Force (Mathematics) / discussion

English of the Definition of Virtual-Force

Something

Could "that something" be replaced with "that condition", "that position", "that state" or some such term in the recursive definition? I find these options more readable and idiomatic than "something". --Hyperpape

RobertJasiek: "something" is the historical choice of a word, see the Japanese 2003 Rules. "condition" is doubtful because it does not have to be a condition; it also be a noun phrase. "position" is as bad because it does not have to be a position. Likewise "state". It has to be as general as "something" and more general than "object". In German mathematics it could be "Aussageform", I think, but "something" is more easily understood!

Hyperpape: The problem is that "satisfies that..." can only be completed by a limited range of complements in English. Some of the restrictions may be idiomatic, some may be grammatical. But you would say "satisfies that condition" but not "satisfies that state", and certainly not "satisfies that position". My inclination would be something like:

For a position, a player can force something (a condition, state, property (you may add something here), hereafter abbreviated "condition") if that player can make a legal move that reaches a second position that satisfies one of the following:

1. The second position satisfies that condition.
2. Any legal move by the opponent from the second position reaches a third position
   • which satisfies that condition
   • or else in which the player can force that condition.

Or slightly more technical:

Let φ be a property of a position. Then for a position p, a player can force φ if that player can make a legal move that reaches a second position q such that:

1. φ(q).
2. Any legal move by the opponent from the q reaches a third position r such that:
   • φ(r)
   • or else in which the player can force φ.

I actually prefer the second.

RobertJasiek: I do not verify right now whether your procedure list maps the definition. When I find time for a formal J2003 version, then I can make you happier...

Hyperpape: Unless I've made a terrible error, I've just substituted φ for "that something", "property" for "something" in general, and added p, q and r as names of the positions.

The Definition

For a ko, a player can virtual-force something if he can force the something while exceptionally he may always make a ko-capture on this ko immediately after the opponent's ko-capture on this ko unless that ko-capture was preceded by the player's ko-capture on a different ko.

Discussion of the English

Without restricting and unrestricting extras, the sentence's core is: "A player can virtual-force something if he can force the something."

A restriction is added: "For a ko,". This means that the core refers to an arbitrarily chosen but then particular ko. Since the core does not itself refer to "ko", the core itself is not restricted for itself yet. Only the extension to the core refers to "ko" and is therefore restricted.

The extension to the sentence's core is: "while exceptionally he may always make a ko-capture on this ko immediately after the opponent's ko-capture on this ko unless that ko-capture was preceded by the player's ko-capture on a different ko."

How is that extension restricted by the restriction "For a ko,"? The extension uses "on this ko", again "on this ko", and "on a different ko". The restriction means that both phrases "on this ko" refer to the same ko and that that ko is arbitrarily chosen but then particular during application of the definition text. I.e., first we choose a ko, then we apply the definition text to it, and during the entire application that ko's intersections are referred to. The same two intersection during the entire application. It is that ko's intersections we are interested in because the phrase "on this ko" is referring to the ko's intersections. - It does not matter if the ko temporarily cedes to be a ko during application because the ko was initially given, from then on we know its two intersections, and then for application of the extension text with respect to its reference to the restriction "For a ko," we are interested in only those intersections. - The phrase "on a different ko" refers to a ko existing on two intersections that differ from the "For a ko,"'s ko two intersections. This is so because of the very meaning of the word "different".

Here "while" means during each moment of the player's attempt to force.

The definition exists in a context (of other definitions) where it awaits the input of given rules. Those are being applied during application of the definition. Such given rules have to have or imply the Basic Ko Rule. Now the "exceptionally" overrides the given rules. More specifically it does so to override the Basic Ko Rule by specifying exceptions.

"he" is "the player".

"the player" is not "the opponent". "the player" and "the opponent" are the exactly two different persons in alternation.

The exception's main clause is: "he may always make a ko-capture on this ko immediately after the opponent's ko-capture on this ko". IOW, whenever the opponent has just made a ko-capture on this ko, then immediately afterwards the player may recapture this ko.

This exception is an unrestriction of the core because it provides the player with extra rights.

The exception itself is in turn restricted by a subclause though: "unless that ko-capture was preceded by the player's ko-capture on a different ko". This is when the "always" of the exception's main clause does not apply.

What do exception and subclause exception of the exception mean in effect? There are two cases:

a) The player has made a ko-capture on a different ko and then the opponent has made a ko-capture on the particular ko. Then the player's next move may not be a ko-capture on the particular ko.

b) The player has made a move that has not been a ko-capture and then the opponent has made a ko-capture on the particular ko. Then the player's next move may be a ko-capture on the particular ko.

Answer-force

The intuition for answer-force is that, if the opponent always moves in such a way to force that something else, then the player can force that something.

Herman: Perhaps an even more intuitive formulation is: Given a goal A that the player wants to achieve, and a different goal B that the opponent wants to achieve, then a player can answer-force A, given B, if:

• If the opponent forces B, he cannot prevent the player from achieving A.
• If the opponent prevents A, then he cannot force B.

