Cyclic positions taxonomy

Charles Matthews There was a rec.games.go thread not long ago about 'Ko taxonomy' - inconclusive as all these discussions tend to be if they concentrate on terminology, but not (I think) fruitless.

I wanted to make a point about the general situation on cyclic positions, now that I have learned the standard words from directed graph theory. The origin of a directed graph based on a Go position on a sub-board is easy to explain: the vertices are the positions reached by legal play, not assumed to be alternating.

There is only going to be a question about potentially repeating play in that part of the board, if there are (directed) cycles in the graph, taking one back to a previous position - necessarily after some captures. From general theory we take the idea of strongly connected component: for us a set of positions each of which can be reached from any other, and to which no further positions can be added while still having that property.

Examples would include: a simple ko position in its two states 'Black to capture' and 'White to capture'. Importantly this is by no means the only type of ko that is included under such a description: any multiple ko (such as a two-stage ko or a double ko) which can be captured 'back and forward' gives rise to an example.

An approach ko is different: its states give rise to at least two distinct strongly connected components. The phenomenon of the approach move means that once it is made, a transition between components has occurred.

Therefore I believe in the long run theorists of the game, at least, will need to integrate into the terminology for cyclic positions of all kinds (going beyond ko) a way of referring more systematically to the components. The underlying acyclic component graph, consisting of the components and the possible transitions between them, is a fundamental object of study, however complex the position. I have proposed that it should be called something like the levels of the cyclic position under study.