Related papers: Smullyan's truth and provability
This paper from 2008 is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the foundations are laid for later results. These foundations consist of a thorough…
Modal logics allow reasoning about various modes of truth: for example, what it means for something to be possibly true, or to know that something is true as opposed to merely believing it. This report describes embeddings of propositional…
We define constructive truth for arithmetic and for intuitionistic analysis, and investigate its properties. We also prove that the set of constructively true (first order) arithmetical statements is Pi-1-2 and Sigma-1-2 hard, and we…
Development of several alternative mathematical models for the biological system in question and discrimination between such models using experimental data is the best way to robust conclusions. Models which challenge existing theories are…
We consider the problem of rational uncertainty about unproven mathematical statements, remarked on by G\"odel and others. Using Bayesian-inspired arguments we build a normative model of fair bets under deductive uncertainty which draws…
This is the third in a series of papers extending Martin-L\"of's meaning explanations of dependent type theory to a Cartesian cubical realizability framework that accounts for higher-dimensional types. We extend this framework to include a…
Supervised machine learning models boast remarkable predictive capabilities. But can you trust your model? Will it work in deployment? What else can it tell you about the world? We want models to be not only good, but interpretable. And yet…
Theaimofthepresentpaperistosuggestthatstatisticalphysicsprovides the correct language to understand the practical behavior of the LLL algorithm, most of which are left unexplained to this day. To this end, we propose sandpile models that…
The aim of this paper is to present a self contained introduction to the Hubbard model and some of its applications.The paper consists of two parts: the first will introduce the basic notions of the Hubbard model starting from the…
Validation is often defined as the process of determining the degree to which a model is an accurate representation of the real world from the perspective of its intended uses. Validation is crucial as industries and governments depend…
Generative AI and Large Language Models (LLMs) hold promise for automating spreadsheet formula creation. However, due to hallucinations, bias and variable user skill, outputs obtained from generative AI cannot be assumed to be accurate or…
Multiple imputation is a straightforward method for handling missing data in a principled fashion. This paper presents an overview of multiple imputation, including important theoretical results and their practical implications for…
This paper presents the first in a series of results that allow us to develop a theory providing finer control over the complexity of normalisation, and in particular of cut elimination. By considering atoms as self-dual non-commutative…
In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a…
We regard explanations as a blending of the input sample and the model's output and offer a few definitions that capture various desired properties of the function that generates these explanations. We study the links between these…
In this paper, we study a new Kripke-style semantics for classical modal logic, named as provability models. We study provability models for the propositional modal logics K, K4, S4 GL, GLP and the interpretability logic ILM. Provability…
Linear logic was conceived in 1987 by Girard and, in contrast to classical logic, restricts the usage of the structural inference rules of weakening and contraction. With this, atoms of the logic are no longer interpreted as truth, but as…
Most recent work on interpretability of complex machine learning models has focused on estimating $\textit{a posteriori}$ explanations for previously trained models around specific predictions. $\textit{Self-explaining}$ models where…
Computability logic is a formal theory of computational tasks and resources. Formulas in it represent interactive computational problems, and "truth" is understood as algorithmic solvability. Interactive computational problems, in turn, are…
A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of…