Natural density and probability, constructively

We give here a constructive account of the frequentist approach to probability, by means of natural density. Using this notion of natural density, we introduce some probabilistic versions of the Limited Principle of Omniscience. Finally we give an attempt general definition of probability structure which is pointfree and takes into account abstractely the process of probability assignment

Models of Intuitionistic Set Theory in Subtoposes of Nested Realizability Toposes

With every pca $\mathcal{A}$ and subpca $\mathcal{A}_\#$ we associate the nested realizability topos $\mathsf{RT}(\mathcal{A},\mathcal{A}_\#)$ within which we identify a class of small maps $\mathcal{S}$ giving rise to a model of intuitionistic set theory within $\mathsf{RT}(\mathcal{A},\mathcal{A}_\#)$. For every subtopos $\mathcal{E}$ of such a nested realizability topos we construct an induced class $\mathcal{S_E}$ of small maps in $\mathcal{E}$ giving rise to a model of intuitionistic set theory within $\mathcal{E}$. This covers relative realizability toposes, modified relative realizability toposes, the modified realizability topos and van den Berg's recent Herbrand topos

A predicative variant of a realizability tripos for the Minimalist Foundation.

open2noHere we present a predicative variant of a realizability tripos validating the intensional level of the Minimalist Foundation extended with Formal Church thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel

An extensional Kleene realizability semantics for the Minimalist Foundation

We build a Kleene realizability semantics for the two-level Minimalist Foundation MF, ideated by Maietti and Sambin in 2005 and completed by Maietti in 2009. Thanks to this semantics we prove that both levels of MF are consistent with the (Extended) formal Church Thesis CT. MF consists of two levels, an intensional one, called mTT and an extensional one, called emTT, based on versions of Martin-L\"of's type theory. Thanks to the link between the two levels, it is enough to build a semantics for the intensional level to get one also for the extensional level. Hence here we just build a realizability semantics for the intensional level mTT. Such a semantics is a modification of the realizability semantics in Beeson 1985 for extensional first order Martin-L\"of's type theory with one universe. So it is formalised in Feferman's classical arithmetic theory of inductive definitions. It is called extensional Kleene realizability semantics since it validates extensional equality of type-theoretic functions extFun, as in Beeson 1985. The main modification we perform on Beeson's semantics is to interpret propositions, which are defined primitively in MF, in a proof-irrelevant way. As a consequence, we gain the validity of CT. Recalling that extFun+ CT+ AC are inconsistent over arithmetics with finite types, we conclude that our semantics does not validate the full Axiom of Choice AC. On the contrary, Beeson's semantics does validate AC, being this a theorem of Martin-L\"of's theory, but it does not validate CT. The semantics we present here appears to be the best Kleene realizability semantics for the extensional level emTT of MF. Indeed Beeson's semantics is not an option for emTT since the full AC added to it entails the excluded middle

Facing Emerging Risks in Carbon Sequestration Networks. A Comprehensive Source Modeling Approach.

Tackling risks in emerging infrastructures is a key point in making them acceptable and safer. Carbon Sequestration pipeline networks, as part of the Carbon Capture and Storage chains, are linked to the handling of large amounts of CO2 and may be subjected to failures and ruptures. This results in large pressurized and multiphase releases of CO2 that behaves as a denser-than-air and asphyxiant gas. The lack of a comprehensive modelling approach in this sense makes employed risk safety procedures often unreliable and lacking. In this work, a comprehensive modelling approach, based on self-collected experimental data, is proposed with the aim of filling existing gaps. Results well match experimental data and the modelling procedure allows for the estimation of characteristic parameters linked to heat transfer phenomena and the incidence of geometry and operative conditions on the release evolution. The occurrence of the solid phase and the applicability of the isothermal hypothesis is discussed showing that specific geometric and operative conditions are required

A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice

We present a Kleene realizability semantics for the intensional level of the Minimalist Foundation, for short mtt, extended with inductively generated formal topologies, Church's thesis and axiom of choice. This semantics is an extension of the one used to show consistency of the intensional level of the Minimalist Foundation with the axiom of choice and formal Church's thesis in previous work. A main novelty here is that such a semantics is formalized in a constructive theory represented by Aczel's constructive set theory CZF extended with the regular extension axiom