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If two paths have the same relevant slice with output being the slicing criteria, then they have the same input-output relation.

I am not sure where we use being. Can we use being after the person or object to describe?

Jim, being a lovely boy, is liked by all his friends.

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The second example is a perfectly fine example of participial phrase.

Being a lovely boy, Jim is liked by all his friends.

I assume the first sentence is missing a the.

If two paths have the same relevant slice with the output being the slicing criteria, then they have the same input-output relation.

I don't understand what the sentence means, but you probably mean the following:

If two paths have the same relevant slice when the output is the slicing criteria, then they have the same input-output relation.

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    I don't see a problem with "the" being omitted in the first sentence. One could consider it elided, or simply that "output" is uncountable. – TecBrat Jun 12 '13 at 15:24
  • Yup, output is fine without the definite article. – Alan Carmack Aug 11 '16 at 22:49
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Both the examples you gave sound natural to me.

Other examples:

I like being at home.

Being kind to each other is important.

Reading, being so important in learning other subjects, must be mastered early.

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