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I am reading the Wikpedia page on discontinuities, it says:

Worth noting is that a large majority of sentences in most languages are projective, i.e. they do not contain discontinuities. Only about 15-25% of actual sentences contain a discontinuity, and the percentage of discontinuous dependencies is even much less, approximately 1-2%.

A sentence contains a discontinuity if and only if it is non-projective sentences, but what does discontinuous dependencies mean?

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If there are crossing lines in the syntactic tree, then one or more discontinuous dependencies is present. A projectivity violation is a discontinuous dependency. There are a lot of terms used to denote such dependencies, many of which are listed at the start of that Wikipedia article (discontinuous dependency, long distance dependency, displaced constituent, discontinuity, projectivity violation, etc.). These terms are denoting the same thing for the most part, although there are preferences acoss various theories.

Looking at some of the trees in that article, the discontinous dependencies are the ones where the solid dependency edge (line) crosses one of the dotted projection lines. Note that each of the trees that contains crossing lines contains just one such discontinuous dependency.

The interesting question concerns how theories of syntax address the crossing lines. Chomskyan syntax has traditionally assumed syntactic movement: https://en.wikipedia.org/wiki/Syntactic_movement. Other grammars (HPSG, LFG, some DGs) assume feature passing instead. In any case, most theories examine the tree structure and consider the path through the tree that one can trace from the displaced unit to its governor/head.

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