Related papers: Predicativity through transfinite reflection
Graph representations have increasingly grown in popularity during the last years. Existing representation learning approaches explicitly encode network structure. Despite their good performance in downstream processes (e.g., node…
We describe a realizability framework for classical first-order logic in which realizers live in (a model of) typed {\lambda}{\mu}-calculus. This allows a direct interpretation of classical proofs, avoiding the usual negative translation to…
We give an arithmetical proof of the strong normalization of the $\lambda$-calculus (and also of the $\lambda\mu$-calculus) where the type system is the one of simple types with recursive equations on types. The proof using candidates of…
The past decade has seen a significant interest in learning tractable probabilistic representations. Arithmetic circuits (ACs) were among the first proposed tractable representations, with some subsequent representations being instances of…
We propose a novel framework for structured prediction via adversarial learning. Existing adversarial learning methods involve two separate networks, i.e., the structured prediction models and the discriminative models, in the training. The…
Mimicking human behavior to actively learning from general experience and achieve artificial general intelligence has always been a human dream. Recent reinforcement learning (RL) based large-thinking models demonstrate impressive…
G\"odel's second incompleteness theorem is standardly understood as showing that no sufficiently strong, consistent theory of arithmetic can prove its own consistency, a result typically interpreted against a model-theoretic background in…
The classical work of Feferman Vaught gives a powerful, constructive analysis of definability in (generalized) product structures, and certain associated enriched Boolean structures. %structures in terms of definability in the component…
The aim of this work is to show that contemporary mathematics, including Peano arithmetic, is inconsistent, to construct firm foundations for mathematics, and to begin building on these foundations.
Primitive recursion, mu-recursion, universal object and universe theories, complexity controlled iteration, code evaluation, soundness, decidability, G\"odel incompleteness theorems, inconsistency provability for set theory, constructive…
We define a class of computable functions over real numbers using functional schemes similar to the class of primitive and partial recursive functions defined by G\"odel and Kleene. We show that this class of functions can also be…
Symbolic Regression (SR) allows for the discovery of scientific equations from data. To limit the large search space of possible equations, prior knowledge has been expressed in terms of formal grammars that characterize subsets of…
Throughout the course of mathematical history, generalizations of previously understood concepts and structures have led to the fruitful development of the hierarchy of number systems, non-euclidean geometry, and many other epochal phases…
We propose a framework for reasoning about programs that manipulate coinductive data as well as inductive data. Our approach is based on using equational programs, which support a seamless combination of computation and reasoning, and using…
In this paper we consider transfinite provability logics where for each ordinal in some recursive well-order we have a corresponding modal provability operator. The modality [xi] will be interpreted as "provable in ACA_0 together with at…
This is an expository article on representation theory of toroidal Lie algebras. We summerize all the results on representation theory of toroidal Lie algebras obtained in the last fifteen years. Apart from that a natural genaralization of…
Representation learning is a key technique in modern machine learning that enables models to identify meaningful patterns in complex data. However, different methods tend to extract distinct aspects of the data, and relying on a single…
It is known that several variations of the axiom of determinacy play important roles in the study of reverse mathematics, and the relation between the hierarchy of determinacy and comprehension are revealed by Tanaka, Nemoto, Montalb\'an,…
Harvey Friedman shows that, over Peano Arithmetic, the consistency statement for a finitely axiomatised theory $A$ can be characterised as the weakest statement $C$ over Peano Arithmetic such that ${\sf PA}+C$ interprets $A$. We study which…
In a forthcoming book, professional computer scientist and physicist Paul Budnik presents an exposition of classical mathematical theory as the backdrop to an elegant thesis: we can interpret any model of a formal system of Peano Arithmetic…