From ca39d24145be90c28f107deb8f1e64ef87a6ff23 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 14 Nov 2025 16:03:59 +0100 Subject: [PATCH] [TI] Catch up and work ahead slightly --- .../02_non-deterministic-complexity.tex | 63 ++++++++++++++++++ .../ti/parts/05_complexity/03_class-np.tex | 2 + semester3/ti/ti-summary.pdf | Bin 996758 -> 1001581 bytes 3 files changed, 65 insertions(+) diff --git a/semester3/ti/parts/05_complexity/02_non-deterministic-complexity.tex b/semester3/ti/parts/05_complexity/02_non-deterministic-complexity.tex index 2c52ad1..dd9ba30 100644 --- a/semester3/ti/parts/05_complexity/02_non-deterministic-complexity.tex +++ b/semester3/ti/parts/05_complexity/02_non-deterministic-complexity.tex @@ -1,2 +1,65 @@ \newpage \subsection{Nichtdeterministische Komplexitätsmasse} + +\begin{definition}[]{Zeit- und Speicherkomplexität} + Sei $M$ eine NMTM oder MTM und $x \in L(M) \subseteq \word$. $\tc_M(x)$ ist die länge einer kürzesten akzeptierenden Berechnung von $M$ auf $x$ + und $\tc_M(n) = \max(\{ \tc_M(x) \divides x \in L(M) \text{ und }|x| = n \} \cup \{ 0 \} )$. + + \vspace{0.25cm} + + $\spc_M(C_i)$ ist die Speicherkomplexität von Konfiguration $C_i$ und $\spc_M(C) = \max\{ \spc_M(C_i) \divides i = 1, 2, \ldots, m \}$. + Zudem ist $\spc_M(x) = \min\{ \spc_M(C) \divides C \text{ ist akzeptierende Berechnung von $M$ auf } x \}$. + Ausserdem ist $\spc_M(n) = \max(\{ \spc_M(x) \divides x \in L(M) \text{ und } |x| = n \} \cup \{ 0 \})$ +\end{definition} + + +\begin{definition}[]{Komplexitätsklassen} + Für alle $f, g : \N \rightarrow \R^+$ definieren wir: + \begin{align*} + \text{NTIME}(f) & = \{ L(M) \divides M \text{ ist eine NMTM mit } \tc_M(n) \in \tco{f(n)} \} \\ + \text{NSPACE}(g) & = \{ L(M) \divides M \text{ ist eine NMTM mit } \spc_M(n) \in \tco{g(n)} \} \\ + \text{NLOG} & = \text{NSPACE}(\log_2(n)) \\ + \text{NP} & = \bigcup_{c \in \N} \text{NTIME}(n^c) \\ + \text{NPSPACE} & = \bigcup_{c \in \N} \text{NSPACE}(n^c) + \end{align*} +\end{definition} + + +\inlinelemma Für alle $t$ und $s$ mit $s(n) \geq \log_2(n)$ gilt: $\text{NTIME}(t) \subseteq \text{NSPACE}(t)$, $\text{NSPACE}(s) \subseteq \bigcup_{c \in \N} \text{NTIME}(c^{s(n)})$ + +\inlinetheorem Für jedes $t : \N \rightarrow \R^+$ und jedes platzkonstruierbare $s$ mit $s(n) \geq \log_2(n)$ gilt: +\rmvspace +\begin{multicols}{2} + \begin{enumerate}[label=(\roman*)] + \item $\text{TIME}(t) \subseteq \text{NTIME}(t)$ + \item $\text{SPACE}(t) \subseteq \text{NSPACE}(t)$ + \item $\text{NTIME}(s(n)) \subseteq \text{SPACE}(s(n)) \subseteq \bigcup_{c \in \N} \text{TIME}(c^{s(n)})$ + \end{enumerate} +\end{multicols} + +\drmvspace +\inlinecorollary $\text{NP} \subseteq \text{PSPACE}$ + +\inlineremark Für jede platzkonstruierbare Funktion $s$ mit $s(n) \geq \log_2(n)$ gilt +\rmvspace +\begin{align*} + \text{NSPACE}(s(n)) \subseteq \bigcup_{c \in \N} \text{TIME}(c^{s(n)}) +\end{align*} + +\drmvspace +\inlinecorollary $\text{NLOG} \subseteq \text{P}$ und $\text{NPSPACE} \subseteq \text{EXPTIME}$ + + +\fancytheorem{Satz von Savitch} Sei $s$ mit $s(n) \geq \log_2(n)$ eine platzkonstruierbare Funktion. Dann gilt: +\rmvspace +\begin{align*} + \text{NSPACE}(s(n)) \subseteq \text{SPACE}(s(n)^2) +\end{align*} + +\drmvspace +\inlinecorollary $\text{PSPACE} = \text{NPSPACE}$ + +Aus den obigen Resultaten resultiert die Komplexitätsklassenhierarchie der sequentiellen Berechnungen: +\begin{align*} + \text{DLOG} \subseteq \text{NLOG} \subseteq \text{P} \subseteq \text{NP} \subseteq \text{PSPACE} \subseteq \text{EXPTIME} +\end{align*} diff --git a/semester3/ti/parts/05_complexity/03_class-np.tex b/semester3/ti/parts/05_complexity/03_class-np.tex index e69de29..338df23 100644 --- a/semester3/ti/parts/05_complexity/03_class-np.tex +++ b/semester3/ti/parts/05_complexity/03_class-np.tex @@ -0,0 +1,2 @@ +\newpage +\subsection{Die Klasse NP und Beweisverifikation} diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 94f0b0f71c8d892f664c5a7dbe8651edd5c53d91..a0e41e992ba22caf70e80f051922f23be3c2d1e2 100644 GIT binary patch delta 36766 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