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# On the Number of Quantifiers Needed to Define Boolean Functions

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LIPIcs.MFCS.2024.34.pdf
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## Acknowledgements

The authors acknowledge Ryan Williams for numerous helpful discussions and conversations, Sebastian Pfau for an observation that improved the statement of the Parallel Play Lemma, and the anonymous reviewers for comments and suggestions that improved the quality of this manuscript.

## Cite As

Marco Carmosino, Ronald Fagin, Neil Immerman, Phokion G. Kolaitis, Jonathan Lenchner, and Rik Sengupta. On the Number of Quantifiers Needed to Define Boolean Functions. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.34

## Abstract

The number of quantifiers needed to express first-order (FO) properties is captured by two-player combinatorial games called multi-structural games. We analyze these games on binary strings with an ordering relation, using a technique we call parallel play, which significantly reduces the number of quantifiers needed in many cases. Ordered structures such as strings have historically been notoriously difficult to analyze in the context of these and similar games. Nevertheless, in this paper, we provide essentially tight bounds on the number of quantifiers needed to characterize different-sized subsets of strings. The results immediately give bounds on the number of quantifiers necessary to define several different classes of Boolean functions. One of our results is analogous to Lupanov’s upper bounds on circuit size and formula size in propositional logic: we show that every Boolean function on n-bit inputs can be defined by a FO sentence having (1+ε)n/log(n) + O(1) quantifiers, and that this is essentially tight. We reduce this number to (1 + ε)log(n) + O(1) when the Boolean function in question is sparse.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Complexity theory and logic
• Theory of computation → Computational complexity and cryptography
##### Keywords
• logic
• combinatorial games
• Boolean functions
• quantifier number

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## References

1. Marco Carmosino, Ronald Fagin, Neil Immerman, Phokion Kolaitis, Jonathan Lenchner, and Rik Sengupta. Multi-structural games and beyond, 2023. URL: https://doi.org/10.48550/arXiv.2301.13329.
2. Marco Carmosino, Ronald Fagin, Neil Immerman, Phokion Kolaitis, Jonathan Lenchner, and Rik Sengupta. On the number of quantifiers needed to define boolean functions, 2024. URL: https://arxiv.org/abs/2407.00688.
3. Andrzej Ehrenfeucht. An application of games to the completeness problem for formalized theories. Fundamenta Mathematicae, 49:129-141, 1961. URL: https://doi.org/10.4064/fm-49-2-129-141.
4. Ronald Fagin, Jonathan Lenchner, Kenneth W. Regan, and Nikhil Vyas. Multi-structural games and number of quantifiers. In 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, Rome, Italy, June 29 - July 2, 2021, pages 1-13, Rome, Italy, 2021. IEEE. URL: https://doi.org/10.1109/LICS52264.2021.9470756.
5. Ronald Fagin, Jonathan Lenchner, Nikhil Vyas, and R. Ryan Williams. On the number of quantifiers as a complexity measure. In Stefan Szeider, Robert Ganian, and Alexandra Silva, editors, 47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022, August 22-26, 2022, Vienna, Austria, volume 241 of LIPIcs, pages 48:1-48:14, Vienna, Austria, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2022.48.
6. Steven Fortune. A note on sparse complete sets. SIAM Journal on Computing, 8(3):431-433, 1979. URL: https://doi.org/10.1137/0208034.
7. Roland Fraïssé. Sur quelques classifications des systèmes de relations. Université d’Alger, Publications Scientifiques, Série A, 1:35-182, 1954. URL: https://doi.org/10.2307/2963939.
8. Martin Grohe and Nicole Schweikardt. The succinctness of first-order logic on linear orders. Log. Methods Comput. Sci., 1(1), 2005. URL: https://doi.org/10.2168/LMCS-1(1:6)2005.
9. L. Hella and K. Luosto. Game characterizations for the number of quantifiers. Mathematical Structures in Computer Science, pages 1-20, 2024.
10. Lauri Hella and Jouko Väänänen. The size of a formula as a measure of complexity. In Asa Hirvonen, Juha Kontinen, Roman Kossak, and Andrés Villaveces, editors, Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, pages 193-214. De Gruyter, Berlin, München, Boston, 2015. URL: https://doi.org/doi:10.1515/9781614516873.193.
11. Neil Immerman. Number of quantifiers is better than number of tape cells. J. Comput. Syst. Sci., 22(3):384-406, 1981. URL: https://doi.org/10.1016/0022-0000(81)90039-8.
12. Neil Immerman. Descriptive Complexity. Springer, New York USA, 1999.
13. Oleg Lupanov. On a method of circuit synthesis. Izvestia VUZ Radiofizika, 1(1):120-140, 1958.
14. Oleg Lupanov. On the realization of functions of logical algebra by formulae of finite classes (formulae of limited depth). Problems of Cybernetics, 6(6):1-14, 1965. Upper bounds on sizes of formulas for all functions (English translation of Problemy Kibernetiki 6 (1961) 5-14.).
15. Stephen R. Mahaney. Sparse complete sets for NP: Solution of a conjecture of berman and hartmanis. Journal of Computer and System Sciences, 25(2):130-143, 1982. URL: https://doi.org/10.1016/0022-0000(82)90002-2.
16. John Riordan and C E Shannon. The number of two-terminal series-parallel networks. Journal of Mathematics and Physics, 21(1-4):83-93, 1942. URL: https://doi.org/10.1002/sapm194221183.
17. Joseph G. Rosenstein. Linear Orderings. Academic Press, New York USA, 1982.
18. Harry Vinall-Smeeth. From quantifier depth to quantifier number: Separating structures with k variables, 2024. URL: https://doi.org/10.48550/arXiv.2311.15885.
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