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In this form of a sentence, we don't put a comma after X.

We define a programming model as X to provide ...

However, what if `X' is a phrase with multiple elements?

We define a programming model as an abstraction of a lower hardware architecture layer, upper software layer, and connections between them, (?) to provide data structures and algorithms ...

I'm not sure because

  1. I need the comma because upper software layer ... between them is an added explanation, so it is enclosed between commas.
  2. I don't need the comma because an abstraction ... between them is just one component X as in We define ... as X to ....

Which is right?

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"the connections between them" is part of the list, and it seems to me that "data structures and algorithms" applies to all three list items.

In that case, I would use the comma after "them".

We define a programming model as an abstraction of a lower hardware architecture layer, upper software layer, and connections between them, to provide data structures and algorithms ...

If you leave the comma out, then it reads like "data structures and algorithms" only applies to "connections between them".
Not sure if it would make logical sense though, since you didn't finish the sentence.

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When in doubt, rather than study rules of punctuation, it is generally better to read the text out loud and consider where it is natural to pause or take a breath. If you put commas at these points, you will rarely go wrong.

  • That is probably better advice for native speakers than for English language learners. – The Photon Jul 20 '16 at 2:12

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