{
  "id": "2401.07235",
  "version": "v1",
  "published": "2024-01-14T09:23:39.000Z",
  "updated": "2024-01-14T09:23:39.000Z",
  "title": "Dynamic Probability Logics: Axiomatization & Definability",
  "authors": [
    "Somayeh Chopoghloo",
    "Massoud Pourmahdian"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-14T09:23:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "axiomatization",
      "definability",
      "infinitary counterpart supplies strong completeness",
      "markov processes",
      "finitary dynamic probability logic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}