Forum for A General Definition of Ko
Comments regarding the ko paper. [#2149]
Hi Robert,
I refer to your ![[ext]](images/extlink.gif)
ko paper. I have two comments:
Comment 1
"Defined elsewhere are in particular: play, move, current-position, basic-ko, (situational) cycle, history-bans."
Where is "elsewhere"?
You don't have to answer this question here. For example, I found "play" and "move" in your Japanese 2003 rules. However, I am not successful in finding what "history-ban" means. But please cite the relevant sources in your paper itself.
Comment 2
"A player's strategy is a set of one or more than one left-parts of move-sequences so that each left-part starts with a move of his, each left-part ends with a move of his, there are not two left-parts so that they without their last move are equal, and the above is not true for the set together with any left-part not in the set."
Okay, let me dissect this sentence and annotate its parts:
"A player's strategy is a set of one or more than one left-parts of move-sequences so that:
(a) each left-part starts with a move of his,
(b) each left-part ends with a move of his,
(c) there are not two left-parts so that they without their last move are equal, and
(d) the above is not true for the set together with any left-part not in the set."
My comment that the word "above" in part (d) is ambiguous: does it refer to part (c) only, or all of parts (a), (b), and (c)? Also, (d) together with (c) implies double negation which is particularly difficult to read.
Assuming "above" in (d) refers to (c) alone, I suggest rewriting the definition as follows. Changes are in italics.
"A player's strategy is a set of one or more left-parts of move-sequences such that:
(a) each left-part starts with a move of his,
(b) each left-part ends with a move of his,
(c) there are no two left-parts such that they without their last move are equal, and
(d) for each left-part not in the set, there is a left-part in the set such that these two left-parts without their last move are equal."
That's all I have for now.
Some definitions are located easily, others not (e.g., if they are common sense among rules experts). current-position, basic-ko, (situational) cycle are so common sense for rules experts that it is hard to locate the best possible reference. Maybe an old rec.games.go article which is now hard to find. Play and move are so very common in rules-related theory that it is becoming dull even to state references. Beyond a certain point, some terms simply have to be assumed as known so that further progress is possible efficiently.
history-bans is still scarce though and I will add a reference in the next update to my types of Basic Kos paper. Thank you for the hint!
"above" in "strategy" refers to (a) to (c). Yes, you are right, this is somewhat ambiguous. Also in the Japanese 2003 Rules. To some extent, the semi-formal definition is a compromise between formalism and readability. I think that for the educated reader the meaning is implicitly clear. Nevertheless, one might be more precise for sure. Also at some other details of J2003. (In a maths annotation, I would define a graph of vertices and their adjacencies for the goban rather than to start with lines.)
"one or more": Someone had suggested that to me. I am not convinced though. It may be better common English but is less precise. I prefer greater precision to better ordinary English style. (Therefore I also like to connect nouns of a term by hyphens.)
"such that": Mathematics uses "so that" more frequently, IIRC.
Some minor points regarding English usage:
"above": a more conventional and elegant mathematical way to describe this would be: "(d) the set is maximal with respect to these properties". (However, this is also a little more abstract, and possibly harder for some people to understand.)
"one or more" vs "one or more than one": as a native speaker, I believe that the meaning of both phrases is identical; I see no loss of precision in using the shorter phrase. But you could avoid the issue entirely by writing "A player's strategy is a non-empty set of left-parts of move-sequences..."
"such that" vs "so that": both are frequently used in mathematics, however with slightly different nuances. Generally, "...A such that B..." refers to object A possessing property B (usually A would be a noun). On the other hand, "...A so that B..." refers to doing action A so that result B is obtained (usually A would be a verb). However, some writers are not consistent with this distinction, and it is always possible to discern the correct meaning from context, so it's not such an important issue.
Empty set: Yes, possible, of course. But it is also a bit harder to understand for non-mathematicians. Therefore I have refrained from using it in J2003 & Co.
Point taken for all the points you say, thanks.
"one or more than one": One alternative is to say "at least one".
"above": xela's "maximal set" suggestion is perfectly good for me. Yet another alternative is to say "(d) this set is not a proper subset of any other set satisfying (a), (b), and (c)".
