From 2806a14365bf205134ced835603c2ee10360d08c Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 8 Oct 2025 18:27:37 +0200 Subject: [PATCH] [TI] Start nonexistance proof section up to lemma 3.3 in the book --- .../02_proofs-of-nonexistance.tex | 22 ++++++++++++++++++ semester3/ti/ti-summary.pdf | Bin 852374 -> 855524 bytes semester3/ti/ti-summary.tex | 2 ++ 3 files changed, 24 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 e69de29..9eb7c77 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 @@ -0,0 +1,22 @@ +\subsection{Beweise der Nichtexistenz} +Im Gegensatz zum Beweis, dass eine bestimmte Klasse von Programmen (Algorithmen) ein Problem lösen kann +(was ein einfacher Existenzbeweis ist, bei welchem man eine korrekte Implementation liefern kann), +ist der Beweis, dass diese Klasse von Programmen (Algorithmen) dies nicht tun kann viel schwieriger, +da man (logischerweise) nicht für alle (undendlich vielen) Programme zeigen kann, dass sie das Problem nicht lösen. + +In diesem Kurs werden wir aber vorerst nur die Klasse der endlichen Automaten behandlen, welche sehr stark eingeschränkt sind, +was diese Beweise verhältnismässig einfach macht. +Falls also ein EA $A$ für zwei unterschiedliche Wörter $x$ und $y$ im gleichen Zustand endet (also $\hdelta(q_0, x) = \hdelta(q_0, y))$), +so heisst das für uns von jetzt an, dass $A$ nicht zwischen $x$ und $y$ unterscheiden kann: + +\begin{lemma}[]{Unterscheidung von Wörtern} + Sei $A$ ein EA über $\Sigma$ und $x \neq y \in \Sigma^*$ so dass + \begin{align*} + (q_0, x) \bigvdash{A}{*} (p, \lambda) \text{ und } (q_0, y) \bigvdash{A}{*} (p, \lambda) + \end{align*} + für ein $p \in Q$ (also $\hdelta_A (q_0, x) = \hdelta(q_0, y) = p(x, y \in \class [p])$). + Dann existiert für jedes $z \in \Sigma^*$ ein $r \in Q$, so dass $xz, yz \in \class[p]$, also gilt insbesondere + \begin{align*} + xz \in L(A) \Longleftrightarrow yz \in L(A) + \end{align*} +\end{lemma} diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 9528e137aab9f4880b45bcc857c1300a3f62e408..4098a6f1407f7fb3a3bb59e8dcbc493c6c67d38e 100644 GIT binary patch delta 52790 zcmZs?V|Zsl_bnKk9ox2Tql1oZ95&?HaoVR4u7}b_s-0nXXgK5)l*fg z>YS(Q(_VY+N(tmV$mFb(frsGaNUiTc!hm37OGJmDOALia15ZgzgJ&aknG9^5Z}9GK zf}LQR=!D6|OWCfQ;LXK0)mUs?;*DocJcVZig6?fP!x(?t(OOOwg-n`~JA4TVR1GlQgb-$tr4uV+Z;kH0tgyoJjD9mSF`IWhuo z;Jhrqg)ICyJ!2t4h%}9=AM|OhvYr4U>_-_xROz4NG7O%#ExRbWqcTXNAJ`xBZ1c{$ ze0J5rscw>Y7~-Q$BVMjMW3`75lJK*ZD< z;#?@4gh|yp4+ffLhfj;10Y7T`WEv-~>Y>58$3ehBj+ZaN;7{(NVT%_nSl9|ZmGPe5 zS&PWaHxV@D@1Ifi6NhxS0G6$nvoBG?t!S}Zd>C#e`pLNuw7TSn zD_4Iv>n#(JOQjPVs+p;_@8qza@nM_2g*6Ic#}0|vQ{Q$JQZyw`2tNN`-bF%Wt|UQj zy5 zn%bGWSdg-_ak3^FL(u}4x;oAq(qDX!YkMiQ38IdLza>|!)@QG-MPD}1d(_>7#m9wx zkz*yL9r0c3=@y~^i%9Ucoap$GTn+AJ*6!7Pb$ZI@;Fhy{aXeO-I2Kr{ZVrPdXX@sZ 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z$~%P?a96bYYbE6JmrjX(HqgZoVHy-)XZFwlu&P|rL=Bvn4^xi8lkZ9*j?v>L2jH}z z>wKiCuvZb(R-@+5EOh2vE_-=zIUIRWJn#>@WvT#|WwSPQHKX0DSuo-O-MH zzsI@%6Bd|r7pmQpzJCwl?O39}EDG4b2m4N)VgKp~crO;rXKY+h59*B?p}hsIzOk3= zv)_;DKhHz{jR3eS!L+}&BgA>s8X)6e_me}5;r*o;*83j$t6(M@v!~+8W diff --git a/semester3/ti/ti-summary.tex b/semester3/ti/ti-summary.tex index ffe377a..a99e258 100644 --- a/semester3/ti/ti-summary.tex +++ b/semester3/ti/ti-summary.tex @@ -7,6 +7,8 @@ \setup{Theoretische Informatik} +\newcommand{\hdelta}{\hat{\delta}} + \begin{document} \startDocument \usetcolorboxes