From 4c485cc6f5b7369198545c439dad494abe7eb160 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 9 Oct 2025 14:07:40 +0200 Subject: [PATCH] [TI] Proofs of Nonexistance: Example --- .../02_proofs-of-nonexistance.tex | 9 ++++++++- semester3/ti/ti-summary.pdf | Bin 856494 -> 857451 bytes 2 files changed, 8 insertions(+), 1 deletion(-) diff --git a/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex b/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex index a9a4b91..01d2a60 100644 --- a/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex +++ b/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex @@ -29,4 +29,11 @@ Intuitiv ist diese Sprache Nichtregulär, da $n$ undendlich gross sein kann, abe Wir müssen hier nur formal ausdrücken, dass das Zählen benötigt wird, dass $L$ akzeptiert wird: Dazu benutzen wir einen indirekten Beweis. Sei $A$ ein EA über $\alphabets{bool}$ und $L(A) = L$. -Die Idee des Beweises ist nun zu zeigen, dass für alle $0, 0^1, \ldots 0^{|Q| + 1}$, man ein $i, j$ hat für welches Lemma 3.3 nicht zutrifft. +Wir betrachten die Wörter $0^1, 0^2, \ldots, 0^{|Q| + 1}$. +Weil wir $|Q| + 1$ Wörter haben, existiert $i, j \in \{ 1, 2, \ldots, |Q| + 1 \}$, so dass $\hdelta_A(q_0, 0^i) = \hdelta_A(q_0, 0^j)$, +also gilt nach Lemma $0^i z \in L \Leftrightarrow 0^j z \in L \smallhspace \forall z \in \wordbool$. +Dies gilt jedoch nicht, weil für jedes $z = 1^i$ zwar jedes $0^i 1^i \in L$ gilt, aber $0^j 1^j \notin L$ + + +Um die Nichtregularität konkreter Sprachen zu beweisen, sucht man nach einfach verifizierbaren Eigenschaften, +denn wenn eine Sprache eine dieser Eigenschaften \textit{nicht} erfüllt, so ist sie nicht regulär. diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index beee4219cc69dc3445fc6751159d40aa2e9a45f9..3ea35bc5dec17848a08f4338f83ad3abe5074a5a 100644 GIT binary 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