ComplexityClass 2002-09-06 15:00:00 2002-09-06 15:00:00 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %\PMlinkescapeword{theory} If $f(n)$ is any function and $T$ is a Turing machine (of any kind) which halts on all inputs, we say that $T$ is time bounded by $f(n)$ if for any input $x$ with length $|x|$, $T$ halts after at most $f(|x|)$ steps. For a decision problem $L$ and a class $K$ of Turing machines, we say that $L\in KTIME(f(n))$ if there is a Turing machine $T\in K$ bounded by $f(n)$ which decides $L$. If $R$ is a search problem then $R\in KTIME(f(n))$ if $L(R)\in KTIME(f(n))$. The most common classes are all restricted to one read-only input tape and one output/work tape (and in some cases a one-way, read-only guess tape) and are defined as follows: \begin{itemize} \item $D$ is the class of deterministic Turing machines. (In this case $KTIME$ is written $DTIME$.) \item $N$ is the class of non-deterministic Turing machines (and so $KTIME$ is written $NTIME$). \item $R$ is the class of positive one-sided error Turing machines and $coR$ the class of negative one-sided error machines \item $BP$ is the class of two-sided error machines \item $P$ is the class of minimal error machines \item $ZP$ is the class of zero error machines \end{itemize} Although $KTIME(f(n))$ is a time complexity class for any $f(n)$, in actual use time complexity classes are usually the union of $KTIME(f(n))$ for many $f$. If $\Phi$ is a class of functions then $KTIME(\Phi)=\bigcup_{f\in\Phi} KTIME(f(n))$. Most commonly this is used when $\Phi=\mathcal{O}(f(n))$. The most important time complexity classes are the polynomial classes: $$K\mathcal{P}=\bigcup_{i\in\mathbb{N}} KTIME(n^i)$$ When $K=D$ this is called just $\mathcal{P}$, the class of problems decidable in polynomial time. One of the major outstanding problems in mathematics is the question of whether $\mathcal{P}=\mathcal{NP}$. We say a problem $\pi\in KSPACE(f(n))$ if there is a Turing machine $T\in K$ which solves $\pi$, always halts, and never uses more than $f(n)$ cells of its output/work tape. As above, if $\Phi$ is a class of function then $KSPACE(\Phi)=\bigcup_{f\in\Phi} KSPACE(f(n))$ The most common space complexity classes are $K\mathcal{L}=KSPACE(\mathcal{O}(\log n))$. When $K=D$ this is just called $\mathcal{L}$. If $\mathcal{C}$ is any complexity class then $\pi\in co\mathcal{C}$ if $\pi$ is a decision problem and $\overline{\pi}\in\mathcal{C}$ or $\pi$ is a search problem and $L(\pi)\in co\mathcal{C}$. Of course, this coincides with the definition of $coR$ above. Clearly $co(co\mathcal{C})=\mathcal{C}$. Since a machine with a time complexity $f(n)$ cannot possibly use more than $f(n)$ cells, $KTIME(f(n))\subseteq KSPACE(f(n))$. If $K\subseteq K^\prime$ then $KTIME(f(n))\subseteq K^\prime TIME(f(n))$ and similarly for space. The following are all trivial, following from the fact that some classes of machines accept and reject under stricter circumstances than others: $$D\subseteq ZP=R\cap coR$$ $$R\cup coR\subseteq BP\cap N$$ $$BP \subseteq P$$