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DOI: 10.4230/LIPIcs.STACS.2012.66
URN: urn:nbn:de:0030-drops-34059
URL: http://drops.dagstuhl.de/opus/volltexte/2012/3405/
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Elberfeld, Michael ; Jakoby, Andreas ; Tantau, Till

Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

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Abstract

An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved efficiently for inputs from $C$. The prime example is Courcelle's Theorem, which states that monadic second-order (MSO) definable problems are linear-time solvable on graphs of bounded tree width. We contribute new algorithmic meta theorems, which state that MSO-definable problems are (a) solvable by uniform constant-depth circuit families (AC0 for decision problems and TC0 for counting problems) when restricted to input structures of bounded tree depth and (b) solvable by uniform logarithmic-depth circuit families (NC1 for decision problems and #NC1 for counting problems) when a tree decomposition of bounded width in term representation is part of the input. Applications of our theorems include a TC0-completeness proof for the unary version of integer linear programming with a fixed number of equations and extensions of a recent result that counting the number of accepting paths of a visible pushdown automaton lies in #NC1. Our main technical contributions are a new tree automata model for unordered, unranked, labeled trees; a method for representing the tree automata's computations algebraically using convolution circuits; and a lemma on computing balanced width-3 tree decompositions of trees in TC0, which encapsulates most of the technical difficulties surrounding earlier results connecting tree automata and NC1.

BibTeX - Entry

@InProceedings{elberfeld_et_al:LIPIcs:2012:3405,
  author =	{Michael Elberfeld and Andreas Jakoby and Till Tantau},
  title =	{{Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{66--77},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{Christoph D{\"u}rr and Thomas Wilke},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2012/3405},
  URN =		{urn:nbn:de:0030-drops-34059},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2012.66},
  annote =	{Keywords: algorithmic meta theorem, monadic second-order logic, circuit complexity, tree width, tree depth}
}

Keywords: algorithmic meta theorem, monadic second-order logic, circuit complexity, tree width, tree depth
Seminar: 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)
Issue Date: 2012
Date of publication: 24.02.2012


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