Branching Programs for Tree Evaluation

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 L ( LogDCFL. We observe that the latter would follow from a proof that k-way branching programs solving FT h d (k) require Ω(k unbounded ) 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) and Θ(k) 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).