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Massoud Pourmahdian

Publications and source records attributed to Massoud Pourmahdian.

At least 19 recordsLinked to original sources
arXiv2026

Goldblatt-Thomason Theorem for Probability Logic

Probability logic (PL) extends propositional logic with countably many probability operators, one for each rational number between 0 and 1. The formulas of this logic are interpreted over the class of Markov processes, i.e., structures of the form $(Ω, Σ, T)$, where$(Ω, Σ)$ is a measurable space and $T$ is a Markov kernel. The main contribution of this paper is the establishment of the Goldblatt-Thomason theorem for probability logic. As an application, we show that the class of Harsanyi type spaces is definable in PL. Moreover, we obtain some variants of the Goldblatt-Thomason theorem for specific subclasses of Markov processes.

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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

A Tame Generic Structure with Non-Algebraic Geometric Closure

By providing a procedure to apply Hrushovski's amalgamation method to the setting of classes of infinite structures, we introduce the notion of \textit{paracollapsed} structures. We show that this approach provides existentially closed generic structures in which the geometric closure is not included in the algebraic closure while the resulting theory is decidable. We show that paracollapsed structures have the strict order property and $\text{TP}_2$.

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arXiv2024

Dynamic Probability Logic: Decidability & Computability

In this article, the decidability and computability issues of dynamic probability logic (DPL) are addressed. Firstly, a proof system $\mathcal{H}_{DPL}$ is introduced for DPL and shown that it is weakly complete. Furthermore, this logic has the finite model property and so is decidable. Secondly, a strongly complete proof system HDPL is presented for DPL and proved that its canonical model is a computable structure.

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arXiv2024

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 $\mathsf{DPL}_{ω_1}\!$. 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.

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arXiv2023

The Craig Interpolation Property in First-order Gödel Logic

In this article, a model-theoretic approach is proposed to prove that the first-order Gödel logic, $\mathbf{G}$, as well as its extension $\mathbf{G}^Δ$ associated with first-order relational languages enjoy the Craig interpolation property. These results partially provide an affirmative answer to a question posed in [Aguilera, Baaz, 2017, Ten problems in Gödel logic].

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arXiv2022

A Dependent Bi-Coloured Field

We have considered a Fraisse class of finitely generated ordered real fields with a colour predicate. A predimension map is defined on finite sets and the Fraisse limit of the class is axiomatized by a theory $T$, which is proved to be dependent. The theory is proved to be non-distal with $dp-rank=\aleph_0$.

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arXiv2022

Bi-Colored Expansions of Geometric Theories

This paper concerns the study of Bi-colored expansions of geometric theories in the light of the Fraïssé-Hrushovski construction method. Substructures of models of a geometric theory $T$ are expanded by a color predicate $p$, and the dimension function associated with the pre-geometry of the $T$-algebraic closure operator together with a real number $0<α\leqslant 1$ is used to define a pre-dimension function $δ_α$. The pair $(\mathcal{K}_α^{+},\leqslant_α)$ consisting of all such expansions with a hereditary positive pre-dimension along with the notion of substructure $\leqslant_α$ associated to $δ_α$ is then used as a natural setting for the study of generic bi-colored expansions in the style of Fraïssé-Hrushovski construction. Imposing certain natural conditions on $T$, enables us to introduce a complete axiomatization $\mathbb{T}_α$ for the class of rich structures in this class. We will show that if $T$ is a dependent theory (NIP) then so is $\mathbb{T}_α$. We further prove that whenever $α$ is rational the strong dependence transfers to $\mathbb{T}_α$. We conclude by showing that if $T$ defines a linear order and $α$ is irrational then $\mathbb{T}_α$ is not strongly dependent.

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arXiv2019

Continuous integration logic

We combine continuous and integral logics and found a logical framework for metric measure spaces equipped with a family of continuous relations and operations. We prove the ultraproduct theorem and deduce compactness and other usual results. We also give applications of the compactness theorem in metric measure theory.

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arXiv2019

Strict Superstablity and Decidability of Certain Generic Graphs

We show that the Hrushovski-\fraisse limit of certain classes of trees lead to strictly superstable theories of various U-ranks. In fact, for each $ α\inω+1\backslash\{0\} $ we introduce a strictly superstable theory of U-rank $ α. $ Furthermore, we show that these theories are decidable and pseudofinite.

