For every $h\geq1$, one-way liveness$h$ (or just owl$h$) 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 three
symbols of $\varSigma_5$.
Every string of graphs from $\varSigma_h$ induces a multi-column graph, the one that we get when we identify adjacent
columns. E.g., the three symbols above from $\varSigma_5$ induce the graph If the multi-column graph contains a path from its leftmost to its rightmost 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 is dead.
The instances of owl$h$ are all strings over $\varSigma_h$. A string is a positive
instance if it induces a live graph; if it induces a dead graph, then it is a negative instance.