In the areas I work in, I almost never see "so that" in definitions. I don't call myself a mathematician, so I presume you guys know better. =)
Using "maximal" in the definition text would make it really hard to read for some because one would have to understand why the maximum is necessarily a constant. Better add some proposition on maximality / completeness in some commentary.
This is the kind of reaction that could be avoided if the "paper" was a wiki page...Now, the page (and this discussion) depends on an external document that can come and go, and become outdated, reducing the value of this page.
And we have come to the point where we are wasting discussion pages on EXTERNAL DOCUMENTS: IMO this is not the way to go forward on SL.
That's why I think you should consider publishing stuff under the umbrella of your home page.
Now, you will have to face synchronisation issues between SL and your own documents. Personally this increasing dependency on external documents of yours feels wrong.
Notwithstanding the fact that your research is important... it feels like the way you share it may become a very thorny issue.
Actually, due to the nature of his contribution, this kind of discussion will occur regardless whether we are referring to SL articles or external documents.
Back in this post Robert gave a very long rant in response to my suggestions to him, and he is obviously unhappy that I complained about his English, and he also said that we should try to understand his research. At the same time it is his desire to post his research results on SL, which most of us have no objections to in principle. Because all of us are getting very tired of having so much meta-discussion, I decided to try something new. Since he plastered links to his ko paper over quite a number of pages on SL, that it becomes a source for ko-related information on SL, I decided to point out how difficult it is to parse the English in his ko paper itself. After all, it takes two hands to clap: the reader must make an effort to understand, but it is also his responsibility to make his research understandable by people. Hopefully, he gets his ko paper improved, more people can understand his ko paper, and his research results can be transferred to SL in a form a wider audience can understand. This way, all wins.
Unkx80, my reply was not only to you. (Otherwise I do not want to prolong related meta-discussion.)
The English of my ko paper is strict so that translation to pure maths is eased: The definitions are written more like maths definitions than like a novel and the comments in the examples use the defined terms instead of informal language. This creates an unpleasent English that is unlike common, everyday English. At the same time, the formalistic English makes it comparatively easy to verify things like "Is this intersection a local-ko-intersection - yes or no?".
Since using strict English in my paper serves important purposes, I will keep it. You might wonder whether additionally I would provide alternative explanations in the paper while using ordinary English. Very unlikely. More likely I would write some books about mathematical Go terms and explain things there suitably for the interested reader.
SL is a different place. There explanations for everybody are more urgent. So, given time, I will also add more explanations here that can be understood more easily than my paper. The first step is to omit some details for experts; in that somewhat simplified manner, I have explaned some things on the Mathematical Ko Terms and the Ko In General pages. Further steps towards making the contents available for readers with less expert knowledge can be made, of course. This might skip yet more details and concentrate on the core and use yet less strict language. Just the contents should remain essentially right though, even when expressed informally.
Regarding "strict English": I am full of admiration for mathematicians such as G.H. Hardy, Paul Halmos, John Conway and others who succeed in writing English that is not only clear and precise, but also concise and elegant, and often a pleasure to read. Such a thing is possible (although lamentably rare). It is not always the case that adding more words to a phrase will improve it, nor that rewriting a phrase more concisely will necessarily cause loss of precision.
In response to axd, I do think it can be constructive to use SL as a discussion forum for external documents. The particular document under discussion here is certainly relevant to many SL pages.
Since I invent or otherwise define a lot of terms, I try to pay particular attention to the choice of words for a term. The more specific a term's meaning is the harder this becomes though. E.g., when writing J2003 I was criticised for the term "hypothetical-strategy" instead of using "strategy". However, I knew that strategy and hypothetical-strategy are two different things. Had I followed the suggestion, I would now have to use a different word for strategy. Horrible thought...
Robert, I agree with everything you say. Since you wrote the ko paper in a definition upon definition style, it implies your desire for mathematical rigour, therefore I am expecting "strict English" and I am okay with it.
The problem is that in some cases, you use words that are too loose, like "above" when referring to multiple clauses. In other cases, you write using very strange sentence constructs, which don't seem to have one single well-defined meaning.
So far I am merely trying to parse the definitions in your ko paper. If I cannot even correctly parse them, there is no way I can understand it.
