68Q15-Constructible.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{Constructible}
\pmcreated{2013-03-22 13:03:20}
\pmmodified{2013-03-22 13:03:20}
\pmowner{Henry}{455}
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\pmtitle{constructible}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{68Q15}
\pmdefines{time constructible}
\pmdefines{space constructible}

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\begin{document}
A function $f:\mathbb{N}\rightarrow\mathbb{N}$ is \emph{time constructible} if there is a deterministic Turing machine $T$ (with alphabet $\{0,1,B\}$) such that when $T$ receives as input the a series of $n$ ones, it halts after exactly $f(n)$ steps.  Similarly $f$ is \emph{space constructible} if there is a similar Turing machine which halts after using exactly $f(n)$ cells.

Most 'natural' functions are both time and space constructible, including constant functions, polynomials, and exponentials, for example.
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