This version of the parser is obtained by storing an index f rather than the full buffer β f in each configuration, and then grouping configurations that share common features, making them equivalent for the side conditions of the system: Instead of storing the full set of dependency links that the algorithm has constructed up to a given point (denoted by V), we only keep track of whether elements in the stack have been assigned a head or not; and we represent this by using a stack of pairs (l, b), where l is the position of a word in the string and b is a flag which is True if the corresponding node has been assigned a head or False if it has not: To obtain a parsing schema from this deduction system, we retain only rules pertaining to the way in which the parser joins dependency structures and builds links between them.
the Reduce step is just a mechanism to select which of a set of possible "linkable words" to link to the word currently being read.