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Research papers in computer science often contain both positive and negative results. A positive result is, usually, an algorithm that solves a certain problem. A negative result is a proof that an algorithm does not exist.

I am looking for a phrase to use when moving from the presentation of a negative to the presentation of a positive result, or vice versa.

Usage example:

The above theorem showed that an exact solution to problem X does not exist.

[requested-phrase-here], an approximate solution does exist, as we show now.

Another example:

We have just proved a lower bound on the run time of any algorithm solving problem Y.

[requested-phrase-here], we now prove an upper bound.

I can use "However", but I want a phrase that emphasizes that we move from the negative part of the paper to the positive part. Maybe something like: "On the positive side", or: "The good news are that...". What do you suggest?

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    This question has potential. It seems the first example is coupling a negative and positive result--not exact, but approximate, while the second example is coupling inversions of a positive result--lower and upper limits. That fundamental difference tends to make choosing a word or phrase significantly more difficult because we cannot be sure exactly what you are after. – ScotM Jul 6 '15 at 19:07
  • To be honest, I'd go with "however"; 'the good news is…' doesn't sound like something I'd put in an academic paper. – anotherdave Jul 6 '15 at 19:23
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I understand the first example, and in that case you are looking for "Fortunately."

Fortunately: 1. Bringing something good and unforeseen; auspicious. 2. Having unexpected good fortune; lucky.

In the second example, I might need some more clarity on what you're looking for, as it seems you are going from negative to positive instead the other way around.

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  • A lower bound on the run-time is considered a negative result. An upper bound is considered positive. – Erel Segal-Halevi Jul 6 '15 at 19:03
  • Ok I think the example stands then! – Philip Devine Jul 6 '15 at 19:43
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Silver lining: a bright prospect

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  • Can you add how you would use it in a connective phrase? Something like "The silver lining is that..."? – herisson Jul 7 '15 at 3:06
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You could say 'by contrast' or 'in contrast'.

Contrast: distinction or emphasis of difference by comparison of opposite or dissimilar things, qualities, etc (esp in the phrases by contrast, in contrast to or with)

CED as found at thefreedictionary.com

The above theorem showed that an exact solution to problem X does not exist.

In contrast, an approximate solution does exist, as we show now.


The asker felt that the phrase 'in contrast' lacked a connotation of going from negative to positive. I do not agree, but nonetheless offer conversely as an alternative suggestion.

In a contrary or opposite way; on the other hand

CED as found at thefreedictionary.com

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  • "In contrast" shows that there is a contrast, but does not have the connotation of going from negative to positive. – Erel Segal-Halevi Jul 6 '15 at 19:02
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    I disagree. However, how about conversely, then? – Charon Jul 6 '15 at 19:04
  • Actually, a bit of research has thrown up some beauties: diametrically or contrariwise? The latter is synonymous and related to my first suggestion. – Charon Jul 6 '15 at 19:11
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From your initial characterization, a negative result precludes a positive result, since after proving there is no algorithm, you can't very well proceed to produce one. So it sounds to me like you're looking for a way to mislead readers. I think you ought not to be doing that.

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    The existence of an algorithm was an example. As was showing that an exact solution couldn't be obtained, but an approximation could. The OP does not state that he is publishing these examples. Even if he had, suggesting that he's looking for way to mislead people is unwarranted mind reading. I think you ought not to be doing that. – deadrat Jul 6 '15 at 19:29
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    Greg, I've noticed you posting comments as if they were answers lately. Posting an Answer should be reserved for a serious answer to the question. – ScotM Jul 7 '15 at 0:43
  • @ScotM, there is no appropriate way to connect contradictory results. So my answer, at least for the first, paradigm case, is: none. There is no appropriate connective. It seems to me that answers the question. – Greg Lee Jul 7 '15 at 4:10

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