I got your point about ambiguity of "above". Please tell me if you see more ambiguity. - Complicated sentence structures: probably, it is so German-like, at least if one likes those sentences with 100 commas. I think though that each definition should have one sentence only to let it be very clear where the definition starts and ends. Therefore it is not always easy to construct simple sentence structures. Anyway, don't hesitate to make suggestions! It might happen that I like some of them. - Nevertheless, I imagine that the real problem of understanding the definitions is the contents. Proofreading their contents was demanding for me, so I imagine that reading for understanding must be demanding, too. Reading definitions is not light literature (although lighter than reading abbreviated proofs in maths journals, where a math student needs 30 minutes per text line to understand what is going on and then he is told by the professor that that was normal...).
axd, research usually starts with external research documents and then sooner or later finds its way to other media like Wikis. E.g., Einstein's relativity theories were not understood by many for years and only therefore he did not get his Nobel prize for his most important theories. Now you find every aspect of his theories in Wikipedia. Today speed of research proliferation is much faster because the internet has made it possible. This does not mean though that research papers would have to be published first on wikis. Rather (some German) universities disallow references from particular scientific works to wikis because of their dynamic nature. Open wikis are a bad place for original research documents but one good place for explaining them (or their core) to everybody. Hence I do not share your doubts.
Comment 3
"A player can answer-force something-1 if the opponent moving second uses an answer-strategy that - regardless of the player's first move - does force something-2 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 the something-1."
I assume this is intended to define the meaning of "answer-force something-1", where "something-1" acts as a kind of variable or property. Can you please explain what is "something-2"? What is its relation to "something-1"?
The defined term is "answer-force" but that term requires a context, which includes the two parameters something-1 and something-2 etc. This already explains what "something-2" is: another parameter. Its relation to something-1 is specified by the definition text, i.e., something-2 is pretty generic quite like something-1 is pretty generic. In 2-value logic, either parameter can be about any aussageform (German; maybe in English that is: expression). However, the following lemma is a direct implication of the definition: If a player can answer-force something-1, then something-2 unequals something-1. (Example: A player cannot answer-force "a cycle" if the opponent prevents "a cycle".)
I understand immediately once you say that both "something-1" and "something-2" are parameters of "answer-force". The problem with your statement is that "something-2" appears so far away from the term "answer-force" and separated by the word "if" that it reads like you are defining "answer-force" with only one parameter "something-1", with "something-2" looking like a free variable but not exactly so.
Let's try rewriting this definition slightly by using "given that" instead of "if". Others might come out with something better.
"A player playing first can answer-force something-1, given that the opponent uses an answer-strategy that forces something-2 (regardless of the player's first move), 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 the something-1."
By the way, I think aussageform means predicate.
I am a fan of "if" in mathematical definitions because always using it for a condition makes it easy to recognize it as such. "given" is used more for statements in presuppositions.
You are missing my point. The way you write this sentence can lead me to interpret it as:
answer-force(something-1) <=> (FORALL something-2, P(something-1, something-2))
where P(something-1, something-2) is the preposition "the opponent moving second uses an answer-strategy that - regardless of the player's first move - does force something-2" AND "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 the something-1".
However, you say you are defining answer-force(something-1, something-2).
That it is a definition is indicated by annotating answer-force in italics.
I know that it is a definition, no problem. Your intended definition translates more or less to predicate logic of the form:
answer-force(something-1, something-2) <=> P(something-1, something-2)
Whereas I can easily interpret your definition as:
answer-force(something-1) <=> (FORALL something-2, P(something-1, something-2))
These two clearly have different semantics.
The "if" is an informal abbreviation for "iff" aka ":<=>". The sentence structure is basically "left of if" | "if" | "right of if". This means that the definition works as follows:
(answer-force, player, something-1) :<=> P(player, something-1, something-2)
Yes, the sentence structure is clear. I think the issue is that something-2 appears on the right as a free variable but does not appear on the left. Either something-2 should also appear on the left, or else a quantifier is needed. From your verbal remarks above I would guess that you intend the former.
If one wanted a logician's precison in the text, then indeed it would make sense to declare also "something-2" on the left side of "if". Like: "For some arbitrary, particular something-2, an arbitrary, particular player can answer-force some arbitrary, particular something-1 if [...]" The text is semi-formal though and not on the level of logicians' strictness.
