Related papers: Recognizable Realizability

Experimental science usually relies on laboratory procedures that, after finitely many steps, terminate with numerical reports on physical quantities. This paper argues that such procedures can be understood as algorithmic once the…

We discuss the possibility of constructing a function that validates the definition or not definition of the partial recursive functions of one variable. This is a topic in computability theory, which was first approached by Alan M. Turing…

In this paper we present an introduction to the area of computability in dynamical systems. This is a fairly new field which has received quite some attention in recent years. One of the central questions in this area is if relevant…

A sequence of non-negative integers is exactly realizable as the fixed point counts sequence of a dynamical system if and only if it gives rise to a sequence of non-negative orbit counts. This provides a simple realizability criterion based…

This article is a fundamental study in computable measure theory. We use the framework of TTE, the representation approach, where computability on an abstract set X is defined by representing its elements with concrete "names", possibly…

We recall from previous work a model-independent framework of computational complexity theory. Notably for the present paper, the framework allows formalization of the issues of precision that present themselves when one considers physical,…

In this paper we present a semantics for a linear algebraic lambda-calculus based on realizability. This semantics characterizes a notion of unitarity in the system, answering a long standing issue. We derive from the semantics a set of…

We study a classical realizability model (in the sense of J.-L. Krivine) arising from a model of untyped lambda calculus in coherence spaces. We show that this model validates countable choice using bar recursion and bar induction.

Learning structured representations of the visual world in terms of objects promises to significantly improve the generalization abilities of current machine learning models. While recent efforts to this end have shown promising empirical…

We study relative precompleteness in the context of the theory of numberings, and relate this to a notion of lowness. We introduce a notion of divisibility for numberings, and use it to show that for the class of divisible numberings,…

A universal Turing machine is a powerful concept - a single device can compute any function that is computable. A universal spin model, similarly, is a class of physical systems whose low energy behavior simulates that of any spin system.…

We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…

We introduce infinite time computable model theory, the computable model theory arising with infinite time Turing machines, which provide infinitary notions of computability for structures built on the reals R. Much of the finite time…

We define a generalization of the Turing machine that computes on general sets. Our main theorem states that the class of generalized Turing machine computable functions and the class of Set Recursive functions coincide.

In this dissertation we provide mathematical evidence that the concept of learning can be used to give a new and intuitive computational semantics of classical proofs in various fragments of Predicative Arithmetic. First, we extend Kreisel…

We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…

We present tools for analysing ordinals in realizability models of classical set theory built using Krivine's technique for realizability. This method uses a conservative extension of $ZF$ known as $ZF_{\varepsilon}$, where two membership…

Relative realizability toposes satisfy a universal property that involves regular functors to other categories. We use this universal property to define what relative realizability categories are, when based on other categories than of the…

In generic realizability for set theories, realizers treat unbounded quantifiers generically. To this form of realizability, we add another layer of extensionality by requiring that realizers ought to act extensionally on realizers, giving…

We provide a novel notion of what it means to be interpretable, looking past the usual association with human understanding. Our key insight is that interpretability is not an absolute concept and so we define it relative to a target model,…