Related papers: Real Hypercomputation and Continuity
In intuitionistic mathematics, the Brouwer Continuity Theorem states that all total real functions are (uniformly) continuous on the unit interval. We study this theorem and related principles from the point of view of Reverse Mathematics…
In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical…
Functors with an instance of the Traversable type class can be thought of as data structures which permit a traversal of their elements. This has been made precise by the correspondence between traversable functors and finitary containers…
We consider the issue of computability at the most fundamental level of physical reality: the Planck scale. To this aim, we consider the theoretical model of a quantum computer on a non commutative space background, which is a computational…
The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the…
In this paper, we investigate the problem of synthesizing computable functions of infinite words over an infinite alphabet (data $\omega$-words). The notion of computability is defined through Turing machines with infinite inputs which can…
HyperLTL model-checking enables the automated verification of information-flow properties for security-critical systems. However, it only provides a binary answer. Here, we introduce two paradigms to compute counterexamples and explanations…
computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…
A word-to-word function is continuous for a class of languages~$\mathcal{V}$ if its inverse maps $\mathcal{V}$_languages to~$\mathcal{V}$. This notion provides a basis for an algebraic study of transducers, and was integral to the…
The paper explores known results related to the problem of identifying if a given program terminates on all inputs -- this is a simple generalization of the halting problem. We will see how this problem is related and the notion of proof…
The paper continues the intriguing theme that many key facts of (single-variable) Real Analysis are not only crucially dependent on the completeness of the real numbers, but are actually equivalent to it. The list of these characterizations…
The notion of a real-valued function is central to mathematics, computer science, and many other scientific fields. Despite this importance, there are hardly any positive results on decision procedures for predicate logical theories that…
We conclude from Goedel's Theorem VII of his seminal 1931 paper that every recursive function f(x_{1}, x_{2}) is representable in the first-order Peano Arithmetic PA by a formula [F(x_{1}, x_{2}, x_{3})] which is algorithmically verifiable,…
Recently we have introduced a new model of infinite computation by extending the operation of ordinary Turing machines into transfinite ordinal time. In this paper we will show that the infinite time Turing machine analogue of Post's…
The diagonal method is often used to show that Turing machines cannot solve their own halting problem. There have been several recent attempts to show that this method also exposes either contradiction or arbitrariness in other theoretical…
We give a detailed treatment of the ``bit-model'' of computability and complexity of real functions and subsets of R^n, and argue that this is a good way to formalize many problems of scientific computation. In the introduction we also…
Within the program of finding axiomatizations for various parts of computability logic, it was proved earlier that the logic of interactive Turing reduction is exactly the implicative fragment of Heyting's intuitionistic calculus. That sort…
In this paper, we extend Busy Beaver function to a class of higher order Busy Beaver functions based on Turing oracle machine. We prove some results about the relation between decidability of number theoretical formula and higher order Busy…
The Functional Machine Calculus (Heijltjes 2022) is a new approach to unifying the imperative and functional programming paradigms. It extends the lambda-calculus, preserving the key features of confluent reduction and typed termination, to…
Holomorphic functions are amazing because their values in an ever so small disk in the complex plane completely determine the function values at arbitrary points in their maximum possible domain. The process of extending such a function…