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A Universal Tree Balancing Theorem

Published:22 October 2018Publication History
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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 TC0. 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 TC0 into equivalent circuits of logarithmic depth and size O(n/ log n); and, (iii) a corresponding result for regular expressions is shown.

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    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 11, Issue 1
      March 2019
      145 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/3287761
      Issue’s Table of Contents

      Copyright © 2018 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 22 October 2018
      • Accepted: 1 August 2018
      • Revised: 1 May 2018
      • Received: 1 October 2017
      Published in toct Volume 11, Issue 1

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