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arXiv · 2402.10293

Parallel Play Saves Quantifiers

Marco Carmosino·Ronald Fagin·Neil Immerman·Phokion Kolaitis·Jonathan Lenchner·Rik Sengupta·Ryan Williams
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Abstract

The number of quantifiers needed to express first-order properties is captured by two-player combinatorial games called multi-structural (MS) games. We play these games on linear orders and strings, and introduce a technique we call "parallel play", that dramatically reduces the number of quantifiers needed in many cases. Linear orders and strings are the most basic representatives of ordered structures -- a class of structures that has historically been notoriously difficult to analyze. Yet, in this paper, we provide upper bounds on the number of quantifiers needed to characterize different-sized subsets of these structures, and prove that they are tight up to constant factors, including, in some cases, up to a factor of $1+\varepsilon$, for arbitrarily small $\varepsilon$.

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Marco Carmosino, Ronald Fagin, Neil Immerman, Phokion Kolaitis, Jonathan Lenchner, Rik Sengupta, Ryan Williams. 2024-02-15. Parallel Play Saves Quantifiers. https://arxiv.org/abs/2402.10293

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