A Universal Tree Balancing Theorem

Abstract

We present a general framework for balancing expressions (terms) in the form of so-called tree straight-line programs. The latter can be seen as circuits over the free term algebra extended by contexts (terms with a hole) and the operations, which insert terms/contexts into contexts. In Ref. [16], it was shown that one can compute for a given term of size n in logspace a tree straight-line program of depth O (log n ) and size O ( n / log n ). In the present article, it is shown that the conversion can be done in DLOGTIME-uniform TC 0 . This allows reducing the term evaluation problem over an arbitrary algebra A to the term evaluation problem over a derived two-sorted algebra F ( A ). Three applications are presented: (i) an alternative proof for a recent result by Krebs et al. [25] on the expression evaluation problem is given; (ii) it is shown that expressions for an arbitrary (possibly non-commutative) semiring can be transformed in DLOGTIME-uniform TC 0 into equivalent circuits of logarithmic depth and size O ( n / log n ); and, (iii) a corresponding result for regular expressions is shown.