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A Game for Counting Logic Formula Size and an Application to Linear Orders

Authors Grégoire Fournier , György Turán

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Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Models of learning
  • Theory of computation → Complexity theory and logic
Keywords
  • Finite Model Theory
  • Logical Aspects of Computational Complexity

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Abstract

Ehrenfeucht-Fraïssé (EF) games are a basic tool in finite model theory for proving definability lower bounds, with many applications in complexity theory and related areas. They have been applied to study various logics, giving insights on quantifier rank and other logical complexity measures. In this paper, we present an EF game to capture formula size in counting logic with a bounded number of variables. The game combines games introduced previously for counting logic quantifier rank due to Immerman and Lander, and for first-order formula size due to Adler and Immerman, and Hella and Väänänen. The game is used to prove an extension of a formula size lower bound of Grohe and Schweikardt for distinguishing linear orders, from 3-variable first-order logic to 3-variable counting logic.

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Grégoire Fournier and György Turán. A Game for Counting Logic Formula Size and an Application to Linear Orders. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 36:1-36:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CSL.2026.36

Author Details

Grégoire Fournier
  • University of Illinois at Chicago, IL, USA
György Turán
  • University of Illinois at Chicago, IL, USA
  • HUN-REN-SZTE Research Group on AI, Szeged, Hungary

Funding

Support from grants NSF 2217023 and NSF 2240532 is acknowledged. Support from Project 2024-1.2.3-HU-RIZONT-2024-00017 is acknowledged, financed by the Ministry of Culture and Innovation of Hungary from the National Research, Development and Innovation Fund, under the a 2024-1.2.3-HU-RIZONT funding scheme.

References

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