Fractional Pebbling and Thrifty Branching Programs

We study the branching program complexity of the {em tree evaluation problem}, introduced in cite{BrCoMcSaWe09} as a candidate for separating nl fromlogcfl. The input to the problem is a rooted, balanced $d$-ary tree of height$h$, whose internal nodes are labelled with $d$-ary functions on$[k]={1,ldots,k}$, and whose leaves are labelled with elements of $[k]$.Each node obtains a value in $[k]$ equal to its $d$-ary function applied to the values of its $d$ children. The output is the value of the root. Deterministic $k$-way branching programs as related to black pebbling algorithms have been studied in cite{BrCoMcSaWe09}. Here we introduce the notion of {em fractional pebbling} of graphs to study non-deterministicbranching program size. We prove that this yields non-deterministic branching programs with $Theta(k^{h/2+1})$ states solving the Boolean problem ``determine whether the root has value 1\u27\u27 for binary trees - this isasymptotically better than the branching program size corresponding toblack-white pebbling. We prove upper and lower bounds on the fractionalpebbling number of $d$-ary trees, as well as a general result relating thefractional pebbling number of a graph to the black-white pebbling number. We introduce a simple semantic restriction called {em thrifty} on $k$-way branching programs solving tree evaluation problems and show that the branchingprogram size bound of $Theta(k^h)$ is tight (up to a constant factor) for all $hge 2$ for deterministic thrifty programs. We show that thenon-deterministic branching programs that correspond to fractional pebbling are thrifty as well, and that the bound of $Theta(k^{h/2+1})$ is tight for non-deterministic thrifty programs for $h=2,3,4$. We hypothesise that thrifty branching programs are optimal among $k$-way branching programs solving the tree evaluation problem - proving this for deterministic programs would separate lspace from logcfl, and proving it for non-deterministic programs would separate nl from logcfl