Related papers: Lifting countable to uncountable mathematics
There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem…
Reversible computation is key in developing new, energy-efficient paradigms, but also in providing forward-only concepts with broader definitions and finer frames of study.Among other fields, the algebraic specification and representation…
By nature, transmissible human knowledge is enumerable: every sentence, movie, audio record can be encoded in a sufficiently long string of 0's and 1's. The works of G\"odel, Turing and others showed that there are inherent limits and…
A novel model of reversible computing, the $\aleph$-calculus, is introduced. It is declarative, reversible-Turing complete, and has a local term-rewriting semantics. Unlike previously demonstrated reversible term-rewriting systems, it does…
The uncountability of the real numbers is one of their most basic properties, known (far) outside of mathematics. Cantor's 1874 proof of the uncountability of the real numbers even appears in the very first paper on set theory, i.e. a…
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…
Every mathematical structure has an elementary extension to a pseudo-countable structure, one that is seen as countable inside a suitable class model of set theory, even though it may actually be uncountable. This observation, proved easily…
In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without…
Counterfactual examples are an appealing class of post-hoc explanations for machine learning models. Given input $x$ of class $y_1$, its counterfactual is a contrastive example $x^\prime$ of another class $y_0$. Current approaches primarily…
In system operations it is commonly assumed that arbitrary changes to a system can be reversed or `rolled back', when errors of judgement and procedure occur. We point out that this view is flawed and provide an alternative approach to…
We present a new approach to proving non-termination of non-deterministic integer programs. Our technique is rather simple but efficient. It relies on a purely syntactic reversal of the program's transition system followed by a…
Given a first-order sentence, a model-checking computation tests whether the sentence holds true in a given finite structure. Data provenance extracts from this computation an abstraction of the manner in which its result depends on the…
Reversible computing can reduce the energy dissipation of computation, which can improve cost-efficiency in some contexts. But the practical applicability of this method depends sensitively on the space and time overhead required by…
While there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far. We propose a notion of computability on the space of…
We consider the following problem for various infinite time machines. If a real is computable relative to large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it…
High-level reversible programming languages are few and far between and in general offer only rudimentary abstractions from the details of the underlying machine. Modern programming languages offer a wide array of language constructs and…
In this survey, my aim has been to discuss the use of sequences and countable sets in general topology. In this way I have been led to consider five different classes of topological spaces: first countable spaces, sequential spaces, Frechet…
The method of using concepts and insight from quantum information theory in order to solve problems in reversible classical computing (introduced in Ref. [1]) have been generalized to irreversible classical computing. The method have been…
We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…