Parallel Play Saves Quantifiers
CoRR(2024)
摘要
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+ε, for arbitrarily small ε.
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