LIPIcs.MFCS.2024.34.pdf
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On the Number of Quantifiers Needed to Define Boolean Functions
Authors
Marco Carmosino 
,
Ronald Fagin ,
Neil Immerman ,
Phokion G. Kolaitis ,
Jonathan Lenchner ,
Rik Sengupta
-
Part of:
Volume:
49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Mathematical Foundations of Computer Science (MFCS) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-08-23
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Document Identifiers
Author Details
- University of California Santa Cruz, CA, USA
- IBM Research-Almaden, San Jose, CA, USA
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 Get BibTex
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
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