Local 2x3 tableau windows and legality

Let $T$ be a bounded computation tableau for a fixed machine $N$. Adjoin a fixed boundary marker # to the left and right of every row. A local $2\times 3$ tableau window is any block of the form $\begin{array}{ccc}{T}_{r,c}& T_{r,c+1}& T_{r,c+2}\\ T_{r+1,c}& T_{r+1,c+1}& T_{r+1,c+2}\end{array}$ cut from two adjacent padded rows and three consecutive padded columns.

Such a window is legal when it is consistent with the local row-to-row rule for bounded tableaux: the boundary markers stay fixed; if the upper row is nonhalting then cells away from the head remain unchanged and the cells in the head's neighborhood change exactly as one allowed transition of $N$ prescribes; and if the upper row is already halting then the lower row is identical to it.

• A legal window is a local consistency check; it does not by itself guarantee that the whole tableau is a valid run.

• On this page, the separate global condition is that the first row is the correct start configuration; once that row is fixed, the local windows govern the rest of the tableau.

• The padded boundary markers let the same local test handle the left edge, the right edge, and very small tableau widths.