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arXiv2018

Pseudofiniteness in Hrushovski Constructions

In a relational language consisting of a single relation $ R, $ we investigate pseudofiniteness of certain Hrushovski constructions obtained via predimension functions. It is notable that the arity of the relation $ R $ plays a crucial role in this context. When $ R $ is ternary, by extending the methods developed in [BL12], we interpret $ \langle\mathbb{Q}^{+},<\rangle $ in the $ \langle\mathcal{K}^{+}_{0},\leq^{*}\rangle $-generic and prove that this structure is not pseudofinite. This provides a negative answer to the question posed in [EW09] (Question 2.6). This result, in fact, unfolds another aspect of complexity of this structure, along with undecidability and strict order property proved in [EW09] and [Bl12]. On the other hand, when $ R $ is binary, it can be shown that the $ \langle\mathcal{K}^{+}_{0},\leq^{*}\rangle $-generic is decidable and pseudofinite.

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arXiv2018

Probability Logic: A Model Theoretic Perspective

In this paper (propositional) probability logic ($PL$) is investigated from model theoretic point of view. First of all, the ultraproduct construction is adapted for $σ$-additive probability models, and subsequently when this class of models is considered it is shown that the compactness property holds with respect to a fragment of $PL$ called basic probability logic ($BPL$). On the other hand, when dealing with finitely-additive probability models, one may extend the compactness property for a larger fragment of probability logic, namely positive probability logic ($PPL$). We finally prove that while the Löwenheim-Skolem number of the class of $σ$-additive probability models is uncountable, it is $\aleph_0$ for the class of finitely additive probability models.

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arXiv2016

Automorphism Groups of Generic Structures: Extreme Amenability and Amenability

We investigate correspondences between extreme amenability and amenability of automorphism groups of Fraïssé-Hrushovski generic structures that are obtained from smooth classes, and their Ramsey type properties of their smooth classes, similar to Kechris, Pestov and Todorcevic, and Tatch Moore. In particular, we focus on some Fraïssé-Hrushovski generic structures that are obtained from pre-dimension functions. Using these correspondences, we prove that automorphism groups of ordered Hrushovski generic graphs are not extremely amenable in both cases of collapsed and uncollapsed. Moreover, we prove that automorphism groups of Fraïssé-Hrushovski generic structures that are obtained from pre-dimension functions with rational coefficients are not amenable.

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arXiv2016

Some Model Theoretic Properties of Non-AC Generic Structures

In the context of Hrushovski constructions we take a language $ \mathcal{L} $ with a ternary relation $ R $ and consider the theory of the generic models $ M^{*}_α, $ of the class of finite $ \mathcal{L}$-structures equipped with predimension functions $ δ_α, $ for $ α\in(0,1]\cap\mathbb{Q} $. The theory of generic structures of non-AC smooth classes have been investigated from different points of view, including decidability and their power in interpreting known structures and theories. For a rational $ α\in(0,1], $ first we prove that the theory of $ M^{*}_α $ admits a quantifier elimination down to a meaningful class of formulas, called \textit{closure formulas}; and on the other hand we prove that $ Th(M^{*}_α) $ does not have the finite model property.

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arXiv2015

Definable tree property for successors of cardinals

It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to indescribables but are much weaker in consistency strength. Also the consistency strength of definable tree property for successor of a singular cardinal is reduced to the existence of a supercompact cardinal and a measurable above it.

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arXiv2011

Computational Models of Certain Hyperspaces of Quasi-metric Spaces

In this paper, for a given sequentially Yoneda-complete T_1 quasi-metric space (X,d), the domain theoretic models of the hyperspace K_0(X) of nonempty compact subsets of (X,d) are studied. To this end, the $ω$-Plotkin domain of the space of formal balls BX, denoted by CBX is considered. This domain is given as the chain completion of the set of all finite subsets of BX with respect to the Egli-Milner relation. Further, a map $ϕ:K_0(X)\rightarrow CBX$ is established and proved that it is an embedding whenever K_0(X) is equipped with the Vietoris topology and respectively CBX with the Scott topology. Moreover, if any compact subset of (X,d) is d^{-1}-precompact, ϕis an embedding with respect to the topology of Hausdorff quasi-metric H_d on K_0(X). Therefore, it is concluded that (CBX,\sqsubseteq,ϕ) is an $ω$-computational model for the hyperspace K_0(X) endowed with the Vietoris and respectively the Hausdorff topology. Next, an algebraic sequentially Yoneda-complete quasi-metric D on CBX$ is introduced in such a way that the specialization order $\sqsubseteq_D$ is equivalent to the usual partial order of CBX and, furthermore, $ϕ:({\cal K}_0(X),H_d)\rightarrow({\bf C}{\bf B}X,D)$ is an isometry. This shows that (CBX,\sqsubseteq,ϕ,D) is a quantitative $ω$-computational model for (K_(X),H_d).

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