From 905b0335fcee655bb1e9c048afa4249084f08313 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 24 Oct 2025 17:28:32 +0200 Subject: [PATCH] [TI] Finish up last parts of non-deterministic EA --- .../02_finite-automata/03_non-determinism.tex | 7 ++++++- semester3/ti/ti-summary.pdf | Bin 958192 -> 958838 bytes 2 files changed, 6 insertions(+), 1 deletion(-) diff --git a/semester3/ti/parts/02_finite-automata/03_non-determinism.tex b/semester3/ti/parts/02_finite-automata/03_non-determinism.tex index ff87845..25247a0 100644 --- a/semester3/ti/parts/02_finite-automata/03_non-determinism.tex +++ b/semester3/ti/parts/02_finite-automata/03_non-determinism.tex @@ -98,8 +98,13 @@ Wir sagen, dass zwei Automaten \bi{äquivalent} sind, falls $L(A) = L(B)$. Eine Folge von Satz 3.2 ist eben, dass $\mathcal{L}_{\text{EA}} = \mathcal{L}_{\text{NEA}}$, also sind die EA genau so stark wie die NEA im Bezug auf die Sprachakzeptierung. Was hingegen ein Problem sein kann, ist dass die durch die Potenzmengenkonstruktion erzeugten Automaten (exponentiell) grösser sind als die NEA. -% Currently on Page 98, below Task 3.22 half way in the text block +Es gibt gewisse NEA, bei welchen man bei der Simulation des Nichtdeterminismus durch Determinismus unausweichlich in exponentiell grösseren EA resultiert. +Man kann beweisen (siehe Seiten 83 und 84 mit Abbildung 3.19 im Buch (= Seiten 98 \& 99 im PDF)), dass man die Potenzmengenkonstruktion nicht allgemein verbessern kann. +\inlinelemma Für alle $k \in \N - \{ 0 \}$ muss jeder EA, der $L_k = \{ x1y \divides x \in \wordbool, y \in (\alphabetbool)^{k - 1} \}$ akzeptiert, mindestens $2^k$ Zustände haben. + +% FIXME: Verify with TA that this is correct too +% Else: Note an example from the worked example, TA's approach from the slides or from the book on P100 (PDF) \fhlc{ForestGreen}{Worked Example} Zeige, das jeder endliche Automat, der die Sprache $L = \{ w \in \{ a , b \}^* 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