Universal Quantification Makes Automatic Structures Hard to Decide

Authors Christoph Haase , Radosław Piórkowski

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Author Details

Christoph Haase
  • Department of Computer Science, University of Oxford, UK
Radosław Piórkowski
  • Department of Computer Science, University of Oxford, UK


We would like to thank the anonymous reviewers for their comments and suggestions, which helped us to improve the presentation of this paper. We are grateful to Alessio Mansutti for spotting a bug in Section 4.

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Christoph Haase and Radosław Piórkowski. Universal Quantification Makes Automatic Structures Hard to Decide. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • automatic structures
  • universal projection
  • state complexity
  • tiling problems


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