Related papers: Computing halting probabilities from other halting…
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,also known as Chaitin's Omega, is an algorithmically random number with many interesting properties. Since Chaitin's seminal work, many popular expositions have appeared, mainly focusing on the…
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…
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…
Chaitin's number Omega is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every…
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…
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,…
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…
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…
The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance)…
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…
The $\Omega$ numbers-the halting probabilities of universal prefix-free machines-are known to be exactly the Martin-L{\"o}f random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-L{\"o}f random left-c.e. real…
The halting problem for Turing machines is decidable on a set of asymptotic probability one. Specifically, there is a set B of Turing machine programs such that (i) B has asymptotic probability one, so that as the number of states n…
Since many real-world problems arising in the fields of compiler optimisation, automated software engineering, formal proof systems, and so forth are equivalent to the Halting Problem--the most notorious undecidable problem--there is a…
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…
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,…
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…
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…
In this paper we consider the time and the crossing sequence complexities of one-tape off-line Turing machines. We show that the running time of each nondeterministic machine accepting a nonregular language must grow at least as n\log n, in…