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I submitted a research paper and the reply has come back. The reviewers are positive overall, but one of them pointed out a flaw in one of my proofs. This happened because I made a wrong assumption regarding the proof. Now I have found a way to get around this and have a better proof. How do I present that my previous proof was wrong, the reviewer was right in saying so, and that now I have a better one?

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Just fix your paper to include the better proof and thank (in the acknowledgments) the anonymous reviewer for pointing out the flaw in the earlier version of the proof. Also, when you submit the revised version of your paper, you should mention that you have corrected the proof following the suggestions of the reviewer in the cover letter (or in the response to referees if you have it separated from the cover letter). –  just-learning Mar 17 at 15:26
    
What's the editor's decision? –  Ri49 Mar 17 at 17:45
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Is this a journal review where you get to submit a revision, or a CS conference "author rebuttal period" where you cannot submit a revision, but only respond to the reviewers comments? –  Aaron Mar 17 at 18:54
    
@Ri49 Editor has asked us to address the reviewer's comments and submit the revision. –  dineshdileep Mar 18 at 3:54
    
@Aaron Yes, it is a journal review with a revision policy –  dineshdileep Mar 18 at 3:54

2 Answers 2

If the reviewers and subsequently the editor does not consider the error enough to warrant rejection, you should simply make the changes necessary and submit the revised manuscript according to instructions. Since the reviewers spotted the errors you should acknowledge their helpful comments (unless this is not the custom in the field for some reason).

It seems to me that you may be lucky to get the chance to revise the manuscript since the type of error you describe, I would think, would likely lead to rejection. But, with a favourable verdict you just revise the manuscript and return it as per instructions.

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I have rewritten the manuscript and have written a response to the reviewers. I was just curious as to how I should be starting the reply in the new proof. Like "I apologize for the apparent mishap in the previous proof which happened due to making this assumption. Now I realize I shouldn't have made that assumption. So and So" or should I just write the new proof, acknowledge their helpful comments and give it back. –  dineshdileep Mar 18 at 1:14
    
@dineshdileep "Thank you for identifying the incorrect assumption in the proof. We have updated the proof in Section x.y." In a good "reply to reviewers" letter, you should outline each point the reviewers make and point them to the correction. That is more important than apologizing. –  Irwin Mar 18 at 22:09

Just fix the proof. If you feel that the reviewer gave helpful suggestions about how to fixe the proof (specifically, going beyond just pointing out that the original version was wrong), you could consider adding an Acknowledgments section thanking the anonymous referee for their suggestions that improved the proof of Theorem X.

You don't need to apologize for your mistakes, either in the paper or your reply to the referees' comments. In your reply, you should of course acknowledge the error, thank the referee for pointing it out and say that you've fixed the problem.

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