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(Remember, this is an {\em abstract\/} computer\dots) The formal definition may not seem to amount to this at first glance. \begin{defn} \label{d:TM} \index{machine, Turing} \index{Turing machine} A {\em Turing machine\/} is a function $M$ such that for some natural number $n$, \begin{eqnarray*} \mathrm{dom}(M) & \subseteq & \{1,\dots,n\} \times \{0,1\} \\ & = & \{\, (s,b) \mid 1 \le s \le n \text{\ and\ } b \in \{0,1\} \,\} \end{eqnarray*} and \begin{eqnarray*} \mathrm{ran}(M) & \subseteq & \{0,1\} \times \{-1,1\} \times \{1,\dots,n\} \\ & = & \{\, (c,d,t) \mid c \in \{0,1\} \text{\ and\ } d \in \{-1,1\} \text{\ and\ }1 \le t \le n \,\}\,. \end{eqnarray*} Note that $M$ does not have to be defined for all possible pairs $$ (s,b) \in \{1,\dots,n\} \times \{0,1\} \,. $$ We will sometimes refer to a Turing machine simply as a {\em machine\/} \index{machine} or {\rm TM\/} \index{TM} Reimagining War in the 21st Century: From Clausewitz to Network-Centric Warfare (Routledge Critical Security Studies) download here.