File:Kolmogorov complexity and computable lower bounds pdf.pdf
Summary
| Description |
English: Shows the Kolmogorov complexity K(s) (red), and two computable lower bound functions, prog1(s) (green), prog2(s) (blue). The horizontal axis (logarithmic scale) enumerates all strings s, ordered by length; the vertical axis (linear scale) measures string length in bits; the depth axis lines up the shown functions. Most strings are incompressible, i.e. their Kolmogorov complexity exceeds their length by a constant amount. Nine compressible strings are shown in the picture, appearing as almost vertical slopes. Due to Chaitin's incompleteness theorem, the output of any program computing a lower bound of the Kolmogorov complexity cannot exceed some fixed limit, which is independent of the input string s; hence both the blue and the green line eventually turn horizontal. |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
| Other versions |
LaTeX source code |
|---|
\documentclass[12pt]{article}
\setlength{\unitlength}{1mm}
\usepackage[pdftex]{color}
\usepackage{amssymb}
\usepackage[paperwidth=210mm,paperheight=55mm]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{210mm}
\setlength{\textheight}{55mm}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}
\pagestyle{empty}
% colors
\definecolor{cCmpr} {rgb}{0.80,0.80,0.80} % compressible string
\definecolor{cClr} {rgb}{1.00,1.00,1.00} % white background
\definecolor{cI} {rgb}{0.00,0.80,0.00} % lwb prog 1
\definecolor{cJ} {rgb}{0.00,0.00,0.99} % lwb prog 2
\definecolor{cKolm} {rgb}{0.99,0.00,0.00} % kolm compl
%\definecolor{cBg} {rgb}{0.00,0.80,0.80} % background
\newcommand{\cmdI}{%
\put(10,2){\line(5,1){60}}%
\put(70,14){\line(1,0){131}}%
\put(40,8){\line(0,-1){4}}%
\put(70,14){\line(0,-1){12}}%
\put(100,14){\line(0,-1){2}}%
\put(110,14){\line(0,-1){10}}%
%\put(120,14){\line(0,-1){}}%
\put(130,14){\line(0,-1){6}}%
\put(170,14){\line(0,-1){4}}%
}
\newcommand{\cmdJ}{%
\put(10,2){\line(5,1){130}}%
\put(140,28){\line(1,0){62}}%
\put(40,8){\line(0,-1){4}}%
\put(70,14){\line(0,-1){12}}%
\put(100,20){\line(0,-1){8}}%
\put(110,22){\line(0,-1){18}}%
\put(120,24){\line(0,-1){4}}%
\put(130,26){\line(0,-1){18}}%
\put(160,28){\line(0,-1){6}}%
\put(170,28){\line(0,-1){18}}%
}
\newcommand{\cmdKolm}{%
\put(10,2){\line(5,1){193}}%
\put(40,8){\line(0,-1){4}}%
\put(70,14){\line(0,-1){12}}%
\put(100,20){\line(0,-1){8}}%
\put(110,22){\line(0,-1){18}}%
\put(120,24){\line(0,-1){4}}%
\put(130,26){\line(0,-1){18}}%
\put(160,32){\line(0,-1){10}}%
\put(170,34){\line(0,-1){24}}%
\put(180,36){\line(0,-1){4}}%
}
\begin{document}
\sf
\begin{picture}(205,50)%
\put(2,2){\line(0,1){5}}%
\put(2,6){\makebox(0,0)[b]{$\sim$}}%
\put(2,8.500){\makebox(0,0)[t]{$\sim$}}%
\put(2,8){\vector(0,1){42}}%
\put(4,50){\makebox(0,0)[tl]{bits}}%
%
\put(2,2){\line(1,0){5}}%
\put(6.500,2){\makebox(0,0)[l]{$\wr$}}%
\put(8.500,2){\makebox(0,0)[r]{$\wr$}}%
\put(8,2){\vector(1,0){197}}%
\put(205,4){\makebox(0,0)[br]{string}}%
%
\put(2,2){\vector(1,1){20}}%
\put(24,22){\makebox(0,0)[tl]{fct}}%
\color{cCmpr}%
\put(42,2){\line(1,1){4}}%
\put(72,2){\line(1,1){4}}%
\put(102,2){\line(1,1){4}}%
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\textcolor{cClr}{\put(42,4){\makebox(0,0)[bl]{\rule{150mm}{3mm}}}}%
\color{cI}%
\put(4.00, 9.00){\cmdI}%
\put(205,21){\makebox(0,0)[tr]{prog$_1$}}%
\thicklines%
\color{cClr}%
\put(2.99, 7.99){\cmdJ}%
\put(3.01, 8.01){\cmdJ}%
\thinlines
\color{cJ}%
\put(3.00, 8.00){\cmdJ}%
\put(205,34){\makebox(0,0)[tr]{prog$_2$}}%
\thicklines%
\color{cClr}%
\put(1.95, 6.95){\cmdKolm}%
\put(1.97, 6.97){\cmdKolm}%
\put(1.99, 6.99){\cmdKolm}%
\put(2.01, 7.01){\cmdKolm}%
\put(2.03, 7.03){\cmdKolm}%
\put(2.05, 7.05){\cmdKolm}%
\color{cKolm}%
\put(2.00, 7.00){\cmdKolm}%
\put(205,45){\makebox(0,0)[tr]{$K$}}%
\end{picture}
\end{document}
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