For every $h\geq1$, compact one-way liveness$h$ (or just cowl$h$) is defined over
the alphabet $\varSigma_h$ of all $2$-column graphs of height $h$ in which all arrows start on the left column and
end on the right column. E.g., the graphs are two
symbols of $\varSigma_5$.
Every ordered pair from $\varSigma_h$ induces the $3$-column graph that we get by identifying the two adjacent columns.
E.g., the pair above induce the graph If this graph
contains a path from the left to the right column, we call it live; if not, we call it dead. E.g., the
graph above is live, because of the bold path. In contrast, the graph induced by the reverse pair
The instances of cowl$h$ are all $2$-long strings over $\varSigma_h$. An instance is
positive if it induces a live graph; otherwise, it induces a dead graph and is negative.