so dass sich jedes Wort $w \in \word$ mit $|w| \geq n_0$ in $w = yxz$ zerlegen lässt, wobei + \rmvspace + \begin{multicols}{2} + \begin{enumerate}[label=\textit{(\roman*)}] + \item $|yx| \leq n_0$ + \item $|x| \geq 1$ + \item Für $X = \{ yx^kz \divides k\in \N \}$ \textit{entweder} $X \subseteq L$ oder $X \cap L = \emptyset$ gilt + \end{enumerate} + \end{multicols} +\end{lemma} + \begin{enumerate}[noitemsep] \item State that according to Lemma 3.4 there exists a constant $n_0$ such that $|w| \geq n_0$. \item Choose a word $w \in L$ that is sufficiently long to enable a sensible decomposition for the next step. @@ -94,6 +113,34 @@ That is a contradiction, which concludes our proof \end{enumerate} +\newpage +\fhlc{Cyan}{Minimum number of states} +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) + \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} +\inlineex Let $L = \{ x1y \divides x \in \wordbool, y \in \{ 0, 1 \}^2 \}$. Show that any FA that accepts $L$ needs at least four states. + +Assume for contradiction that there exists EA $A = (Q, \alphabetbool, \delta_A, q_0, F)$ with $|Q| < 4$. +Let's take the $4$ words $00, 01, 10, 11$. Then according to Lemma 3.3, there needs to exist a $z$ such that $xz \in L(A) \Longleftrightarrow yz \in L(A)$ +with $\hdelta_A(q_0, x) = \hdelta_A(q_0, y)$ for $x, y \in \{ 00, 01, 10, 11 \}$. + +This however is a contradiction, as we can find a $z$ for each of the pairs $(x, y)$, such that $xz \in L(A)$, but $yz \notin L(A)$. +See for reference the below table (it contains suffixes $z$ fulfilling prior condition): + +\begin{tables}{c|cccc}{ & $00$ & $01$ & $10$ & $11$} + $00$ & - & $00$ & $0$ & $0$ \\ + $01$ & & - & $0$ & $0$ \\ + $10$ & & & - & $00$ \\ + $11$ & & & & - \\ +\end{tables} +Thus, all four words have to lay in pairwise distinct states and we thus need at least $4$ states to detect this language. + + + + \subsection{Non-determinism} The most notable differences between deterministic and non-deterministic FA is that the transition function maps is different: $\delta: Q \times \Sigma \rightarrow \mathcal{P}(Q)$. I.e., there can be any number of transitions for one symbol from $\Sigma$ from each state. @@ -115,5 +162,3 @@ For each state, the set of states $P = \hdelta(q_0, z)$ for $|z| = n$ represents Correspondingly, we add new states if there is no other state that is in the same branch of the calculation tree $\mathcal{B}_M(x)$. 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