Simple generic structures

AbstractA study of smooth classes whose generic structures have simple theory is carried out in a spirit similar to Hrushovski (Ann. Pure Appl. Logic 62 (1993) 147; Simplicity and the Lascar group, preprint, 1997) and Baldwin–Shi (Ann. Pure Appl. Logic 79 (1) (1996) 1). We attach to a smooth class 〈K0,≺〉 of finite L-structures a canonical inductive theory TNat, in an extension-by-definition of the language L. Here TNat and the class of existentially closed models of (TNat)∀=T+,EX(T+), play an important role in description of the theory of the 〈K0,≺〉-generic. We show that if M is the 〈K0,≺〉-generic then M∈EX(T+). Furthermore, if this class is an elementary class then Th(M)=Th(EX(T+)). The investigations by Hrushovski (preprint, 1997) and Pillay (Forking in the category of existentially closed structures, preprint, 1999), provide a general theory for forking and simplicity for the nonelementary classes, and using these ideas, we show that if 〈K0,≺〉, where ≺∈{⩽,⩽∗}, has the joint embedding property and is closed under the Independence Theorem Diagram then EX(T+) is simple. Moreover, we study cases where EX(T+) is an elementary class. We introduce the notion of semigenericity and show that if a 〈K0,≺〉-semigeneric structure exists then EX(T+) is an elementary class and therefore the L-theory of 〈K0,≺〉-generic is near model complete. By this result we are able to give a new proof for a theorem of Baldwin and Shelah (Trans. AMS 349 (4) (1997) 1359). We conclude this paper by giving an example of a generic structure whose (full) first-order theory is simple

Dynamic Probability Logics: Axiomatization & Definability

We first study probabilistic dynamical systems from logical perspective. To this purpose, we introduce the finitary dynamic probability logic} ($\mathsf{DPL}$), as well as its infinitary extension $\mathsf{DPL}_{\omega_1}\!$. Both these logics extend the (modal) probability logic ($\mathsf{PL}$) by adding a temporal-like operator $\bigcirc$ (denoted as dynamic operator) which describes the dynamic part of the system. We subsequently provide Hilbert-style axiomatizations for both $\mathsf{DPL}$ and $\mathsf{DPL}_{\omega_1}\!$. We show that while the proposed axiomatization for $\mathsf{DPL}$ is strongly complete, the axiomatization for the infinitary counterpart supplies strong completeness for each countable fragment $\mathbb{A}$ of $\mathsf{DPL}_{\omega_1}\!$. Secondly, our research focuses on the (frame) definability of important properties of probabilistic dynamical systems such as measure-preserving, ergodicity and mixing within $\mathsf{DPL}$ and $\mathsf{DPL}_{\omega_1}$. Furthermore, we consider the infinitary probability logic $\mathsf{InPL}_{\omega_1}$ (probability logic with initial probability distribution) by disregarding the dynamic operator. This logic studies {\em Markov processes with initial distribution}, i.e. mathematical structures of the form $\langle \Omega, \mathcal{A}, T, \pi\rangle$ where $\langle \Omega, \mathcal{A}\rangle$ is a measurable space, $T: \Omega\times \mathcal{A}\to [0, 1]$ is a Markov kernel and $\pi: \mathcal{A}\to [0, 1]$ is a $\sigma$-additive probability measure. We prove that many natural stochastic properties of Markov processes such as stationary, invariance, irreducibility and recurrence are $\mathsf{InPL}_{\omega_1}$-definable