So A and B will either both be achieved, or neither can be achieved.

Example:

answer-force example  

Given that Black has the goal "Seki", and White has the goal "Not die", then Black can answer-force seki with -. Note that Black cannot force seki if White has a different goal than "not die", because in that case white would play elsewhere with , and there would not be seki.

RobertJasiek: I have not thought through the "either...or" or "or" logic completely yet. What if the opponent cannot even force B as a task for itself? This lets me think that "or" is correct rather than "either...or". Also compare the "or" construction in the definition of semi-stable ko. I am not really convinced in general yet that the "or" construction equals the answer-force usage. I have found it easier to invent answer-force than to prove equivalence of the two constructions. "either...or" introduces undesired side-effects though.

Herman: I had assumed that answer-force was mutually exclusive with force, i.e that if a player can answer-force something, it is assumed he cannot force the same thing, but I see that that is not necessarily the case. Can you clarify? Example:

corner  

Is it valid to say that Black can answer-force the death of the white group in the upper left, given that White has an answer-strategy to kill the black group in the upper right?

RobertJasiek: Consider a player's two-eye-formation covering the whole board. The player can force having a two-eye-formation. Besides the player can answer-force having a two-eye-formation (for whichever force-objective the opponent might have). So force and answer-force are not mutually exclusive. - As to your example's question, an answer-strategy does not have any parameter-like objective. Also a strategy does not have any such objective. strategy and answer-strategy are a definition logic level lower than the terms force and answer-force, which do use such objectives. I.e., you should formulate your question differently as "[...] White can force to kill the black group in the upper right". Now we can study whether indeed White (moving second) can force this. White cannot. What you really want to know is whether Black can answer-force death of the white string in the upper left if White forces death of the black upper right string. You will find out by applying the definitions carefully, I hope.

Herman: Ok, thanks, that clarifies a lot. Another question, then. In your paper you give the definition:

"A player can answer-force A if the opponent moving second uses an answer-strategy that - regardless of the player's first move - does force B and if there is at least one strategy of the player so that each move-sequence that is compatible with the player's strategy and answer-compatible with the opponent's answer-strategy fulfils A. (emphasis added).

Is it required that "the player's first move" be compatible with one of the player's strategies? (My intuition would say yes, but the definition does seem to require it).

RobertJasiek: The condition "that is compatible with the player's strategy" applies to move-sequences (not to move-sequences without their first move). Compatible checks left-parts ending with a move of the player. Since those move-sequences start with that player's move, a left-part consisting of exactly one move does end with that player's move, too. So also it has to be checked for compatible.

Herman: Ok, thanks. That could do with a small rewrite, perhaps. As it stands, the condition "is compatible with" seems to be restricted to the second clause, starting "if there is at least", but it also applies to the first clause, starting "if the opponent". I'll think about it some more.

unkx80: I'm quite sleepy now, but as I wade through this discussion, I can't help to question something. Earlier you asked "What if the opponent cannot even force B as a task for itself?" But I quote your definition: "A player can answer-force A if the opponent moving second uses an answer-strategy that - regardless of the player's first move - does force B and if there is at least one strategy of the player so that each move-sequence that is compatible with the player's strategy and answer-compatible with the opponent's answer-strategy fulfils A." And then "A player does force something if he uses a strategy of his so that each compatible move-sequence fulfils that something." I take it to mean that your definition says that the opponent has a (answer-)strategy that allows him to always force B?

RobertJasiek: Thanks for being less sleepy than me today:) Yes, as you read it I have also applied it to my examples, IIRC.

unkx80: By the way, may I suggest rewording that definition as follows:

"A player can answer-force A, given that the opponent moving second uses an answer-strategy that - regardless of the player's first move - does force B, if there is at least one strategy of the player so that each move-sequence that is compatible with the player's strategy and answer-compatible with the opponent's answer-strategy fulfils A."

That's because your definition as it stands uses a "... if ... and if ..." structure which some people will parse as "... [if ... and if ...]" when you intended "[... if ...] and if ...". For me, the "and" suggests the first parse.

RobertJasiek: What is the difference between "... [if ... and if ...]" and "[... if ...] and if ..."?

unkx80: The structure "X if Y" means that "X" is being defined, and its meaning is "Y". Now, you are defining the term "can answer-force A", but in the context that "opponent uses an answer-strategy that does force B". Therefore that "X" should be the context plus the term. How does it read like, "something-being-defined if it belongs to some context and if it together with its context has this meaning", versus "in some context, something-being-defined if it has this meaning"?

RobertJasiek: Without doubt, it is possible to write the definition more in maths fashion:) "For some player P, for some A, for some B, there exists strategy S, there exists answer-strategy T ...blahblah..." I suggest you write down the maths version, then we have the best idea how to express it in English with the least remaining ambiguity.