arXiv subjects

Åsa Hirvonen

Publications and source records attributed to Åsa Hirvonen.

7 recordsLinked to original sources

arXiv2026

Approximate Functional Dependencies---Implication Problem Revisited

Functional dependencies are an important and well-studied class of database constraints that correspond to a notion expressed by dependence atoms in team logic. In practice, data often contain errors, so in some cases it might be useful to allow the database to have a small number of tuples that violate the desired dependency. Väänänen (2017) studied the axiomatisation of a notion of approximate dependence that specifies for each dependence atom how much of the database can be disregarded. We demonstrate that the interaction of approximate dependence atoms is more complicated than previously thought in the sense that there is a semantic consequence that is not captured by the inference rules introduced before. We show that Väänänen's axiomatisation is still complete in the restricted case of unary dependencies. We also consider the complexity of model checking for approximate dependence: it is NP-complete for disjunctions of two atoms and LOGSPACE-hard for individual atoms.

cs.LO↗

arXiv2024

Ehrenfeucht-Fraïssé Games for Continuous First-Order Logic

We define a version of the Ehrenfeucht-Fraïssé game in the setting of metric model theory and continuous first-order logic and show that the second player having a winning strategy in a game of length $n$ exactly corresponds to being elementarily equivalent up to quantifier rank $n$. We then demonstrate the usefulness of the game with some examples. Finally, we discus connections between the game of length $ω$ and infinitary logic.

math.LO↗

arXiv2022

On ultraproducts, the spectral theorem and rigged Hilbert spaces

We start by showing how to approximate unitary and bounded self-adjoint operators by operators in finite dimensional spaces. Using ultraproducts we give a precise meaning for the approximation. In this process we see how the spectral measure is obtained as an ultralimit of counting measures that arise naturally from the finite dimensional approximations. Then we see how generalized distributions can be interpreted in the ultraproduct. Finally we study how one can calculate kernels of operators $K$ by calculating them in the finite dimensional approximations and how one needs to interpret Dirac deltas in the ultraproduct in order to get the kernels as propagators $\langle x_{1}|K|x_{0}\rangle$.

math.LO↗

arXiv2021

Games and Scott sentences for positive distances between metric structures

We develop various Ehrenfeucht-Fra\"ıssé games for distances between metric structures. We study two forms of distances: pseudometrics stemming from mapping spaces onto each other with some form of approximate isomorphism, and metrics stemming from measuring the distances between two spaces isometrically embedded into a third space. Using an infinitary version of Henson's positive bounded logic with approximations, we form Scott sentences capturing fixed distances to a given space. The Scott sentences of separable spaces are in $\mathcal{L}_{ω_1ω}$ for 0-distances and in $\mathcal{L}_{ω_2ω}$ for positive distances.

math.LO↗

arXiv2019

Facets of Distribution Identities in Probabilistic Team Semantics

We study probabilistic team semantics which is a semantical framework allowing the study of logical and probabilistic dependencies simultaneously. We examine and classify the expressive power of logical formalisms arising by different probabilistic atoms such as conditional independence and different variants of marginal distribution equivalences. We also relate the framework to the first-order theory of the reals and apply our methods to the open question on the complexity of the implication problem of conditional independence.

cs.LO↗

arXiv2018

On Eigenvectors, Approximations and the Feynman Propagator

Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product $\langle x_{0}| K^{t}| x_{1}\rangle $ of the eigenvector $| x_{0}\rangle $ of the position operator with eigenvalue $x_{0}$ and $K^{t}(| x_{1}\rangle )$, where $K^{t}$ is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and without heavy case-by-case scaling, it gives the wrong value. We look at this phenomenon, and then complement this by showing how to instead calculate the kernel of the time evolution operator (for both the free particle and the harmonic oscillator) in the limit model. We believe that our method of calculating these is new.

math.LO↗

arXiv2016

Measuring dependence in metric abstract elementary classes with perturbations

We define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is $d^p$-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces and we give a new example of such a class based on B. Zilber's approximations of Weyl algebras. We introduce a way to measure the dependence of a tuple $a$ from a set $B$ over another set $A$. We prove basic properties of the notion, e.g. that $a$ is independent of $B$ over $A$ in the usual sense of homogeneous model theory if and only if the measure of dependence is $<\varepsilon$ for all $\varepsilon >0$. As an example of our measure of dependence we show a connection between the measure and entropy in models from quantum mechanics in which the spectrum of the observable is discrete. As an application, we show that weak simplicity implies a very strong form of simplicity and study the question of when the dependence inside a set of all realisations of some type can be seen to arise from a pregeometry in cases when the type is not regular. In the end of the paper, we demonstrate our notions and results in one more example: a class built from the $p$-adic integers.

math.LO↗