Robert, thanks for the clarification. Actually, your own more formal suggestion is so much clearer. Alternatively, you might want to use the following style:
"A player can answer-force something-1 against something-2 if...
When something-2 is obvious from the context, we simply say that a player can answer-force something-1."
If you don't like "against something-2", replace it with "after something-2" or any other phrase of your liking. Make sure that "something-2" appears together with "answer-force" before the first "if" to make it clear that "something-1" and "something-2" are both arguments of "answer-force".
Bass: A more readable naming scheme for parameters are the upper case alphabets, sometimes written in a bold and/or italic typeface to distinguish them from other text: A player can answer-force A, if the opponent forces B, and so on.
I agree, as far as my taste in concerned. With only one paramater (as in "force"), "something" may still be the best understood phrase. Maybe I copied too much wording from J2003. OTOH, not everybody would easily understand mere letters as paramaters. One can never find a wording that is easily accesible for all readers...
Version 7a corrects typos and examples 27+28 and slightly changes the wording (not the intended contents) of three definitions. See the paper's changes log at its end.
These comments refer to version 7a of your ko paper. All are minor.
Comment 4
"Under default restriction rules without history-bans, a local-ko-intersection is an intersection if a set of cycles exists so that
- each of the cycles starts from the start-position,
- each of the cycles has at least one play creating the current-position,
- the intersection belongs to each of the cycles' cycle-sets, and
- a player can answer-force one of the cycles by moving first in it if the opponent moving second does prevent local-area-improvement of the player on the cycle-set."
(a) For what I know, usually "... an intersection is a local-ko-intersection if ..." is preferred. The meaning seems a bit different.
(b) The "it" in the last clause can refer to "the intersection" or "the cycle" (more specifically, "an intersection in the cycle"). I presume you mean the former.
Ditto for the definition for global-ko-intersection.
Comment 5
The second sentence on page 10:
"One counter-example that should be a positive example would let the definition of ko-intersection be a failure since there is no general classification of all non-ko-intersections yet."
This sentence is a garden path sentence, because the initial part "One counter-example that should be a positive example" suggests something else. I suggest this instead:
"Since there is no general classification of all non-ko-intersections yet, any counter-example that is classified as a positive example by the definition of ko-intersection would make this definition a failure."
local-ko-intersection is being defined. Therefore it has to stand before the "is".
"moving first in it" cannot refer to "intersection" because one never moves "in" an intersection. The phrase can refer to "one of the cycles", and so it does. Moving first in a cycle means to make its first move.
Comment 5: ok, I will think about getting a better style for the sentence for maybe the next version of the paper.
local-ko-intersection is being defined. Therefore it has to stand before the "is".
Then the "if" should be replaced by "for which", ie:
Under default restriction rules without history-bans, a local-ko-intersection is an intersection for which a set of cycles exists so that
This is not a contest for best common English style. A mathematical definition written as a text almost requires an "if[f]" for its conditions. The consistency of standard words in maths definitions supports universal understanding, eh, among mathematicians. This I have seen from all my maths professors and in every English language maths book I read.
The phrase as it stands is wrong, plain and simple. That has nothing to do with "best common English style", it is just wrong. Using "for which" fixes it.
Robert, I have to agree with Herman here. What you were saying is that something of a more specialized concept is something of a more generalized concept if some conditions are satisfied, and you are defining the more specialized concept. What you actually want to say is that something of a more specialized concept is something of a more generalized concept for which certain properties are satisfied. Therefore, I suggested turning the two things around, and somebody else suggested substituting "if" with "for which". I believe either fixes the problem.
In my understanding, the semantics of the definition are the same regardless of whether "if" or "for which" is used to attach the conditions. The English grammatical construction differs but does not affect the contents.
Nope. The current version means:
Given a local-ko-intersection, it is true that it is a member of the set of intersections when the following conditions apply.
This definition assumes that the reader is already aware of what a local-ko-intersection is, and is now learning what intersections are.
What you want to say (presumably):
Given an intersection, it is true that it is a member of the set local-ko-intersections when the following conditions apply.
ie, this assumes that the reader is aware of what an intersection is, and is now learning what local-ko-intersections are.
The if(f) construct simply does not bind the way you want in that context.
