The impact of state merging on predictive accuracy in probabilistic tree automata: Dietze's conjecture revisited

Dietze's conjecture concerns the problem of equipping a tree automaton M with weights to make it probabilistic, in such a way that the resulting automaton N predicts a given corpus $C$ as accurately as possible. The conjecture states that the accuracy cannot increase if the states in M are merged with respect to an equivalence relation ∼ so that the result is a smaller automaton $M^{\sim }$. Put differently, merging states can never improve predictions. This is under the assumption that both M and $M^{\sim }$ are bottom-up deterministic and accept every tree in $C$. We prove that the conjecture holds, using a construction that turns any probabilistic version $N^{\sim }$ of $M^{\sim }$ into a probabilistic version N of M, such that N assigns at least as great a weight to each tree in $C$ as $N^{\sim }$ does.