From 14070f6839bf4f010d8c1248881473836106f891 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sat, 6 Dec 2025 14:44:10 +0100 Subject: [PATCH] [TI] Compact: Basically done up to (including) 6.5 --- semester3/ti-compact/parts/05_complexity.tex | 111 ++++++++++++++++++- semester3/ti-compact/ti-compact.pdf | Bin 354637 -> 373508 bytes 2 files changed, 108 insertions(+), 3 deletions(-) diff --git a/semester3/ti-compact/parts/05_complexity.tex b/semester3/ti-compact/parts/05_complexity.tex index 1ff627b..41a0cb9 100644 --- a/semester3/ti-compact/parts/05_complexity.tex +++ b/semester3/ti-compact/parts/05_complexity.tex @@ -23,12 +23,18 @@ For the TM $M$ itself, we have $\text{Time}_M(n) = \max\{ \text{Time}_M(x) \divi \inlinedef The big-O-notation is defined as in A\&D, we however write $\text{Time}_A(n) \in \tco{g(n)}$, etc +\inlinetheorem There exists decision problem $(\alphabetbool, L)$, such that for each MTM $A$ that decides it, +there exists an MTM $B$ that also decides it and for which $\text{Time}_B(n) \leq \log_2(\text{Time}_A(n))$ + \inlinedef An MTM $C$ is \bi{optimal} for $L$, if $\text{Time}_C(n) \in \tco{f(n)}$ and $\tcl(f(n))$ is a lower bound for the time complexity of $L$ +% ──────────────────────────────────────────────────────────────────── + \subsection{Complexity classes} Below is a list of complexity classes \begin{definition}[]{Complexity classes} + \drmvspace\rmvspace \begin{align*} \text{TIME}(f) & = \{ L(B) \divides B \text{ is an MTM with } \tc_B(n) \in \tco{f(n)} \} \\ \text{SPACE}(g) & = \{ L(A) \divides A \text{ is an MTM with } \spc_A(n) \in \tco{g(n)} \} \\ @@ -39,7 +45,8 @@ Below is a list of complexity classes \end{align*} \end{definition} -For any function $t : \N \rightarrow \R^+$, we have $\text{TIME}(t(n)) \subseteq \text{SPACE}(t(n))$. +\inlinelemma For any function $t : \N \rightarrow \R^+$, we have $\text{TIME}(t(n)) \subseteq \text{SPACE}(t(n))$. + A list of relationships for these classes: \rmvspace \begin{multicols}{2} @@ -47,8 +54,106 @@ A list of relationships for these classes: \item $P \subseteq \text{PSPACE}$ \item $\text{DLOG} \subseteq P$ \item $\text{PSPACE} \subseteq \text{EXPTIME}$ - \item $\text{DLOG} \subseteq P \subseteq \text{PSPACE} \subseteq \text{EXPTIME} + \item $\text{DLOG} \subseteq P \subseteq \text{PSPACE} \subseteq \text{EXPTIME}$ \end{itemize} \end{multicols} -\inlinedef +\drmvspace +\begin{definition}[]{Space- and time-constructible} + Let $s, t : \N \rightarrow \N$. $s$ is called \bi{space-constructible} if there exists $1$-Band-TM $M$, such that + \begin{enumerate} + \item $\text{Space}_M(n) \leq s(n) \ \forall n \in \N$ + \item for each input $0^n$ with $n \in \N$, $M$ generates the word $0^{s(n)}$ on its memory tape and stops in $\qacc$ + \end{enumerate} + + $t$ is called \bi{time-constructible}, if there exists an MTM $A$, such that + \begin{enumerate} + \item $\text{Time}_A(n) \in \tco{t(n)}$ + \item For each input $0^n$ with $n \in \N$, $A$ generates $0^{t(n)}$ on its first memory tape and stops in $\qacc$ + \end{enumerate} +\end{definition} + +\inlinelemma Let $s$ be space-constructible, $M$ an MTM with $\text{Space}_M(x) \leq s(|x|) \ \forall x \in L(M)$. +Then exists MTM $A$ with $L(A) = L(M)$ and $\text{Space}_A(n) \leq s(n)$, i.e. we have +$\text{Space}_A(y) \leq s(|y|) \ \forall y \in \Sigma_M$. + +\inlinelemma Let $t$ be time-constructible, $M$ an MTM with $\text{Time}_M(x) \leq t(|x|) \ \forall x \in L(M)$. +Then exists MTM $A$ with $L(A) = L(M)$ and $\text{Time}_A(n) \in \tco{t(n)}$ + +\inlinetheorem $\forall s : \N \rightarrow \N$ with $s(n) \geq \log_2(n)$, we have $\text{SPACE}(s(n)) \subseteq \bigcup_{c \in \N} \text{TIME}(c^{s(n)})$ + +\inlinetheorem Given $s_1, s_2: \N \rightarrow \N$ with properties $s_2(n) \geq \log_2(n)$, $s_2$ is space-constructible and $s_1(n) = o(s_2(n))$ +($s_2(n)$ grows asymptotically faster than $s_1$). +Then we have $\text{SPACE}(s_1) \subsetneq \text{SPACE}(s_2)$ + + +\inlinetheorem Given $t_1, t_2: \N \rightarrow \N$ with properties $t_2$ is time-constructible and $t_1(n) \cdot \log_2(t_1(n)) = o(t_2(n))$ +Then we have $\text{TIME}(s_1) \subsetneq \text{TIME}(s_2)$ + + + +% ──────────────────────────────────────────────────────────────────── + +\subsection{Non-deterministic measurements of complexity} +\inlinedef For NMTM or NTM, the time complexity is the length of the shortest accepting calculation of $M$ on $x$ and the same applies to space complexity as well. +The rest of the definition is equivalent to the one for deterministic TM and MTM. + +\begin{definition}[]{Complexity classes} + For all $f, g: \N \rightarrow \R^+$, we define + \rmvspace + \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 For all $t$ and $s$ with $s(n) \geq \log_2(n)$: $\text{NTIME} \subseteq \text{NSPACE}(t)$ and +$\text{NSPACE}(s) \subseteq \bigcup_{c \in \N} \text{NTIME}(c^{s(n)})$ + +For $t: \N \rightarrow \R^+$ and every space-constructible $s$ with $s(n) \geq \log_2(n)$, we have: +\rmvspace +\begin{multicols}{2} + \begin{enumerate} + \item $\text{TIME}(t) \subseteq \text{NTIME}(t)$ + \item $\text{SPACE}(t) \subseteq \text{NSPACE}(t)$ + \item $\text{NTIME}(s(n)) \subseteq \text{SPACE}(s(n)) \subseteq \bigcup_{c \in \N} \text{TIME}(c^{s(n)})$ + \item $\text{NP} \subseteq \text{PSPACE}$ + \item $\text{NSPACE}(s(n)) \subseteq \bigcup_{c \in \N} \text{TIME}(c^{s(n)})$ + \item $\text{NLOG} \subseteq P$ + \item $\text{NPSPACE} \subseteq \text{EXPTIME}$ + \item $\text{NSPACE}(s(n)) \subseteq \text{SPACE}(s(n)^2)$ (Savitch) + \item $\text{PSPACE} = \text{NPSPACE}$ + \end{enumerate} +\end{multicols} + +\rmvspace +If we combine some of the above results, we get: +\rmvspace +\begin{align*} + \text{DLOG} \subseteq \text{NLOG} \subseteq P \subseteq NP \subseteq \text{PSPACE} \subseteq \text{EXPTIME} +\end{align*} + + +% ──────────────────────────────────────────────────────────────────── + +\subsection{Proof verification} +\begin{definition}[]{$p$-Verifier} + And MTM $A$ is a $p$-Verifier (for $p : \N \rightarrow \N$) and $V(A) = L$ for $L \subseteq \word$, if $A$ has the following properties and works on all inputs $\word \times \wordbool$: + \begin{enumerate}[label=\textit{(\roman*)}] + \item $\text{Time}_A(w, x) \leq p(|w|)$ for each input $(w, x) \in \word \times \wordbool$ + \item $\forall w \in L, \exists x \in \wordbool$, such that $|x| \leq p(|w|)$ and $(w, x) \in L(A)$. $x$ is \bi{proof} of the claim $w \in L$ + \item $\forall y \notin L$ we have $(y, z) \notin L(A)$ for all $z \in \wordbool$ + \item If $p(n) \in \tco{n^k}, k \in \N$, then $p$ is a polynomial time verifier. 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