On the Number of Quantifiers as a Complexity Measure

Authors Ronald Fagin , Jonathan Lenchner , Nikhil Vyas , Ryan Williams

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Ronald Fagin
  • IBM Research, Almaden, CA, USA
Jonathan Lenchner
  • IBM T.J. Watson Research Center, Cambridge, MA, USA
Nikhil Vyas
  • MIT EECS, Cambridge, MA, USA
Ryan Williams
  • MIT CSAIL and EECS, Cambridge, MA, USA

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Ronald Fagin, Jonathan Lenchner, Nikhil Vyas, and Ryan Williams. On the Number of Quantifiers as a Complexity Measure. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


In 1981, Neil Immerman described a two-player game, which he called the "separability game" [Neil Immerman, 1981], that captures the number of quantifiers needed to describe a property in first-order logic. Immerman’s paper laid the groundwork for studying the number of quantifiers needed to express properties in first-order logic, but the game seemed to be too complicated to study, and the arguments of the paper almost exclusively used quantifier rank as a lower bound on the total number of quantifiers. However, last year Fagin, Lenchner, Regan and Vyas [Fagin et al., 2021] rediscovered the game, provided some tools for analyzing them, and showed how to utilize them to characterize the number of quantifiers needed to express linear orders of different sizes. In this paper, we push forward in the study of number of quantifiers as a bona fide complexity measure by establishing several new results. First we carefully distinguish minimum number of quantifiers from the more usual descriptive complexity measures, minimum quantifier rank and minimum number of variables. Then, for each positive integer k, we give an explicit example of a property of finite structures (in particular, of finite graphs) that can be expressed with a sentence of quantifier rank k, but where the same property needs 2^Ω(k²) quantifiers to be expressed. We next give the precise number of quantifiers needed to distinguish two rooted trees of different depths. Finally, we give a new upper bound on the number of quantifiers needed to express s-t connectivity, improving the previous known bound by a constant factor.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • number of quantifiers
  • multi-structural games
  • complexity measure
  • s-t connectivity
  • trees
  • rooted trees


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