The probability of an SCFG rule instance computed by Algorithm 1 can be written in this functional form: where and the MRF has one factor f i for each child nonterminal X i in the grammar rule R. The factor's value is the probability of the child nonterminal, which can be expressed as a function of its four boundaries: For reasons that are explained in the following section, we augment our Markov Random Fields with a dummy factor for the completed parent nonterminal's chart item.
we have shown in the exponent in the complexity of polynomial-time parsing algorithms for synchronous context-free grammars grows linearly with the length of the grammar rules.