Dynamic Probability Logics: Axiomatization & Definability
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913196329140224 |
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| author | Chopoghloo, Somayeh Pourmahdian, Massoud |
| author_facet | Chopoghloo, Somayeh Pourmahdian, Massoud |
| contents | 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 $\mathsf{DPL}_{ω_1}\!$. 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 $\mathsf{DPL}_{ω_1}\!$. 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 $\mathsf{DPL}_{ω_1}\!$. 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 $\mathsf{DPL}_{ω_1}$. Furthermore, we consider the infinitary probability logic $\mathsf{InPL}_{ω_1}$ (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 $\langle Ω, \mathcal{A}, T, π\rangle$ where $\langle Ω, \mathcal{A}\rangle$ is a measurable space, $T: Ω\times \mathcal{A}\to [0, 1]$ is a Markov kernel and $π: \mathcal{A}\to [0, 1]$ is a $σ$-additive probability measure. We prove that many natural stochastic properties of Markov processes such as stationary, invariance, irreducibility and recurrence are $\mathsf{InPL}_{ω_1}$-definable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07235 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamic Probability Logics: Axiomatization & Definability Chopoghloo, Somayeh Pourmahdian, Massoud Logic 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 $\mathsf{DPL}_{ω_1}\!$. 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 $\mathsf{DPL}_{ω_1}\!$. 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 $\mathsf{DPL}_{ω_1}\!$. 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 $\mathsf{DPL}_{ω_1}$. Furthermore, we consider the infinitary probability logic $\mathsf{InPL}_{ω_1}$ (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 $\langle Ω, \mathcal{A}, T, π\rangle$ where $\langle Ω, \mathcal{A}\rangle$ is a measurable space, $T: Ω\times \mathcal{A}\to [0, 1]$ is a Markov kernel and $π: \mathcal{A}\to [0, 1]$ is a $σ$-additive probability measure. We prove that many natural stochastic properties of Markov processes such as stationary, invariance, irreducibility and recurrence are $\mathsf{InPL}_{ω_1}$-definable. |
| title | Dynamic Probability Logics: Axiomatization & Definability |
| topic | Logic |
| url | https://arxiv.org/abs/2401.07235 |