From 1525a3e1a1f9643854ff677b2b8b8c18b1393986 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 25 Nov 2025 12:34:44 +0100 Subject: [PATCH] [TI] Start NP class and proof verification section --- .../ti/parts/05_complexity/03_class-np.tex | 29 ++++++++++++++++++ semester3/ti/ti-summary.pdf | Bin 1001579 -> 1006300 bytes 2 files changed, 29 insertions(+) diff --git a/semester3/ti/parts/05_complexity/03_class-np.tex b/semester3/ti/parts/05_complexity/03_class-np.tex index 338df23..0ee3acf 100644 --- a/semester3/ti/parts/05_complexity/03_class-np.tex +++ b/semester3/ti/parts/05_complexity/03_class-np.tex @@ -1,2 +1,31 @@ \newpage \subsection{Die Klasse NP und Beweisverifikation} +Da praktische Lösbarkeit eines Problems mit polynomieller Zeit verbunden wird, ist es wichtig zu wissen, welche Probleme in polynomieller Zeit lösbar sind und welche nicht. + +Der Vergleich zwischen den Klassen $P$ und $NP$ ist äquivalent zu der Frage, ob es einfacher ist, gegebene Beweise zu verifizieren, als sie herzustellen. + +Betrachten wir folgendes: Sei $L = SAT$, wobei +\rmvspace +\begin{align*} + SAT = \{ x \in \words{logic} \divides x \text{ kodiert eine erfüllbare Formel in CNF} \}. +\end{align*} + +\drmvspace +Dann ist die Aussage $\Phi \in SAT$ äquivalent zu der Behauptung ``$\Phi$ ist eine erfüllbare Formel in CNF'' + +Für nichtdeterministische Berechnungen nennen wir $\alpha_1, \ldots, \alpha_n$ \bi{Zertifikate} für eine Aussage $\Xi$, falls für diese $\Xi(\alpha_1, \ldots, \alpha_n)$ hält. + +\begin{definition}[]{Verifizierer} + Sei $L \subseteq \word$ und $p : \N \rightarrow \N$. + Eine MTM $A$ ist ein $p$-Verifizierer und $V(A) = L$, falls $A$ mit folgenden Eigenschaften auf allen Eingaben aus $\word \times \wordbool$ arbeitet: + \begin{enumerate}[label=(\roman*)] + \item $\tc_A(w, x) \leq p(|w|)$ für jede Eingabe $(w, x) \in \word \times \wordbool$ + \item Für jedes $w \in L$ existiert ein $x \in \wordbool$, so dass $|x| \leq p(|w|)$ und $(w, x) \in L(A)$. $x$ ist \bi{Zeugen} (oder \bi{Beweis}) der Behauptung $w \in L$ + \item Für jedes $y \notin L$ gilt $(y, z) \notin L(A)$ für alle $z \in \wordbool$ + \item Falls $p(n) \in \tco{n^k}$ für ein $k \in \N$, so ist $p$ ein \bi{Polynomialzeit-Verifizierer}. Die Klasse ist + \vspace{-0.8pc} + \begin{align*} + VP = \{ V(A) \divides A \text{ ist ein Polynomialzeit-Verifizierer } \} + \end{align*} + \end{enumerate} +\end{definition} diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 5f8b37f5927b3f93273851b63cd9ab1858ce5bff..e38b93cb32329edb93faadede8dce62277a9fc8f 100644 GIT binary patch delta 131211 zcmZs>b8K(X7ByPic6(}@r?zd|Hcs1LZQDMzZQHhOo8P(j<>lqeyMK&iX0N?R=9*c1 zWlUVe@oqNp)`~-eFfk=cLQ4S(ipirW1YeiQWAwwXqFBddHt{UNnJpttq?N`)5Z)!S#Tn3%2`JM{)E0o>sV7tT&= z;tvdh`1J><RRiiI;L)`@G=L0+>A!cCf=9_RH{It(I;;0!z|{ds^rBK?&ZHu&_`>~a zwwnBdH}!UENv^x2DO(WY4+)|orua#33Vp-27mBo1`C{5AphgKmKRsX2myGEN(O*kO zs{DDZrFJ73qMn{N{7=Z^60n~UfgK@Q*ag35I^}!SO&bajO}J>{GR*N&boCk~p7cB+ zVt5ob+pt>wEyV$sOg|Z^6ZDMlNWFm7i+duaiuE^lL@paoj52nXmR_o06AKt<> z92J@|#;>#wg?(jSYAS0eKV46E8t_JGJVfXIlo#2FhMfyRKCR=H9?ND_>=iu`^A5+~-> z@R^*ND<2QTB$_(t)Z%JLd!W22f#BV}9rf3?q2Bf*U}eN@`_K#O3S18^Gf@IYrvvO0 zHlz6g(>r`w8KtP%8QaMSQ$5H-nUKL}ERRq#S#%5rw()||-vn{Je#O3Qr>|+~RSk8O 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