Papers by Massoud Pourmahdian
arXiv (Cornell University), Jan 13, 2024
Our aim in this paper is twofold. We first study probabilistic dynamical systems from logical per... more Our aim in this paper is twofold. We first study probabilistic dynamical systems from logical perspective. To this purpose, we introduce the finitary dynamic probability logic (DPL), as well as its infinitary extension DPL ω1 . Both these logics extend the (modal) probability logic (PL) by adding a temporal-like operator (denoted as dynamic operator) which describes the dynamic part of the system. We subsequently provide Hilbert-style axiomatizations for both DPL and DPL ω1 . We show that while the proposed axiomatization for DPL is strongly complete, the axiomatization for the infinitary counterpart supplies strong completeness for each countable fragment A of DPL ω1 . Secondly, our research focuses on the (frame) definability of important properties of probabilistic dynamical systems such as measure-preserving, ergodicity and mixing within DPL and DPL ω1 . Furthermore, we consider the infinitary probability logic InPL ω1 (probability logic with initial probability distribution) by disregarding the dynamic operator. This logic studies Markov processes with initial distribution, i.e. mathematical structures of the form ⟨Ω, A, T, π⟩ where ⟨Ω, A⟩ is a measurable space, T ∶ Ω × A → [0, 1] is a Markov kernel and π ∶ A → [0, 1] is a σ-additive probability measure. We verify that the expressive power of InPL ω1 is strong enough for showing that the n-step transition probability T n of Markov kernel T , is InPL ω1 -definable. This fact implies that many natural stochastic properties of Markov processes such as stationary, invariance, irreducibility and recurrence are InPL ω1 -definable. These facts justify the importance of studying these particular extensions of PL.
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Papers by Massoud Pourmahdian