A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]-like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal self-delimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number is-like. In this paper we show that the converse implication is true as well: any Ω-like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.

Abstract not found

Gödel's Incompleteness Theorem asserts that any sufficiently rich, sound, and recursively axiomatizable theory is incomplete. We show that, in a quite general topological sense, incompleteness is a rather common phenomenon: With respect to any reasonable topology the set of true and unprovable statements of such a theory is dense and in many cases even co-rare.

A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly non-computable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 63 bits of a Chaitin Omega: 000000100000010000100000100001110111001100100111100010010011100. Full description of programs and proofs will be given elsewhere. 1

Are there `biologically computing agents' capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically negative. Our results will be formulated in the language of membrane computing (P systems). Some mathematical results presented here are interesting in themselves. In contrast with most speed-up methods which are based on non-determinism, our results rest upon some universality results proved for deterministic P systems. These results will be used for building "accelerated P systems". In contrast with the case of Turing machines, acceleration is a part of the hardware (not a quality of the environment) and it is realised either by decreasing the size of "reactors" or by speeding-up the communication channels.

This paper is a subjective, short overview of algorithmic information theory. We critically discuss various equivalent algorithmical models of randomness motivating a #randomness hypothesis". Finally some recent results on computably enumerable random reals are reviewed. 1 Randomness: An Informal Discussion In which we discuss some di#culties arising in de#ning randomness. Suppose that one is watching a simple pendulum swing back and forth, recording 0 if it swings clockwise at a given instant and 1 if it swings counterclockwise. Suppose further that after some time the record looks as follows: 10101010101010101010101010101010: At this point one would like to deduce a #theory" from the experiment. 1 The #theory" should account for the data presently available and make #predictions" about future observations. How should one proceed? It is obvious that there are many #theories" that one could write-down for the given data, for example: 10101010101010101010101010101010000000000...

Quantum cryptography is the only approach to privacy ever proposed that allows two parties (who do not share a long secret key ahead of time) to communicate with provably perfect secrecy under the nose of an eavesdropper endowed with unlimited computational power and whose technology is limited by nothing but the fundamental laws of nature. This essay provides a personal historical perspective on the field. For the sake of liveliness, the style is purposely that of a spontaneous after-dinner speech.

Complexity, non-predictability and randomness not only occur in quantum mechanics and non-linear dynamics, they also occur in pure mathematics and shed new light on the limitations of the axiomatic method. In particular, we discuss a Diophantine equation exhibiting randomness, and how it yields a proof of Godel's incompleteness theorem. Our view of the physical world has certainly changed radically during the past hundred years, as unpredictability, randomness and complexity have replaced the comfortable world of classical physics. Amazingly enough, the same thing has occurred in the world of pure mathematics, 2 G. J. Chaitin in fact, in number theory, a branch of mathematics that is concerned with the properties of the positive integers. How can an uncertainty principle apply to number theory, which has been called the queen of mathematics, and is a discipline that goes back to the ancient Greeks and is concerned with such things as the primes and their properties? Following Davis (...

The article presents a new interpretation for Zipf’s law in natural language which relies on two areas of information theory. We reformulate the problem of grammar-based compression and investigate properties of strongly nonergodic stationary processes. The motivation for the joint discussion is to prove a proposition with a simple informal statement: If an n-letter long text describes n β independent facts in a random but consistent way then the text contains at least n β /log n different words. In the formal statement, two specific postulates are adopted. Firstly, the words are understood as the nonterminal symbols of the shortest grammar-based encoding of the text. Secondly, the texts are assumed to be emitted by a nonergodic source, with the described facts being binary IID variables that are asymptotically predictable in a shift-invariant way. The proof of the formal proposition applies several new tools. These

Efforts to calculate values of the noncomputable Busy Beaver function are discussed in the light of algorithmic information theory. I would like to talk about some impossible problems that arise when one combines information theory with recursive function or computability theory. That is to say, I'd like to look at some unsolvable problems which arise when one examines computation unlimited by any practical 2 G. J. Chaitin bound on running time, from the point of view of information theory. The result is what I like to call "algorithmic information theory" [5]. In the Computer Recreations department of a recent issue of Scientific American [7], A. K. Dewdney discusses efforts to calculate the Busy Beaver function \Sigma. This is a very interesting endeavor for a number of reasons. First of all, the Busy Beaver function is of interest to information theorists, because it measures the capability of computer programs as a function of their size, as a function of the amount of informatio...