Related papers: Computability of Brjuno-like functions
We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic processes is considered…
It is known that there exist mathematical problems of practical relevance which cannot be computed on a Turing machine. An important example is the calculation of the first derivative of continuously differentiable functions. This paper…
Computational problems are classified into computable and uncomputable problems. If there exists an effective procedure (algorithm) to compute a problem then the problem is computable otherwise it is uncomputable. Turing machines can…
This paper provides a new and more direct proof of the assertion that a Turing computable function of the natural numbers is primitive recursive if and only if the time complexity of the corresponding Turing machine is bounded by a…
By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous…
The study of computability has its origin in Hilbert's conference of 1900, where an adjacent question, to the ones he asked, is to give a precise description of the notion of algorithm. In the search for a good definition arose three…
Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary…
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…
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schr\"odinger…
Turing's famous 'machine' model constitutes the first intuitively convincing framework for computing with real numbers. Kleene's computation schemes S1-S9 extend Turing's approach and provide a framework for computing with objects of any…
Due to common misconceptions about the Church-Turing thesis, it has been widely assumed that the Turing machine provides an upper bound on what is computable. This is not so. The new field of hypercomputation studies models of computation…
One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…
We revisit the question (most famously) initiated by Turing: can human intelligence be completely modeled by a Turing machine? We show that the answer is \emph{no}, assuming a certain weak soundness hypothesis. More specifically we show…
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…
For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While…
There are growing uncertainties surrounding the classical model of computation established by G\"odel, Church, Kleene, Turing and others in the 1930s onwards. The mismatch between the Turing machine conception, and the experiences of those…
We propose a definition of quantum computable functions as mappings between superpositions of natural numbers to probability distributions of natural numbers. Each function is obtained as a limit of an infinite computation of a quantum…
The hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine. Several numerical simulations of a possible hypercomputational algorithm based on…
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…
A number of examples have been given of physical systems (both classical and quantum mechanical) which when provided with a (continuously variable) computable input will give a non-computable output. It has been suggested that these systems…