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What are the composition rules of these sentences?
For example: the C-Rule of "bring home the bacon" is: VC("the bacon",NP,AJ2)VA([home],A,AJ1);
Here is also a good reference to explain the C-Rules: "C-rule" in the UNL Wiki.
Many people seem to be under the impression that context free phrase structure grammar (cfpsg) cannot describe discontinuous constituents. It can, and these are good examples to show that. Suppose that we can derive a new phrase structure rule (psr) by composing two psrs, according to the scheme [A -> x B y](B -> z) = A -> xzy, where A, B are non-terminal symbols and x, y, z are strings of non-terminal and terminal symbols. The notation [rule i](rule j) = rule k means that rule i applies to rule j to derive rule k.
Then, for your examples, we have:
a. [VP -> let NP out of the bag](NP -> the cat) = VP -> let the cat out of the bag
b. [VP -> take off NP](NP -> clothes) = VP -> take off clothes
c. [VP -> burn NP at both ends](NP -> the candle) = VP -> burn the candle at both ends
Example b., as it stands, is of course not a discontinuous constituent, but with a particle movement rule, [VP -> take off NP] ==> [VP -> take NP off], can be converted into one. (A constituent is a string of terminal symbols such as "take off", and a discontinuous constituent is a string of terminal symbols interrupted by a non-terminal, such as "take NP off".)
This shows that the composition of discontinuous constituents, such as the ones involved in your examples, can be simply represented within a very well known theory, cfpsg, without appealing to an idiosyncratic framework.
