Due to the collapsing of transitive predictions, this step can be implemented in a very efficient and straightforward manner. As explained in Section 4.5, one has to perform a single pass over the current state set, identifying all nonterminals Z occurring to the right of dots, and add states corresponding to all productions that are reachable through the left-corner relation . As indicated in equation (3), contributions to the forward probabilities of new states have to be summed when several paths lead to the same state. However, the summation in equation (3) can be optimized if the values for all old states with the same nonterminal Z are summed first, and then multiplied by . These quantities are then summed over all nonterminals Z, and the result is once multiplied by the rule probability to give the forward probability for the predicted state.