相关论文: Aspects of Chaitin's Omega
A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random…
The halting probability of a Turing machine is the probability that the machine will halt if it starts with a random stream written on its one-way input tape. When the machine is universal, this probability is referred to as Chaitin's omega…
A fruitful way of obtaining meaningful, possibly concrete, algorithmically random numbers is to consider a potential behaviour of a Turing machine and its probability with respect to a measure (or semi-measure) on the input space of binary…
Chaitin [G. J. Chaitin, J. Assoc. Comput. Mach., vol.22, pp.329-340, 1975] introduced \Omega number as a concrete example of random real. The real \Omega is defined as the probability that an optimal computer halts, where the optimal…
It would be a heavenly reward if there were a method of weighing theories and sentences in such a way that a theory could never prove a heavier sentence (Chaitin's Heuristic Principle). Alas, no satisfactory measure has been found so far,…
In 1975, Chaitin introduced his celebrated Omega number, the halting probability of a universal Chaitin machine, a universal Turing machine with a prefix-free domain. The Omega number's bits are {\em algorithmically random}--there is no…
We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin's Omega number, halting probability, and program-size complexity. A classification of Turing machines…
In 1975 Chaitin introduced his \Omega number as a concrete example of random real. The real \Omega is defined based on the set of all halting inputs for an optimal prefix-free machine U, which is a universal decoding algorithm used to…
This book presents a personal account of the mathematics and metamathematics of the 20th century leading up to the discovery of the halting probability Omega. The emphasis is on history of ideas and philosophical implications.
This article discusses what can be proved about the foundations of mathematics using the notions of algorithm and information. The first part is retrospective, and presents a beautiful antique, Godel's proof, the first modern incompleteness…
Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a…
The series solution of the behavior of a finite number of physical bodies and Chaitin's Omega number share quasi-algorithmic expressions; yet both lack a computable radius of convergence.
Chaitin's work, in its depth and breadth, encompasses many areas of scientific and philosophical interest. It helped establish the accepted mathematical concept of randomness, which in turn is the basis of tools that I have developed to…
We investigate the continuous function $f$ defined by $$x\mapsto \sum_{\sigma\le_L x }2^{-K(\sigma)}$$ as a variant of Chaitin's Omega from the perspective of analysis, computability, and algorithmic randomness. Among other results, we…
The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the…
The halting probabilities of universal prefix-free machines are universal for the class of reals with computably enumerable left cut (also known as left-c.e. reals), and coincide with the Martin-Loef random elements of this class. We study…
Some Goedel centenary reflections on whether incompleteness is really serious, and whether mathematics should be done somewhat differently, based on using algorithmic complexity measured in bits of information. [Enriques lecture given…
To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert's tenth problem, we consider two further classes of mathematically non-decidable problems, those of a modified version of the Hilbert's tenth…
So far, following the works of A.M. Turing, the algorithms were considered as the mathematical abstraction from which we could write programs for computers whose principle was based on the theoretical concept of Turing machine. We start…
In 1686 in his Discours de Metaphysique, Leibniz points out that if an arbitrarily complex theory is permitted then the notion of "theory" becomes vacuous because there is always a theory. This idea is developed in the modern theory of…