We generalize standard Turing machines working in time ω on a tape of length ω to abstract machines with time α and tape length α, for α some limit ordinal. This model of computation determines an associated computability theory: α-computability theory. We compare the new theory to α-recursion theory, which was developed by G. Sacks and his school. For α an admissible ordinal, the basic notions...

Every recursively enumerable set of integers (r.e. set) is enumerable by a primitive recursive function. But if the enumeration is required to be one-one, only a proper subset of all r.e. sets qualify. Starting from a collection of total recursive functions containing the primitive recursive functions, we thus define a sub-computability as an enumeration of the r.e. sets that are themselves one...

The notion of computability closure has been introduced for proving the termination of the combination of higher-order rewriting and beta-reduction. It is also used for strengthening the higher-order recursive path ordering. In the present paper, we study in more details the relations between the computability closure and the (higher-order) recursive path ordering. We show that the first-order ...

We present the different constructive definitions of real number that can be found in the literature. Using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them. This definition of computability turns out to be equivalent to other definitions given in the litera...

As suggested by the title, it has recently become clear that theorems of Nonstandard Analysis (NSA) give rise to theorems in computability theory (no longer involving NSA). Now, the aforementioned discipline divides into classical and higher-order computability theory, where the former (resp. the latter) sub-discipline deals with objects of type zero and one (resp. of all types). The aforementi...

Despite the recursive non-computability of Hilbert’s tenth problem, we outline and argue for a quantum algorithm that is based on the Quantum Adiabatic Theorem. It is explained how this algorithm can solve Hilbert’s tenth problem. The algorithm is then considered in the context of several “no-go” arguments against such hypercomputation. Logical arguments are usually based on Cantor’s diagonal t...

Characteristic properties of majorant-computable real-valued functions are studied. A formal theory of computability over the reals which satisses the requirements of numerical analysis used in Computer Science is constructed on the base of the deenition of majorant-computa-bility proposed in 13]. A model-theoretical characterization of majorant-computability real-valued functions and their dom...

A Programming Approach to Computability. Enumeration and Universality of the Computable Functions. 4 The FOR in real programming languages. Computability and complexity are in some way the basics of what computer science is about.Computability and Complexity: From a Programming Perspective, Neil D. 1Dana Scott was an early proponent of programming approach to automata.Computability unit of the ...

In this paper, we highlight a new computational aspect of Nonstandard Analysis relating to higher-order computability theory. In particular, we prove that the Gandy-Hyland functional equals a primitive recursive functional involving nonstandard numbers inside Nelson’s internal set theory. From this classical and ineffective proof in Nonstandard Analysis, a term from Gödel’s system T can be extr...

We define the computability of probability distributions on the real line as well as that of distribution functions. Mutual relationships between the computability notion of a probability distribution and that of the corresponding distribution function are discussed. It is carried out through attempts to effectivize some classical fundamental theorems concerning probability distributions. We th...