My goodness, which mathematics books have you been reading? (It's a rhetorical question, no need for you to provide an answer ;-)
If you want your work to gain a wider acceptance, then you have to pay at least a little bit of attention to style. Bad or clumsy style often serves as a warning that mistakes are likely to happen (in mathematics as in go). On the point under discussion here, unkx80 and Herman are absolutely correct. The text as it stands is a definition of the word "intersection", assuming that "local-ko-intersection" is already understood. I'm sure that you don't intend this.
Regarding the use of the word "if": here are two common mathematical definitions:
http://en.wikipedia.org/wiki/Derivative
http://en.wikipedia.org/wiki/Group_(mathematics)
In each case you have to scroll past the introductory remarks and context to find the actual definition. In the first case, the word if occurs only in a subordinate clause; in the second, it is absent. I simply chose these particular definitions as the first things that came to mind. I could provide many similar examples. If you browse through the wikipedia pages on mathematics, or through a selection of books, you'll find a great variety in the sentence structures used for definitions.
I'm quoting wikipedia only because it's easily accessible. The style of those pages is consistent with many standard textbooks on the same subjects.
Please understand that I intend these comments in a constructive and supportive spirit (and I believe that unkx80 and Herman have the same attitude). I have great respect for your knowledge and experience as a rules expert. I would love to see this knowledge presented in a form that others can more easily benefit from.
In the short term I'm unlikely to continue following this discussion, as it's taking a large amount of time that I should be spending on other things. But it's very likely that I'll return to the topic in the future.
Best wishes.
The majority of the definitions in Robert's ko paper doesn't contain the word "if" either. =)
I understand your concern that local-ko-intersection rather than intersection is supposed to be defined. I just don't get it yet why the current sentence structure suggests the latter. Need to reflect it, or you might provide clearer explanations if you want me to understand that faster.
Ok, I'll try one more time:
Here I'll give two ways you could describe local-ko-intersection mathematically (where I is the set of intersections, L is the set of local-ko-intersections and P(x) is the set of conditions that must apply to an intersection x for it to be a local-ko-intersection)
First way:
∀i ∈ I (P(i) → i ∈ L)
- In English
- For all intersection in the set of all intersections, if the following conditions hold, then the intersection is a member of the set of local-ko-intersections.
- Simplified
- For each intersection, if the following conditions hold, the intersection is a local-ko-intersection.
- Change the order of the subclauses
- For each intersection, the intersection is a local-ko-intersection, if the following conditions hold.
- Simplify further
- An intersection is a local-ko-intersection if the following conditions hold.
Second way:
L = { i ∈ I : P(i) }
- In English
- The set of local-ko-intersections is the set of intersections in all intersections for which the following conditions hold.
- Simplify
- local-ko-intersections are intersections for which the following conditions hold.
- Singular
- a local-ko-intersection is an intersection for which the following conditions hold.
So as you can see, the constructs "if" and "for which" have completely different mathematical roots.
Note: to see that for which is used for the secong notation, see http://en.wikipedia.org/wiki/Set_notation#Focusing_on_the_membership_of_a_set (sixth paragraph, starting "A more powerful mechanism for denoting a set...")
Let me try this. Assume a set INTERSECTIONS is defined and you want to define a set LOCAL-KO-INTERSECTIONS using some predicate P(.). In set notation, you write it as follows:
"LOCAL-KO-INTERSECTIONS = {x ∈ INTERSECTIONS | P(x)}"
1
This is equivalent to saying:
"x ∈ LOCAL-KO-INTERSECTIONS <=> x ∈ INTERSECTIONS and P(x)"
Making it more English-like:
"x is a local-ko-intersection iff x is an intersection and P(x)"
From the clause after the iff, x is an intersection. Therefore, we replace x by intersection. After the replacement, the first clause after iff becomes redundant, and we remove it. We get:
"An intersection is a local-ko-intersection iff P(intersection)"
2
Consider how you want to express the set "{x ∈ INTERSECTIONS | P(x)}" in English:
"the set of elements x in INTERSECTIONS where P(x)"
But each element in INTERSECTIONS is an intersection. Therefore:
"the set of intersections where P(intersection)"
We now go back to "LOCAL-KO-INTERSECTIONS = {x ∈ INTERSECTIONS | P(x)}" and try to express the elements of LOCAL-KO-INTERSECTIONS:
"A local-ko-intersection is an intersection where P(intersection)"
(I believe "where" and "for which" means the same thing.)
