Related papers: Aspects of Chaitin's Omega
We describe the Turing Machine, list some of its many influences on the theory of computation and complexity of computations, and illustrate its importance.
Probabilistic omega-automata are variants of nondeterministic automata for infinite words where all choices are resolved by probabilistic distributions. Acceptance of an infinite input word can be defined in different ways: by requiring…
Algorithmic information theory roots the concept of information in computation rather than probability. These lecture notes were constructed in conjunction with the graduate course I taught at Universit\`a della Svizzera italiana in the…
People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at…
The aim of this expository paper is to present a nice series of results, obtained in the papers of Chaitin (1976), Solovay (1975), Calude et al. (1998), Kucera and Slaman (2001). This joint effort led to a full characterization of lower…
The problem of Turing pattern formation has attracted much attention in nonlinear science as well as physics, chemistry and biology. So far all Turing patterns have been observed in stationary and oscillatory media only. In this letter we…
We answer two questions posed by Castro and Cucker, giving the exact complexities of two decision problems about cardinalities of omega-languages of Turing machines. Firstly, it is $D_2(\Sigma_1^1)$-complete to determine whether the…
Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then…
We prove that there is no algorithm to tell whether an arbitrarily constructed Quantum Turing Machine has same time steps for different branches of computation. We, hence, can not avoid the notion of halting to be probabilistic in Quantum…
This paper, which is dedicated to Alan Turing on the 50th anniversary of his death, gives an overview and discusses the philosophical implications of incompleteness, uncomputability and randomness.
The Turing machine halting problem can be explained by several factors, including arithmetic logic irreversibility and memory erasure, which contribute to computational uncertainty due to information loss during computation. Essentially,…
The origin of the uncertainty inherent in quantum measurements has been discussed since quantum theory's inception, but to date the source of the indeterminacy of measurements performed at an angle with respect to a quantum state's…
This paper introduces a new computing model based on the cooperation among Turing machines called orchestrated machines. Like universal Turing machines, orchestrated machines are also designed to simulate Turing machines but they can also…
We describe an alternative method (to compression) that combines several theoretical and experimental results to numerically approximate the algorithmic (Kolmogorov-Chaitin) complexity of all $\sum_{n=1}^82^n$ bit strings up to 8 bits long,…
In a recent historical overview, Cristian S. Calude, Elena Calude, and Solomon Marcus identify eight stages in the development of the concept of a mathematical proof in support of an ambitious conjecture: we can express classical…
A real number \alpha is called recursively enumerable if there exists a computable, increasing sequence of rational numbers which converges to \alpha. The randomness of a recursively enumerable real \alpha can be characterized in various…
The paper considers the halting scheme for quantum Turing machines. The scheme originally proposed by Deutsch appears to be correct, but not exactly as originally intended. We discuss the result of Ozawa as well as the objections raised by…
Through a straightforward Bayesian approach we show that under some general conditions a maximum running time, namely the number of discrete steps performed by a computer program during its execution, can be defined such that the…
The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong…
We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that "a real is c.e. and random iff it is the halting…