EXPTIME Hardness of an n by n Custodian Capture Game

Round 1

Reviewer 1 Report

MInor erro 

Line 40; PITME should be PTIME

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Dear authors

 

thank you for the changes made. I am recommending the paper for acceptance. 

 

There are a number of issues which I expect the production office to address with you. I have not compiled a complete list, some examples are

line 119: at which the only BLACK can stop

line 242: the sentence is missing an "and" or a "which",

line 245: something is off with the brackets here (is there something missing?)

line 332: fee man --> free man. Also: you use both, "free-men" as well as "free men" throughout the document

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors performed an extensive rewriting and significantly improved their paper. The final proofs are still quite hard to follow, and would definitely benefit from the usage of an automated checking tool; but can be accepted also in their current form.

The authors clarified that the intended victory condition R.5 involves the total number of pieces captured during a whole game. This suggests that the number of pieces captured so far should be part of a game configuration; however, the authors never consider this (basically assuming that no pieces have been captured so far). It seems to me that the configuration they build is so that you won't capture any piece until when the game is won all at once; so the result should still hold. In any case, some further discussion on this point is needed (and should be done before the first dead-lock proofs).

There are several typos across the paper, and an overall revision pass is also needed, to fix them and improve some involved sentences. A (non comprehensive) list of few of them follows:

• Line 18: GAMES WITH INTERCEPTION... the all-caps looks very badly here; I would suggest using italic and quotes such as: ``Games with Interception...''
• Line 19: "May-yek Siam" => "May-yek in Siam" (and similarly with the following examples)
• Line 46: PITME => PTIME
• Line 48: EXPTIME => EXPSPACE
• Line 51: "position for the problem" => "position encoding the problem"
• Line 68: "...proves that of the custodian capture game between EXPTIME and EXPSPACE" is very hard to parse.
• Line 241: "...the number of times...", does this refer to the cyclic sequence or to some (unknown) history before the start of the cyclic sequence? Please clarify.
• Line 383: "fee-men" => "free-men"
• Line 563: "repetion" => "repetition"

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

The authors present proof that the Custodian Capture board game is EXPTIME. They claim this helps us understand that ancient cultures were playing board games with a time complexity similar to modern games such as Chess and Go. The authors claim this game is historically relevant because it is played in many different cultures, but do not present further explanations of the cultural variants of this game, do not explain how we know that ancient cultures play this game, and do not give references to help the reader. 

 

Here are some major points:

1. The title is misspelled ”EXPTME” -> “EXPTIME”
2. The authors should explain how the game was played in distant historic cultures and should explain how we are able to infer this knowledge. Most importantly even if this is not relevant for the proof, the authors should explain how the game usually would start and should give some explanation and or example of middle and end games.
3. The authors do not explain how the G-series games are defined in general and how the proof of their EXPTIME completeness works in general. The authors should give to the less-experienced reader sufficient background of what are the different time complexities classes and give a broad explanation of the time hierarchy theorem, this is fundamental contextual information required to help the reader understand if the theorems presented are correct or not. The authors should give some references to these fundamental aspects and theorems. The authors should also give references to similar papers that used the polynomial reduction to prove the complexity of other games.
4. The authors should define certain concepts as for example 12DNF and give a brief explanation of disjunctive normal forms before they use these concepts in the definitions.
5. Figure 2 is not a concrete example. Especially considered that at the moment we do not understand how conjunctions will be used to describe the gameplay. I would like an actual on the board game example.
6. The concept of rigid is not previously defined.
7. The proof of lemma 3 is wrong and it refers to the wrong figures, the authors copied lemma4 probably because it has a similar structure.
8. In conclusion, while I understand that in games like chess or go we want to show that they are EXPTIME, because, at the time, we knew that these games are hard for humans, but we did not know how hard they are for computers. I do not understand what we gain from an EXPTIME proof of a game we actually do not know how it is played. The reason I say this is that traditional board games are not usually played at different board sizes n and since n is finite we should somehow estimate also the constant factor of the time complexity, which is unfortunately strongly dependent on the initial game state, otherwise it is hard to know if the players during their middle game phase actually created a situation that allowed the kind context that makes the game EXPTIME. Furthermore, the EXPTIME proof only requires a to show a single game situation that makes the game EXPTIME but does not tell us much of what happens on average. For instance, one can imagine situations where the player avoids strategically to get into EXPTIME situations. For this reason, I do not think that this proof actually tells us much of the actual complexity of the game. In my point of view for this paper to be relevant, it should answer these other questions as well or at least should point them out.

Reviewer 2 Report

Dear authors

I very much enjoyed reading the manuscript, but some of the details of the proofs are beyond me, at least beyond what I can state with confidence.

I will ask the editors to consider my review only supplementary.

I found the content to be well written and easy to follow (up to the point where I got lost in the proofs). I actually enjoyed reading the paper and I do recommend it for acceptance.

 

 

Reviewer 3 Report

The paper presents a reduction of a custodian capture-based game to a known EXPTIME-complete game.

Even though the results seem plausible, their presentation is often vague and incomplete, making it impossible to be fully confident in the correctness of the result. In general, since the proof requires to handle a large number of cases, it feels like it may greatly benefit from the use of an automated proof assistant (e.g., Coq). Otherwise, it needs to be more clearly presented in order to be fully convincing.

Furthermore, the title of the paper itself (and corresponding parts in abstract and introduction) seem to misleadingly suggest that all custodian capture games are EXPTIME-complete. Custodian capture games vary wildly, and the authors' result only apply to a specific one, which is in fact artificially made by them and does not seem to be a natural generalisation of any existing one. For example, they assume (i) rook-like movement, which is a feature of about half of those games, with one-cell-at-a-time movement being equally popular; (ii) custodial capture of blocks of opponents, which is a pretty rare feature across those games (I found it in Hasami Shogi and not much else); (iii) a winning condition of capturing 5 pieces, which is not present in any version of the game as far as I know; (iv) arbitrary starting configurations, which also doesn't apply to existing games. I think the authors need to change the title to more accurately represent their contribution, and accordingly fix the presentation in the paper. For example, something like "EXPTIME Completeness of a Custodial Capture Based Game" sounds more honest.

Finally, the reference to related works and background is very limited, as witnessed by the only 3 citations present, and would benefit from being expanded. Overall, I think the paper needs some extensive work to make it into a proper contribution to a journal. More detailed comments follow.

SPECIFIC COMMENTS

• Page 1: The impact of the contribution is highly over-stated.
  The authors should rephrase the title, abstract and introduction to be honest about the real nature of their contribution, clarifying that they prove EXPTIME-completeness for a specific custodial capture based game which they invented.
  
  
• Page 1: The rules of the game are not clear.
  I didn't get that block of pieces were captured in rows until when I reached the examples. I am currently still uncertain about the interpretation of R5: is it five men from the start of the game, or five men in a single move? The later proofs seem to be based on the latter interpretation, but I'm not sure it is what the authors intended. Finally, there is no mention to the game setup: the authors assume an arbitrary starting configuration, and indeed they create one that could not be produced as a result of any legal evolution of a game. This is not what usually happens with custodial capture games and it should be explicitly stated.
  
  
• Page 2: the presentation of Figure 1 at this point of the paper is very hard to follow. I don't know whether it is reasonable to improve it; but the authors can also consider to postpone it to the end of the article, replacing it with very few sentences hinting at the ideas without details.
  
  
• In this paper's proofs, figures play a crucial role. Thus, they should closely follow the text where they are used, being in its immediate vicinity. Instead, they are often several pages ahead of the text, forcing the reader to continuously go back and forth. Furthermore, several separately-numbered figures are used to represent a single sequence of moves. This makes it more difficult for the reader to find them.
  In the revision, the authors should group figures related to a single sequence of moves (where needed, they could use sub-numberings like 13a,13b..), and make sure that every figure is very close to the point where it is used.
  
  
• Line 73: Figure 9 is not clear and doesn't help (it may as well be removed).
  
  
• Line 89-90: the proof of Lemma 4 is erroneously copy-pasted as the proof of Lemma 3 as well (you should refer to figures 30-33).
  
  
• Lemma 6: I think the case of the central black piece moving up by 1 cell is missing.
  
  
• Line 122: it is not clear why the upper part of a timer should be considered as a slight modification of a weak point; it looks significantly different. This also makes the proof of Lemma 8 not very convincing.
  
  
• Figures 67-68: the description given of figure 68 does not seem to match the detailed content in figure 67. I have the impression that Figure 68-69 are probably rotated by 180 degrees with respect with Figure 67, and they also assume more obstacles than in Figure 67. These discrepancies make it impossible to follow the text in lines 139-143 and cast doubts on later proofs. The authors should ensure that the figures match.
  
  
• Lemma 12: the proof is very hard to follow.
  The authors should be more explicit when they consider subsequent moves or alternative moves. Furthermore, their reasoning seem to assume that intermediate position in between of segments are possible, which is in contrast with Lemma 2. Given lemma 2, there are only few discrete points that are actually reachable by players: most of them are labelled with letters and few aren't; I suggest to label them all.
  Given the discreteness of allowed positions, "\bar{KH} - {K,H}", "\bar{CT1} - {T1}" and "\bar{AB} - {B}" seem to be only a confusing way to say "G", "C" and "A" respectively (or am I missing something?). Similarly, "\bar{BD} - {D}" could just be {C, C'} where C' is the unlabelled node next to C.
  These improvements would need to be propagated also to following proofs.
  
  
• Line 194: it is not clear to me which is the "gadget over \bar{AB}", and why it should prevent W to go upwards.
  
  
• Line 197: I don't think the same argument holds for the second part of the gadget. In the proof of the first part, the presence of w2 in the second part was crucial; and there is no w3 to be correspondingly used in the proof of the second part.
  
  
• Lemma 14: the proof is too vague and doesn't sound convincing to me.

 

TYPOS AND MINOR COMMENTS

• Title: "EXPTME" => "EXPTIME"
• Lines 2 and 10: "the opponent pieces" => "an orthogonal line of opponent pieces" could help with the unclarity of rule R3.
• Line 33: a reference or footnote about 12-DNF formulae could be useful. Also, it is not clear whether the fact that conjunctions are bounded to 12 terms is exploited in any way by this paper.
• Figure 2: x-WIN should be (black circle)-WIN.
• Line 140: the reference to Figure 67 should probably be Figure 68.
• Line 160: "we distinct a line..." is a convoluted sentence. Something more direct like: "We use AB to denote.... We use \bar{AB} to denote...." would be better.
• Line 190: I believe that "w2" should be "w1".