From decbd55c7e8fe314fc880518ccfe949765389df0 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 9 Oct 2025 12:39:46 +0200 Subject: [PATCH] [TI] Continue proofs of nonexistance --- .../02_proofs-of-nonexistance.tex | 10 ++++++++++ semester3/ti/ti-summary.pdf | Bin 855524 -> 856494 bytes 2 files changed, 10 insertions(+) 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 9eb7c77..a9a4b91 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 @@ -20,3 +20,13 @@ so heisst das für uns von jetzt an, dass $A$ nicht zwischen $x$ und $y$ untersc xz \in L(A) \Longleftrightarrow yz \in L(A) \end{align*} \end{lemma} +Das obenstehende Lemma 3.3 ist ein Spezialfall einer Eigenschaft, die für jedes (deterministische) Rechnermodell gilt. + +Mithilfe von Lemma 3.3 kann man für viele Sprachen deren Nichtregularität beweisen. + +\inlineex Sei $L = \{ 0^n1^n \divides n \in \N \}$. +Intuitiv ist diese Sprache Nichtregulär, da $n$ undendlich gross sein kann, aber ein EA logischerweise endlich ist. +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. diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 4098a6f1407f7fb3a3bb59e8dcbc493c6c67d38e..beee4219cc69dc3445fc6751159d40aa2e9a45f9 100644 GIT binary patch delta 37035 zcmX7PV{o8N({*faY-eNJwl>bjw#_Tf#2&ie+ z6gWt|R=0(Y9{uirE_Okm==6p<3{P9s7*f|-H2fCjqpJY;7FXKJk!yks7M5CEo2HXx zhQ)O#q~Vg8+kP?7To(uhOUH4gQU_x4mTJ!+yo#v@pB7^(a?@uI!+AlnI&2Mg>2YOF%YG;6 zpOME7ZaW~P(#tKiXfLTJ6_Q}zN%Y*gi4(ui;S}+?#Y>c`s&JAN3tF!kRknOGww(H^ z(n+~W-Bf&jEn(a9`p-1-F*I6Lp9Rq!N&O>vzI#~z{vRcUyQ@@R1!D#bm_k&tMfvkp zhi=WN3h0U?Cy_^CW>L^&dFk^k9WWbSB+>_5N_n83+neMK#qdv*+PfJTscq|w+r-3V zP-2X95USdmHGiy5+@|x4eUQgw=pr`#_t~zHE4<7r7B;SJz3(^le)vQ#vs*hK#aC^? z6f%-kIk^ep2|f~(;YOBzm7J92fXEI0&Cxxi- 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