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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.
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.
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.
