相关论文: MathZero, The Classification Problem, and Set-Theo…

We consider first the zero-nonzero determination problem, which consists in determining the list of zero-nonzero conditions realized by a finite list of polynomials on a finite set Z included in C^k with C an algebraic closed field. We…

In set theory without the axiom of regularity, we consider a game in which two players choose in turn an element of a given set, an element of this element, etc.; a player wins if its adversary cannot make any next move. Sets that are…

This work is motivated by the papers [EG85] and [Ngu15] in which the following two problems are solved. Let $\mathcal{O}$ is a finitely generated $\mathbb{Z}$-algebra that is an integrally closed domain of characteristic zero, consider the…

The algebraic intersection type unification problem is an important component in proof search related to several natural decision problems in intersection type systems. It is unknown and remains open whether the algebraic intersection type…

We investigate the problem of type isomorphisms in the presence of higher-order references. We first introduce a finitary programming language with sum types and higher-order references, for which we build a fully abstract games model…

We study two-player games with alternating moves played on infinite trees. Our main focus is on the case where the trees are full (regular) and the winning set is open (with respect to the product topology on the tree). Gale and Stewart…

What is learned by sophisticated neural network agents such as AlphaZero? This question is of both scientific and practical interest. If the representations of strong neural networks bear no resemblance to human concepts, our ability to…

Type-free systems of logic are designed to consistently handle significant instances of self-reference. Some consistent type-free systems also have the feature of allowing the sort of general abstraction or comprehension principle that…

Zero-sum games have long guided artificial intelligence research, since they possess both a rich strategy space of best-responses and a clear evaluation metric. What's more, competition is a vital mechanism in many real-world multi-agent…

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…

The objective of this article is to formalize the definition of NP problems. We construct a mathematical model of discrete problems as independence systems with weighted elements. We introduce two auxiliary sets that characterize the…

This work investigates the adaptation of the AlphaZero reinforcement learning algorithm to Tablut, an asymmetric historical board game featuring unequal piece counts and distinct player objectives (king capture versus king escape). While…

The standard axioms of set theory, the Zermelo-Fraenkel axioms (ZFC), do not suffice to answer all questions in mathematics. While this follows abstractly from Kurt G\"odel's famous incompleteness theorems, we nowadays know numerous…

Subzero automata is a class of tree automata whose acceptance condition can express probabilistic constraints. Our main result is that the problem of determining if a subzero automaton accepts some regular tree is decidable.

This paper addresses the task of zero-shot image classification. The key contribution of the proposed approach is to control the semantic embedding of images -- one of the main ingredients of zero-shot learning -- by formulating it as a…

A new algebraic treatment of dependent type theory is proposed using ideas derived from topos theory and algebraic set theory.

This paper introduces a class of objects called decision rules that map infinite sequences of alternatives to a decision space. These objects can be used to model situations where a decision maker encounters alternatives in a sequence such…

Using a model of the environment, reinforcement learning agents can plan their future moves and achieve superhuman performance in board games like Chess, Shogi, and Go, while remaining relatively sample-efficient. As demonstrated by the…

Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…

Solving a problem of Diestel and Pott, we construct a large class of infinite matroids. These can be used to provide counterexamples against the natural extension of the Well-quasi-ordering-Conjecture to infinite matroids and to show that…