For x, z ∈ Σ and y ∈ Σ + we have that is the weight that the grammarḠ assigns to a string xyz that has a cut or bracketing around y. Technically, it corresponds to the sum of the weights of all derivations that have a constituent spanning y.
we have that If x is a string of length m, and is the substring of x from positions t to t , it also happens that for any t between 1 and m. In this paper we will make frequent use of inside and outside quantities.