License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2017.27
URN: urn:nbn:de:0030-drops-81435
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8143/
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Rossman, Benjamin

An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity

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LIPIcs-ITCS-2017-27.pdf (0.7 MB)

Abstract

Previous work of the author [Rossmann'08] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence of quantifier-rank poly(k). Quantitatively, this improves the result of [Rossmann'08], where the upper bound on quantifier-rank is a non-elementary function of k.

BibTeX - Entry

@InProceedings{rossman:LIPIcs:2017:8143,
  author =	{Benjamin Rossman},
  title =	{{An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{27:1--27:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Christos H. Papadimitriou},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8143},
  URN =		{urn:nbn:de:0030-drops-81435},
  doi =		{10.4230/LIPIcs.ITCS.2017.27},
  annote =	{Keywords: circuit complexity, finite model theory}
}

Keywords: circuit complexity, finite model theory
Collection: 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)
Issue Date: 2017
Date of publication: 28.11.2017

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