Simple Models of Randomization and Preservation Theorems
Karim Khanaki, Massoud Pourmahdian
TL;DR
The paper analyzes which model-theoretic properties are preserved under the randomization construction $T o T^R$, proving that $NIP$ is preserved from a theory $T$ to its randomization $T^R$ and developing a uniform, combinatorial proof using simple models, indiscernible arrays, and the $Rademacher$ mean width. It also discusses stability preservation via a Littlestone-dimension approach and establishes a Shelah-style dichotomy for $T^R$ in terms of $NIP$ and $NSOP$, with a parallel treatment for continuous theories. The results bridge model theory and learning-theoretic tools, offering a quantitative, combinatorial account that avoids heavy analytic machinery. The work further sketches extensions to continuous theories, showing how the core ideas adapt beyond the classical discrete setting and outlining future directions for a full stability-preserving treatment in that context.
Abstract
The main purpose of this paper is to present a new and more uniform model-theoretic/combinatorial proof of the theorem ([5]): The randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$ is a (complete) first-order continuous theory with $NIP$. The proof method is based on the significant use of a particular type of models of $T^{R}$, namely simple models, certain indiscernible arrays, and Rademacher mean width. Using simple models of $T^R$ gives the advantage of re-proving this theorem in a simpler and quantitative manner. We finally turn our attention to $NSOP$ in randomization. We show that based on the definition of $NSOP$ given [13], $T^R$ is stable if and only if it is $NIP$ and $NSOP$.