# A universal tree balancing theorem

## TL;DR

This paper introduces a universal framework for balancing tree expressions, enabling efficient evaluation and transformation of complex algebraic and regular expression structures within strict computational bounds.

## Contribution

It provides a general method to compute balanced tree straight-line programs in DLOGTIME-uniform TC$^0$, reducing evaluation complexity for various algebraic structures and regular expressions.

## Key findings

- Balanced tree programs can be computed in DLOGTIME-uniform TC$^0$.
- Expression evaluation over arbitrary algebras can be reduced to a derived algebra.
- Transformations of expressions into shallow circuits are possible for semirings and regular expressions.

## Abstract

We present a general framework for balancing expressions (terms) in 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. It is shown that for every term one can compute in DLOGTIME-uniform TC$^0$ a tree straight-line program of logarithmic depth and size $O(n/\log n)$. This allows reducing the term evaluation problem over an arbitrary algebra $\mathcal{A}$ to the term evaluation problem over a derived two-sorted algebra $\mathcal{F}(\mathcal{A})$. Several applications are presented: (i) an alternative proof for a recent result by Krebs, Limaye and Ludwig 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.