相关论文: A Brief History of Future Set Theory

One of the biggest criticisms of the Set Shaping Theory is the lack of a practical application. This is due to the difficulty of its application. In fact, to apply this technique from an experimental point of view we must use a table that…

Smart premise selection is essential when using automated reasoning as a tool for large-theory formal proof development. A good method for premise selection in complex mathematical libraries is the application of machine learning to large…

This article describes a Turing machine which can solve for $\beta^{'}$ which is RE-complete. RE-complete problems are proven to be undecidable by Turing's accepted proof on the Entscheidungsproblem. Thus, constructing a machine which…

Decision support systems for classification tasks are predominantly designed to predict the value of the ground truth labels. However, since their predictions are not perfect, these systems also need to make human experts understand when…

While counterfactual fairness of point predictors is well studied, its extension to prediction sets--central to fair decision-making under uncertainty--remains underexplored. On the other hand, conformal prediction (CP) provides efficient,…

We start by presenting a theory of finite sets using the approach which is essentially that taken by Whitehead and Russell in Principia Mathematica}, and which does not involve the natural numbers (or any other infinite set). This theory is…

Why do modern language models, trained to do well on next-word prediction, appear to generate coherent documents and capture long-range structure? Here we show that next-token prediction is provably powerful for learning longer-range…

This paper develops a methodology for representing machine learning models as models of formal theories, grounded in the perspective that machine learning models are a form of database and that databases are models of theories in coherent…

Modern learning algorithms excel at producing accurate but complex models of the data. However, deploying such models in the real-world requires extra care: we must ensure their reliability, robustness, and absence of undesired biases. This…

Are generative pre-trained transformer (GPT) models, trained only to predict the next token, implicitly learning a world model from which sequences are generated one token at a time? We address this question by deriving a causal…

Given an initial family of sets, we may take unions, intersections and complements of the sets contained in this family in order to form a new collection of sets; our construction process is done recursively until we obtain the last family.…

The theory of finitely supported algebraic structures represents a reformulation of Zermelo-Fraenkel set theory in which every construction is finitely supported according to the action of a group of permutations of some basic elements…

Foundation models are premised on the idea that sequence prediction can uncover deeper domain understanding, much like how Kepler's predictions of planetary motion later led to the discovery of Newtonian mechanics. However, evaluating…

A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the…

Native type systems are those in which type constructors are derived from term constructors, as well as the constructors of predicate logic and intuitionistic type theory. We present a method to construct native type systems for a broad…

F-systems are digraphs that enable to model sentences that predicate the falsity of other sentences. Paradoxes like the Liar and Yablo's can be analyzed with that tool to find graph-theoretic patterns. In this paper we present the F-systems…

Conceiving of premises as collected into sets or multisets, instead of sequences, may lead to triviality for classical and intuitionistic logic in general proof theory, where we investigate identity of deductions. Any two deductions with…

When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin's 2004 survey article on Auction Theory using the Isabelle/HOL…

There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…

We give an analysis and generalizations of some long-established constructive completeness results in terms of categorical logic and pre-sheaf and sheaf semantics. The purpose is in no small part conceptual and organizational: from a few…