Related papers: The Arithmetical Hierarchy: A Realizability-Theore…
Based on an analysis of the inference rules used, we provide a characterization of the situations in which classical provability entails intuitionistic provability. We then examine the relationship of these derivability notions to uniform…
Axiomatizing mathematical structures is a goal of Mathematical Logic. Axiomatizability of the theories of some structures have turned out to be quite difficult and challenging, and some remain open. However axiomatization of some…
In which a review of the concept of countability is done in mathematics, subjecting review some of the theorems so far accepted, showing their inconsistency and also taking concrete elements on the countability of all the powers of the set…
Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.
We investigate computability in the lattice of equivalence relations on the natural numbers. We mostly investigate whether the subsets of appropriately defined subrecursive equivalence relations -for example the set of all polynomial-time…
Understanding the functional architecture of complex systems is crucial to illuminate their inner workings and enable effective methods for their prediction and control. Recent advances have introduced tools to characterise emergent…
This paper presents a theory of systemic undecidability, reframing incomputability as a structural property of systems rather than a localized feature of specific functions or problems. We define a notion of causal embedding and prove a…
This paper explores the relationship of artificial intelligence to the task of resolving open questions in mathematics. We first present an updated version of a traditional argument that limitative results from computability and complexity…
The rationalizability concept was introduced in \cite{Ber84} and \cite{Pea84} to assess what can be inferred by rational players in a non-cooperative game in the presence of common knowledge. However, this notion can be defined in a number…
Reproducibility is one of the core dimensions that concur to deliver Trustworthy Artificial Intelligence. Broadly speaking, reproducibility can be defined as the possibility to reproduce the same or a similar experiment or method, thereby…
There is a cognitive limit in Human Mind. This cognitive limit has played a decisive role in almost all fields including computer sciences. The cognitive limit replicated in computer sciences is responsible for inherent Computational…
We define a variant of realizability where realizers are pairs of a term and a substitution. This variant allows us to prove the normalization of a simply-typed call-by-need $$\lambda$-$calculus with control due to Ariola et al. Indeed, in…
A standard tool for classifying the complexity of equivalence relations on $\omega$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce…
We introduce a new way of quantifying the degrees of incompatibility of two ob- servables in a probabilistic physical theory and, based on this, a global measure of the degree of incompatibility inherent in such theories, across all…
In Hayashi and Leigh (2024), the authors formulate classical number realisability for first-order arithmetic and a corresponding axiomatic system based on Krivine's classical realisability interpretation. This paper presents a…
Hierarchical task decomposition is a method used in many agent systems to organize agent knowledge. This work shows how the combination of a hierarchy and persistent assertions of knowledge can lead to difficulty in maintaining logical…
We consider basic conceptual questions concerning the relationship between statistical estimation and causal inference. Firstly, we show how to translate causal inference problems into an abstract statistical formalism without requiring any…
We discuss views about whether the universe can be rationally comprehended, starting with Plato, then Leibniz, and then the views of some distinguished scientists of the previous century. Based on this, we defend the thesis that…
Due to common misconceptions about the Church-Turing thesis, it has been widely assumed that the Turing machine provides an upper bound on what is computable. This is not so. The new field of hypercomputation studies models of computation…
A problem is a multivalued function from a set of \emph{instances} to a set of \emph{solutions}. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness…