Simple Models of Randomization and Preservation Theorems

2024/08/27 by Karim Khanaki, Khanaki, Karim, Massoud Pourmahdian +1
Computer Science · #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO) #Machine Learning and Algorithms

paper · pdf · doi:10.48550/arxiv.2408.15014

openalex publication_date 2024/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Independence in randomizations
• Independence relations in randomizations
• Definable closure in randomizations
• Generic stability, randomizations, and NIP formulas
• Algebraic Independence Relations in Randomizations