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Karim Khanaki

Publications and source records attributed to Karim Khanaki.

18 recordsLinked to original sources
arXiv2026

Keisler Measures and Generically Stable Random Types

We introduce the notions of $rgs$ and $irgs$ for Keisler measures, motivated by the study of generically stable random types and their associated Morley sequences. We obtain characterizations of these notions in terms of averages of classical first-order formulas over suitable probabilistic partitions (Theorems 3.2 and 3.3). We compare these notions with $fim$, $fam$, and self-averaging, and show that for types the notions $fim$, $irgs$, and $rgs$ coincide. We prove that every $irgs$ measure is dependent (Theorem 4.5); consequently, such measures are symmetric (Corollary 4.8). Furthermore, we show that for $irgs$ measures the model-theoretic instability events $\mathbf{O}^φ$, $\mathbf{I}^φ$, and $\mathbf{L}^φ$ have $\mathbb{P}_μ$-measure zero (Theorem 5.4), extending results from [8] beyond the $fim$ case.

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arXiv2026

On classification of continuous first order theories

We give several new characterizations of $IP$ (the independence property) and $SOP$ (the strict order property) for continuous first order logic and study their relations to the function theory and the Banach space theory. We suggest new dividing lines of unstable theories by the study of subclasses of Baire-1 functions and argue why one should not expect a perfect analog of Shelah's theorem, namely a theory is unstable iff it has $IP$ or $SOP$, for real-valued logics, especially for continuous logic.

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arXiv2025

Simple Models of Randomization and Preservation Theorems

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 $NSOP$.

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arXiv2025

Dependent measures in independent theories

We introduce the notion of dependence, as a property of a Keisler measure, and generalize several results of [HPS13] on generically stable measures (in $NIP$ theories) to arbitrary theories. Among other things, we show that this notion is very natural and fundamental for several reasons: (i) all measures in $NIP$ theories are dependent, (ii) all types and all $fim$ measures in any theory are dependent, and (iii) as a crucial result in measure theory, the Glivenko-Cantelli class of functions (formulas) is characterized by dependent measures.

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arXiv2025

Generic Stability and Modes of Convergence

We study generically stable types/measures in both classical and continuous logics, and their connection with randomization and modes of convergence of types/measures.

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arXiv2024

Glivenko-Cantelli classes and NIP formulas

We give several new equivalences of $NIP$ for formulas and new proofs of known results using [T87] and [HOR91]. We emphasize that Keisler measures are more complicated than types (even in $NIP$ context), in an analytic sense. Among other things, we show that, for a first order theory $T$ and formula $ϕ(x,y)$, the following are equivalent: (i) $ϕ$ has $NIP$ (for theory $T$). (ii) For any global $ϕ$-type $p(x)$ and any model $M$, if $p$ is finitely satisfiable in $M$, then $p$ is generalized $DBSC$ definable over $M$. In particular, if $M$ is countable, $p$ is $DBSC$ definable over $M$. (Cf. Definition 3.3, Fact 3.4.) (iii) For any global Keisler $ϕ$-measure $μ(x)$ and any model $M$, if $μ$ is finitely satisfiable in $M$, then $μ$ is generalized Baire-1/2 definable over $M$. In particular, if $M$ is countable, $p$ is Baire-1/2 definable over $M$. (Cf. Definition 3.5.) (iv) For any model $M$ and any Keisler $ϕ$-measure $μ(x)$ over $M$, \begin{align*} \sup_{b\in M}|\frac{1}{k}\sum_1^kϕ(p_i,b)-μ(ϕ(x,b))|\to 0 \end{align*} for almost every $(p_i)\in S_ϕ(M)^{\Bbb N}$ with the product measure $μ^{\Bbb N}$. (Cf. Theorem 4.3.) (v) Suppose moreover that $T$ is countable, then for any countable model $M$, the space of global $M$-finitely satisfied types/measures is a Rosenthal compactum. (Cf. Theorem A.1.)

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arXiv2023

Remarks on convergence of Morley sequences

We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on equivalences of convergent Morley sequences. We then introduce the notion of eventual $NIP$, as a property of a model, and give a variant of [KP18, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of Pillay [P18] on the model-theoretic meaning of Grothendieck's double limit theorem.

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arXiv2023

Grothendieck's Double Limit Theorem and Model Theory

This is an expository paper in Persian on Grothendieck's double limit theorem and its connection with the (neo-)stability project. We review recent results/observations and discuss historical and philosophical issues.

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arXiv2022

Dividing lines in unstable theories and subclasses of Baire 1 functions

We give a new characterization of $SOP$ (the strict order property) in terms of the behaviour of formulas in any model of the theory as opposed to having to look at the behaviour of indiscernible sequences inside saturated ones. We refine a theorem of Shelah, namely a theory has $OP$ (the order property) if and only if it has $IP$ (the independence property) or $SOP$, in several ways by characterizing various notions in functional analytic style. We point out some connections between dividing lines in first order theories and subclasses of Baire 1 functions, and give new characterizations of some classes and new classes of first order theories.

