Given a fixed set of productions, G is the set of WCFGs with exactly those productions (i.e., they vary only in the production weights), G Z<∞ is the subset of G that defines (joint) probability distributions over trees (i.e., that have a finite partition function Z) and P Z<∞ is the set of probability distributions defined by grammars in G Z<∞ .
chi (1999) and Abney, McAllester, and Pereira (1999) proved that P Z<∞ is the same as P PcFG , the set of probability distributions defined by the PcFG G PcFG with the same productions as G. even though the set of WcFGs properly includes the set of PcFGs, WcFGs define exactly the same probability distributions over trees as PcFGs.