A Unified Method for Placing Problems in Polylogarithmic Depth

In this work we present a generic approach to obtain upper bounds. For this purpose we consider the term evaluation problem which is, given a term over some algebra and a valid input to the term, computing the value of the term on that input. In contrast to previous methods we allow the algebra to be completely general and consider the problem of obtaining an efficient upper bound for this problem. To that end we present a generic term evaluation algorithm that works in polylogarithmic depth. This allows us to obtain the same bounds as in the classical setting, and at the same time reach out to capture new problems. The term evaluation problem is a classical problem studied under many names such as formula evaluation problem, formula value problem etc.. Many variants of the problems where the algebra is well behaved have been studied. For example, the problem over the Boolean semiring or over the semiring (N,+,×). We extend this line of work. Our efficient term evaluation algorithm then serves as a tool for showing polylogarithmic depth upper bounds for various well-studied problems. To underline the utility of our result we show new bounds and reprove known results using our approach and thereby present a unified proof approach for problems of this nature. The spectrum of problems for which we apply our term evaluation algorithm is wide: in particular, the application of the algorithm we consider include (but are not restricted to) arithmetic formula evaluation, word problems for tree and visibly pushdown automata, and various problems related to bounded tree-width and clique-width graphs.