Edit: I realized that Herman is trying to explain this along exactly the same lines as I do!
Yes, we posted these very similar explanations within a few minutes of each other. Talk about coincidence :)
Herman, Unkx80, the maths you explain is clear. In math annotation, you do it for the set of all intersections or the set of all local-ko-intersections while (in English) I do it for an arbitrary, particular (local-ko-)intersection. Set or element is not the problem, I think, because the set consists of all the elements, i.e., one can be derived from the other or vice versa.
We seem to agree that "where" or "for which" are suitable conjunctions here. I also think that "if" is suitable here and I do not understand (yet) why you disagree about the "if".
I think that "if" is equally possible here because a local-ko-intersection is an intersection only if it indeed is a local-ko-intersection, i.e., if it also meets the predicate's conditions. Otherwise what was considered to possibly be a local-ko-intersection does not exist as a local-ko-intersection. Either both "intersection" and the predicate are fulfilled (so we have some local-ko-intersection) or not both are fulfilled (so we do not have some local-ko-intersection).
I have an additional pure English language question related to "for which": Here we need a restrictive clause rather than a non-restrictive clause. Apart from no comma versus comma, restrictive clauses use "that" rather than "which", if we trust my Oxford grammer bible. So it should be "for that" rather than ",for which [...],".
You can use "that" instead of "for which", but not "for that".
a local-ko-intersection is an intersection for which the following conditions hold.
is equivalent to:
a local-ko-intersection is an intersection that satisfies the following conditions.
The difference is which part is the subject of the sentence. So in the first sentence, "conditions" is the subject, ie:
"The conditions hold for the intersection".
In the second, "intersection" is the subject, ie:
"The intersection satisfies the conditions".
In the first case, because intersection is not the subject, there is a preposition involved: "for". After a preposition English always uses which (or whom if it concerns a person), never that. See: http://en.wikipedia.org/wiki/English_relative_clauses#Use_with_preposition
As to why I object to the "if", I assumed that was clear from my explanation, but apparently it was not. The phrase:
a local-ko-intersection is an intersection if the following conditions hold
is equivalent to:
∀x ∈ L (P(x) → x ∈ I)
which is equivalent to:
I = { x ∈ L : P(x) }
which is equivalent to:
The set of all intersections consists of those local-ko-intersections for which the following conditions hold.
So with your construct, you are defining the set of all intersection based on the set of local-ko-intersections (which has presumably been given)
You assume too much. It is simpler. What I mean is: "...is i | P(i)". Or in English: "...is i so that P(i)". I have not used "so that" because that phrase occurs already later in the sentence.
I know what you mean, I was explaining what your sentence means. The sentence you wrote does not mean what you want it to mean. English simply does not work that way. You could perhaps use "so that" instead of "for which", but it would be a very clunky expression. Using "for which" is clear, correct, and unambiguous. I really don't understand why you object so strenuously to using correct English.
The parse for "A is B if C" is
[
[
[A]
[is
[B]
]
]
[if
[C]
]
]
whereas the parse for "A is B where C" is
[
[A]
[is
[
[B]
[where
[C]
]
]
]
]
That requires a parsing convention... I have meant the other parsing, which you state for "where". Sure, this ambiguity can happen when one uses text instead of symbols. Even "where" can be global. But now I see the point in using "for which": the parsing context becomes unambiguous.
So far my comments have been all about the English, I have not even started discussing the actual contents behind this set of definitions. Thus, these are the comments from someone who has no background in rules theory, but with a bit on training in discrete structures. However, I have expended far more effort in giving the feedback than I have anticipated. Therefore, I have to stop, and come back to the ko paper again probably another time.
I have one final question for now.
Can someone please explain to me what a "ko ban" is? Thanks.
Perhaps the person answering this question can create an article titled ko ban on SL.
"for which" replaces "if", a sentence introducing examples has a new wording, 3 diagrams have received a previously missing, trivial stone.
![[Welcome to Sensei's Library!]](images/stone-hello.png "Sensei's Library")