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What is the difference between "underlay", "underlie" and their present continuous forms (and adjectives) "underlying" and "underlaying"?

And what is the right word to say something like:

The actual behavior of something depends on the underlying platform.

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Here's how I understand it:

Underlie means to lie under or to serve as a basis for something else.

There are many fundamental problems which underlie our failed relationship

Underlying refers to that which underlies, in this case the problems.

Underlay means to place something under something else or to provide something as support for something else; it can be contrasted with overlay. It may also be the past tense form of underlie.

The sound engineer decided to underlay the main track with some special effects

Underlaying refers to that which underlays, in this case the sound engineer.

So I hope this makes it clear that in your example sentence underlying is the correct word. The platforms lie under (and do not lay something else under) the actual behavior of something.

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It's the same situation as lie and lay. Like lie, underlie is intransitive; like lay, underlay is transitive. – John Lawler May 9 '13 at 22:56
@JohnLawler Hmmm, I don't think that's true. Some sources list underlie as transitive. The object of underlie is what the subject lies under, as in my example sentence, if you reduce it to: "Many problems underlie our failed relationship". – p.s.w.g May 9 '13 at 23:06
Is our failed relationship a direct object, then? How about DO tests? Not every NP that follows a verb is a DO; for instance, He weighs 100 kg is an intransitive sentence, because 100 kg is a measure phrase, not a DO. – John Lawler May 10 '13 at 0:01
@JohnLawler Isn't it? What else would our failed relationship be in this context? And what DO test are you referring to? – p.s.w.g May 10 '13 at 6:57
There are a number. Passivization, Extraction, and Insertion, for instance. Respectively, ??Our relationship is underlain by many problems, ?*It's our relationship that many problems underlie, ??Many problems underlie deeply our relationship. Fairly ambivalent results there; the first two should be good and the last bad for a normal DO. I suspect what we have here is some variation in transitivity. Not surprising, given the widespread variation between lie and lay, and the fact that underlie and underlay are both rare and technical. – John Lawler May 10 '13 at 14:34

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