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arXiv2021

$\aleph_0$-categorical Banach spaces contain $\ell_p$ or $c_0$

This paper has three parts. First, we establish some of the basic model theoretic facts about $M_{\mathcal{T}}$, the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In~particular, we show: (1) $M_{\mathcal{T}}$ has the \emph{non independence property} (NIP); (2) every Banach space that is $\aleph_0$-categorical up to small perturbations embeds $c_0$ or $\ell_p$ ($1\leqslant p<\infty$) almost isometrically; consequently the (continuous) first-order theory of $M_{\mathcal{T}}$ does not characterize $M_{\mathcal{T}}$, up to almost isometric isomorphism.

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arXiv2019

Stability, NIP, and NSOP; Model Theoretic Properties of Formulas via Topological Properties of Function Spaces

We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, `Talagrand's stability', and explain the relationship between this property and NIP in continuous logic. Using a result of Bourgain, Fremlin and Talagrand, we prove the `almost definability' and `Baire~1 definability' of coheirs assuming NIP. We show that a formula $ϕ(x,y)$ has the strict order property if and only if there is a convergent sequence of continuous functions on the space of $ϕ$-types such that its limit is not continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein-Šmulian theorem.

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arXiv2018

Correspondences between model theory and Banach space theory

In \cite{K3} we pointed out the correspondence between a result of Shelah in model theory, i.e. a theory is unstable if and only if it has IP or SOP, and the well known compactness theorem of Eberlein and Šmulian in functional analysis. In this paper, we relate a {\em natural} Banach space $V$ to a formula $ϕ(x,y)$, and show that $ϕ$ is stable (resp NIP, NSOP) if and only if $V$ is reflexive (resp Rosenthal, weakly sequentially complete) Banach space. Also, we present a proof of the Eberlein-Šmulian theorem by a model theoretic approach using Ramsey theorems which is illustrative to show some correspondences between model theory and Banach space theory.

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arXiv2018

Remarks on Banach spaces determined by their finite dimensional subspaces

A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ such that $X$ is finitely representable (f.r.) in $Y$ and $Y$ is f.r. in $X$ then $Y$ is isometric to $X$. We provide a direct proof (without model theory) of the fact that every finitely determined space $X$ (isometrically) contains every (separable) space $Y$ which is finitely representable in $X$. We also point out how a similar argument proves the Krivine-Maurey theorem on stable Banach spaces, and give the model theoretic interpretations of some results.

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arXiv2017

Remarks on NIP in a model

We define the notion $ϕ(x,y)$ has $NIP$ in $A$, where $A$ is a subset of a model, and give some equivalences by translating results from [1]. Using additional material from [11] we discuss the number of coheirs when $A$ is not necessarily countable. We also revisit the notion "$ϕ(x,y)$ has $NOP$ in a model $M$" from [8].

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arXiv2017

NIP formulas and Baire 1 definability

In this short note, using results of Bourgain, Fremlin, and Talagrand \cite{BFT}, we show that for a countable structure $M$, a saturated elementary extension $M^*$ of $M$ and a formula $ϕ(x,y)$ the following are equivalent: (i) $ϕ(x,y)$ is NIP on $M$ (in the sense of Definition 2.1). (ii) Whenever $p(x)\in S_ϕ(M^*)$ is finitely satisfiable in $M$ then it is Baire 1 definable over $M$ (in sense of Definition 2.5).

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arXiv2016

Amenability, extreme amenability, model-theoretic stability, and dependence property in integral logic

This paper has three parts. First, we study and characterize amenable and extremely amenable topological semigroups in terms of invariant measures using integral logic. We prove definability of some properties of a topological semigroup such as amenability and the fixed point on compacta property. Second, we define types and develop local stability in the framework of integral logic. For a stable formula $ϕ$, we prove definability of all complete $ϕ$-types over models and deduce from this the fundamental theorem of stability. Third, we study an important property in measure theory, Talagrand's stability. We point out the connection between Talagrand's stability and dependence property (NIP), and prove a measure theoretic version of definability of types for NIP formulas.

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arXiv2015

Eberlein-Smulian compactness and Kolmogorov extension theorems; a model theoretic approach

This paper has two parts. First, we complete the proof of the Kolmogorov extension theorem for unbounded random variables using compactness theorem of integral logic which was proved for bounded case in [8]. Second, we give a proof of the Eberlein-Smulian compactness theorem by Ramsey's theorem and point out the correspondence between this theorem and a result in Shelah's classification theory.

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