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Sorry if this is the wrong stackexchange site for this question.

I'm studying for a final for my english linguistics class and going through example sentences that we should be able to draw syntax trees for. This sentence: "He looked at the dog with one eye" was marked as ambiguous. The rules we were given for a generative English grammar are as follows:

Sentence [S] -> Noun Phrase (Auxiliary) Verb Phrase

Verb Phrase [VP] -> (Adverb) Verb (Prepositional Phrase) (Noun Phrase)

Noun Phrase [NP] -> (Determiner/Noun Phrase) (Adjective) Noun (Prepositional Phrase)

Prepositional Phrase [PP] -> Preposition Noun Phrase

X Phrase -> X Phrase Conjunction X Phrase (X being a placeholder for V, N or P)

X -> X Conjunction X

In this grammar, variables in parenthesis are optional for the phrase. So for the example sentence I have the tree

syntax tree

(made with http://mshang.ca/syntree/ if you find it helpful)

However, I'm having a hard time working out another tree. The phrase "with one eye" could refer to either be attached to "the dog" or directly to the verb phrase (and thus mean that 'He' only has one eye).

It seems to me that in order to represent the second meaning in a syntax tree, VP -> V (PP) (PP) would have to be a rule in the grammar.

Any help is greatly appreciated, thanks.

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    Is your question whether VP -> V (PP) (PP) would also have to be included in the rules, or are you asking whether your rule system implies such a rule? I.e. what is the question? – user6726 Dec 19 '15 at 21:32
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The problem is that the generative rule for VP is too weak. To handle 'at the dog with one eye' as two adverbial phrases, the rule should be something like VP -> [adverb] verb [[PP]] [NP] [[PP]] {allowing any number of sequential adverbial phrases}. Any PP generated under the NP formula would be restrictive to the NP, not adverbial. This new VP rule is still weak, as adverbial phrases include adverbs as well as prepositional phrases, and they can occur before the subject, as well as in the predicate: before or after the verb, or even after the object.

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