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55 lines
2.7 KiB
TeX
55 lines
2.7 KiB
TeX
\newsection
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\section{Complexity}
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\label{sec:complexity}
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\stepcounter{subsection}
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\subsection{Measurements of Complexity}
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\compactdef{Time complexity} For a computation $D = C_1, \ldots, C_k$ of $M$ on $x$ is defined by $\text{Time}_M(x) = k - 1$.
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For the TM $M$ itself, we have $\text{Time}_M(n) = \max\{ \text{Time}_M(x) \divides x \in \Sigma^n \}$
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\begin{definition}[]{Space complexity}
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Let $C = (q, x, i, \alpha_1, i_1, \ldots, \alpha_k, i_k)$,
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with $0 \leq i \leq |x| + 1$ and $0 \leq i_j \leq |\alpha_j|$ for $j = 1, \ldots, k$ be a configuration.
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The space complexity of configuration $C$ is $\text{Space}_M(C) = \max\{ |\alpha_i| \divides i = 1, \ldots, k \}$.
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The space complexity of a calculation $D = C_1, \ldots, C_l$ on $x$ is $\text{Space}_M(x) = \max\{ \text{Space}_M(C_i) \divides i = 1, \ldots, l \}$
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The space complexity of a TM $M$ is $\text{Space}_M(n) = \max\{ \text{Space}_M(x) \divides x \in \Sigma^n \}$
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\end{definition}
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\inlinelemma For every $k$-tape-TM $A$, there exists an equivalent $1$-tape-TM $B$ such that $\text{Space}_B(n) \leq \text{Space}_A(n)$
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\inlinelemma For every $k$-tape-TM $A$, $\exists$ a $k$-tape-TM such that $L(A) = L(B)$ and $\text{Space}_B(n) \leq \frac{\text{Space}_A(n)}{2} + 2$
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\inlinedef The big-O-notation is defined as in A\&D, we however write $\text{Time}_A(n) \in \tco{g(n)}$, etc
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\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 $L$
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\subsection{Complexity classes}
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Below is a list of complexity classes
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\begin{definition}[]{Complexity classes}
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\begin{align*}
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\text{TIME}(f) & = \{ L(B) \divides B \text{ is an MTM with } \tc_B(n) \in \tco{f(n)} \} \\
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\text{SPACE}(g) & = \{ L(A) \divides A \text{ is an MTM with } \spc_A(n) \in \tco{g(n)} \} \\
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\text{DLOG} & = \text{SPACE}(\log_2(n)) \\
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\text{P} & = \bigcup_{c \in \N} \text{TIME}(n^c) \\
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\text{PSPACE} & = \bigcup_{c \in \N} \text{SPACE}(n^c) \\
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\text{EXPTIME} & = \bigcup_{d \in \N} \text{TIME}(2^{n^d})
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\end{align*}
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\end{definition}
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For any function $t : \N \rightarrow \R^+$, we have $\text{TIME}(t(n)) \subseteq \text{SPACE}(t(n))$.
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A list of relationships for these classes:
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\rmvspace
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\begin{multicols}{2}
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\begin{itemize}
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\item $P \subseteq \text{PSPACE}$
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\item $\text{DLOG} \subseteq P$
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\item $\text{PSPACE} \subseteq \text{EXPTIME}$
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\item $\text{DLOG} \subseteq P \subseteq \text{PSPACE} \subseteq \text{EXPTIME}
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\end{itemize}
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\end{multicols}
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\inlinedef
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