Branching Programs for Tree Evaluation

Abstract

The problem $FT^{h}_{d}(k)$ consists in computing the value in [k] = {1,...,k} taken by the root of a balanced d-ary tree of height h whose internal nodes are labelled with d-ary functions on [k] and whose leaves are labelled with elements of [k]. We propose ${FT^{h}_{d}(k)}$ as a good candidate for witnessing ${\mathbf{L}} \subsetneq{\mathbf{LogDCFL}}$. We observe that the latter would follow from a proof that k-way branching programs solving ${FT^{h}_{d}(k)}$ require $\Omega(k^{\mbox{\scriptsize unbounded function}(h)})$ size. We introduce a “state sequence” method that can match the size lower bounds on $FT^{h}_{d}(k)$ obtained by the Nec̆iporuk method and can yield slightly better (yet still subquadratic) bounds for some nonboolean functions. Both methods yield the tight bounds Θ(k ³) and Θ(k ^5/2) for deterministic and nondeterministic branching programs solving $FT^{3}_{2}(k)$ respectively. We propose as a challenge to break the quadratic barrier inherent in the Nec̆iporuk method by adapting the state sequence method to handle $FT^{4}_{d}(k)$.