Given a tape position $(s,i,\mathbf{a})$, we will refer to cell $i$ as the {\em scanned cell\/}\index{scanned cell}\index{cell, scanned} and to $s$ as the {\em state\/}\index{state}. \end{defn} Note that if $(s,i,\mathbf{a})$ is a tape position, then the corresponding Turing machine's scanner is presently reading $a_i$ (which is one of $0$ or $1$). \subsection*{Conventions for tapes} Unless stated otherwise, we will assume that all but finitely many cells of any given tape are blank, and that any cells not explicitly described or displayed are blank.

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