Gianluca Paolini

Quantum Team Logic and Bell’s Inequalities

A logical approach to Bells Inequalities of quantum mechanics has been introduced by Abramsky and Hardy [2]. We point out that the logical Bells Inequalities of [2] are provable in the probability logic of Fagin, Halpern and Megiddo [4]. Since it is now considered empirically established that quantum mechanics violates Bells Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bells Inequalities are not provable, and prove a Completeness Theorem for this logic. For this end we generalise the team semantics of dependence logic [7] first to probabilistic team semantics, and then to what we call quantum team semantics.

DEPENDENCE LOGIC IN PREGEOMETRIES AND ω -STABLE THEORIES

We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences concerning basic atomic independence statements are complete with respect to a variety of semantics. Our results show that the uses of independence concepts in as different areas as database theory, algebra, and model theory, can be completely characterized by the same axioms. We also consider concepts related to independence, such as dependence.

A logic for arguing about probabilities in measure teams

We use sets of assignments, a.k.a. teams, and measures on them to define probabilities of first-order formulas in given data. We then axiomatise first-order properties of such probabilities and prove a completeness theorem for our axiomatisation. We use the Hardy–Weinberg Principle of biology and the Bell’s Inequalities of quantum physics as examples.

Independence logic and abstract independence relations

We continue the work on the relations between independence logic and the model-theoretic analysis of independence, generalizing the results of [15] and [16] to the framework of abstract independence relations for an arbitrary AEC. We give a model-theoretic interpretation of the independence atom and characterize under which conditions we can prove a completeness result with respect to the deductive system that axiomatizes independence in team semantics and statistics.

No Uncountable Polish Group Can be a Right-Angled Artin Group

We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 < k < ω , then l g ( x ) ≤ l g ( x k ) ; (ii) if l g ( y ) < k < ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups.

Reduction of database independence to dividing in atomless Boolean algebras

We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless Boolean algebras. This establishes interesting connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov (Isr. J. Math. 194(2):957–1012, 2013), the former case of independence can be seen as the discrete version of the latter.

Polish topologies for graph products of cyclic groups

We give a complete characterization of the graph products of cyclic groups admitting a Polish group topology, and show that they are all realizable as the group of automorphisms of a countable structure. In particular, we characterize the right-angled Coxeter groups (resp. Artin groups) admitting a Polish group topology. This generalizes results from [8], [9] and [4].

Beyond abstract elementary classes: On the model theory of geometric lattices

Based on Crapos theory of one point extensions of combinatorial geometries, we find various classes of geometric lattices that behave very well from the point of view of stability theory. One of them,

A Finite Axiomatization of G-Dependence

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Group metrics for graph products of cyclic groups

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