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Suppose we have a rules derived from a treebank. And in order to get a syntax tree of a given sentence we use the cyk algorithm. In order to use the cyk we should convert the rules into chomsky normal form, Such that all of my rules are A -> B C or A -> a (A,B,C are non-terminals and a is a terminal). From the cyk we get the most probable tree, constituted from the cnf rules. But what if we want to de-transform the tree to be constituted from the original rules? We can expand very easily the Binary rules into the original. But what about unit productions? for example; A->B->a transformed into A->a in cnf, how do we get the original unary rules path A->B->a? Any ideas will be helpful.

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I don't think you can recover a unit production, unless (and this is probably the common case) the unit production is unambiguous. If there is only one derivation from a non-terminal to a particular terminal -- and it should be easy to figure that out while you are doing the transformation to CNF -- then you can record that derivation and reinsert it when you find the collapsed derivation in the CYK parse.

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