Related papers: Is the Halting probability a Dedekind real number?
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by being based on two…
In this paper we present the probabilistic typed natural deduction calculus TPTND, designed to reason about and derive trustworthiness properties of probabilistic computational processes, like those underlying current AI applications.…
Mechanisms for the automation of uncertainty are required for expert systems. Sometimes these mechanisms need to obey the properties of probabilistic reasoning. A purely numeric mechanism, like those proposed so far, cannot provide a…
There is increasing interest within the research community in the design and use of recursive probability models. Although there still remains concern about computational complexity costs and the fact that computing exact solutions can be…
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…
It is nowadays common to consider that proof must be part of the learning of mathematics from Kindergarten to University1. As it is easy to observe, looking back to the history of mathematical curricula, this has not always been the case…
Classical models of computation traditionally resort to halting schemes in order to enquire about the state of a computation. In such schemes, a computational process is responsible for signalling an end of a calculation by setting a halt…
It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an…
In this paper we prove that given a black box assumed to generate bits of a given non-recursive real $\Omega$ there is no computable decision procedure generating sequences of decisions such that if the output is indeed $\Omega$ the process…
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…
The benchmark for computation is typically given as Turing computability; the ability for a computation to be performed by a Turing Machine. Many languages exploit (indirect) encodings of Turing Machines to demonstrate their ability to…
We present a formulation of quantum circuits where the focus is set on whether a given circuit (made of unitary operators and projective measurements with definite outcomes) does reflect an actually realizable physical experiment. In order…
We consider a randomised version of Kleene's realisability interpretation of intuitionistic arithmetic in which computability is replaced with randomised computability with positive probability. In particular, we show that (i) the set of…
Our main result is a new proof of correctness of Euclid's algorithm. The proof is conducted in algorithmic theory of natural numbers Th3. A formula H is constructed that expresses the halting property of the algorithm. Next, the proof of H…
There have been many attempts to solve the P versus NP problem. However, with a new proof method, P not equal NP can be proved. A time limit is set for an arbitrary Turing machine and an input word is rejected on a timeout. The time limit…
The extended Riemann hypothesis (ERH) for Dedekind zeta functions remains one of the most elusive open problems in number theory. Over the last century, many equivalent statements to the classical Riemann hypothesis alone have been…
There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…
The proofs that the real numbers are denumerable will be shown, i.e., that there exists one-to-one correspondence between the natural numbers $N$ and the real numbers $\Re$. The general element of the sequence that contains all real numbers…
Since the early twentieth century, it has been understood that mathematical definitions and proofs can be represented in formal systems systems with precise grammars and rules of use. Building on such foundations, computational proof…
We study the question of what is computable by Turing machines equipped with time travel into the past; i.e., with Deutschian closed timelike curves (CTCs) having no bound on their width or length. An alternative viewpoint is that we study…