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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…
We present an illative system I_s of classical higher-order logic with subtyping and basic inductive types. The system I_s allows for direct definitions of partial and general recursive functions, and provides means for handling functions…
The current work introduces the notion of pdominant sets and studies their recursion-theoretic properties. Here a set A is called pdominant iff there is a partial A-recursive function {\psi} such that for every partial recursive function…
This paper proposes bimorphic recursion, which is restricted polymorphic recursion such that every recursive call in the body of a function definition has the same type. Bimorphic recursion allows us to assign two different types to a…
The Ackermann function is a famous total recursive binary function on the natural numbers. It is the archetypal example of such a function that is not primitive recursive, in the sense of classical recursion theory. However, and in seeming…
Partiality is a natural phenomenon in computability that we cannot get around. So, the question is whether we can give the areas where partiality occurs, that is, where non-termination happens, more structure. In this paper we consider…
In this note we study the convergence of recursively defined infinite series. We explore the role of the derivative of the defining function at the origin (if it exists), and develop a comparison test for such series which can be used even…
This paper presents a new foundational approach to information theory based on the concept of the information efficiency of a recursive function, which is defined as the difference between the information in the input and the output. The…
Can a problem undecidable with classical resources be decidable with quantum ones? The answer expected is no; as both being Turing theories, they should not solve the Halting problem - a problem unsolvable by any Turing machine. Yet, we…
Decision tree learning is increasingly being used for pointwise inference. Important applications include causal heterogenous treatment effects and dynamic policy decisions, as well as conditional quantile regression and design of…
The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since…
We address the decision problem for a fragment of real analysis involving differentiable functions with continuous first derivatives. The proposed theory, besides the operators of Tarski's theory of reals, includes predicates for…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what…
Undecidability of various properties of first order term rewriting systems is well-known. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of…
Replication studies estimate the replicability rate of scientific results by aggregating binary verdicts of experiments. Exact replications are rarely attainable, so most replication sequences are non-exact. Experiments differ in ways that…
We study the possibility to reconstruct the primordial function for some periodic function. The procedure includes an analytical continuation of a discrete function for Fourier coefficients computation, that introduces an ambiguity. To…
The classical decision problem, as it is understood today, is the quest for a delineation between the decidable and the undecidable parts of first-order logic based on elegant syntactic criteria. In this paper, we treat the concept of…
In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy…
Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify…