License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CONCUR.2023.13
URN: urn:nbn:de:0030-drops-190075
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Haase, Christoph ; Piórkowski, Radosław

Universal Quantification Makes Automatic Structures Hard to Decide

LIPIcs-CONCUR-2023-13.pdf (1 MB)


Automatic structures are structures whose universe and relations can be represented as regular languages. It follows from the standard closure properties of regular languages that the first-order theory of an automatic structure is decidable. While existential quantifiers can be eliminated in linear time by application of a homomorphism, universal quantifiers are commonly eliminated via the identity ∀x.Φ≡¬(∃x.¬Φ). If Φ is represented in the standard way as an NFA, a priori this approach results in a doubly exponential blow-up. However, the recent literature has shown that there are classes of automatic structures for which universal quantifiers can be eliminated by different means without this blow-up by treating them as first-class citizens and not resorting to double complementation. While existing lower bounds for some classes of automatic structures show that a singly exponential blow-up is unavoidable when eliminating a universal quantifier, it is not known whether there may be better approaches that avoid the naïve doubly exponential blow-up, perhaps at least in restricted settings.
In this paper, we answer this question negatively and show that there is a family of NFA representing automatic relations for which the minimal NFA recognising the language after eliminating a single universal quantifier is doubly exponential, and deciding whether this language is empty is ExpSpace-complete.

BibTeX - Entry

  author =	{Haase, Christoph and Pi\'{o}rkowski, Rados{\l}aw},
  title =	{{Universal Quantification Makes Automatic Structures Hard to Decide}},
  booktitle =	{34th International Conference on Concurrency Theory (CONCUR 2023)},
  pages =	{13:1--13:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-299-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{279},
  editor =	{P\'{e}rez, Guillermo A. and Raskin, Jean-Fran\c{c}ois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-190075},
  doi =		{10.4230/LIPIcs.CONCUR.2023.13},
  annote =	{Keywords: automatic structures, universal projection, state complexity, tiling problems}

Keywords: automatic structures, universal projection, state complexity, tiling problems
Collection: 34th International Conference on Concurrency Theory (CONCUR 2023)
Issue Date: 2023
Date of publication: 07.09.2023

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