Fixed Ko Rule: Continued Discussion

[RobertJasiek]

Robert: So you wanted to mean "afterwards" as "any time afterwards".

Of course. This is what superko and fixed ko rules are all about.

Even in that case, you artificially selected a specific subset of moves that might trigger the respective restriction.

Please explain!

This holds even if from the current position there is no play available to do so. [...] This holds even if the current position is one of 'A' or 'B', but no play is available to create the other one.

Sure. It means that it is a ko rule that does not only apply to basic ko captures and single stone suicides.

Your distinction is just bogus.

Of course not. A proof has worked for PSK and has been refuted for doubled fixed ko.

The similarities that you state do not remove that difference at all. In fact, nothing can remove the difference, as proven.

Whether a prohibition is in place when no prohibited play is theoretically available is a purely philosophical question.

"purely" is an exaggeration, but philosophy is important here indeed. FYI, professionals would never like a rule with such a weak philosophy as the double fixed ko rule; they have great doubts even with good philosophy new inventions. (For theoretical study, double fixed ko or cycle removal rule is perfectly valid, of course.)

[HermanHiddema]

I think it is mostly just a matter of representation, and Double Fixed Ko (DFK) can be as consistent as Positional Super Ko (PSK).

We can represent a game of go as a graph (V, E), where each position is a Vertex (V), and each transition from one position to another is an Edge (E).

For the edges we can choose whether all edges must be unidirectional (→), or whether bidirectional edges (↔) are also allowed.

If only unidirectional edges are allowed, then two vertices A and B that can reach each other in a single move (i.e. basic ko) must have two edges between them (A→B and B→A). If bidirectional edges are allowed, we can represent this with a single edge (A↔B).

With PSK, when we make a play to move from A to B, we remove vertex A from the graph (and by extension all attached edges).

With normal Fixed Ko, we use unidirectional edges only, and we remove each edge as we use it.

With DFK, we allow bidirectional edges, and we remove each edge as we use it.

With this representation, DFK is consistent, nothing is removed that did not exist.

Note that to implement only basic ko, using bidirectional edges has the advantage that we can mark the last used edge as unavailable for the next move, no need to find the reverse direction edge.

[Harleqin]

Robert: all you complain about is that not all edges actually are bidirectional (in fact, only "basic ko" edges are---exactly those that are not covered under simple fixed ko). To appease your aesthetic sense, it should perhaps be worded such that not "all edges are seen as bidirectional" but "it applies to both uni- and bidirectional edges".

Two wording proposals:

"A move, which transforms one position to another, is prohibited if any previous move transformed one of those positions to the other."

"A move that transforms position A to position B is prohibited if any previous move transformed either A to B or B to A."

[RobertJasiek]

Herman, with your representation, DFK makes more aesthetical sense (but is hardly understood by non-mathematicians).

[HermanHiddema]

Herman, with your representation, DFK makes more aesthetical sense (but is hardly understood by non-mathematicians).

True, though much of the research texts behind all sorts of rules are not readily understood, of course.

It is certainly an advantage of superko that it is not hard to grasp, intuitively, for any player. I do doubt most of them will immediately grasp all the strategic implications, such as the one shown at http://senseis.xmp.net/?RulesBeast1

To make DFK more easily understood, you can also phrase it as a combination of basic ko and fixed ko, something like:

A player may not make a move that returns the position to that immediately prior to the opponent's previous move (i.e. basic ko).
If any other earlier position is reached, you may not make the same move as you made the last time in that position.

But of course that is no longer as elegant as Harleqin's phrasing.