De-Quantising the Solution of Deutsch's Problem

Probably the simplest and most frequently used way to illustrate the power of quantum computing is to solve the so-called {\it Deutsch's problem}. Consider a Boolean function $f: \{0,1\} \to \{0,1\}$ and suppose that we have a (classical) black box to compute it. The problem asks whether $f$ is constant (that is, $f(0) = f(1)$) or balanced ($f(0) \not= f(1)$). Classically, to solve the problem seems to require the computation of $f(0)$ and $f(1)$, and then the comparison of results. Is it possible to solve the problem with {\em only one} query on $f$? In a famous paper published in 1985, Deutsch posed the problem and obtained a ``quantum'' {\em partial affirmative answer}. In 1998 a complete, probability-one solution was presented by Cleve, Ekert, Macchiavello, and Mosca. Here we will show that the quantum solution can be {\it de-quantised} to a deterministic simpler solution which is as efficient as the quantum one. The use of ``superposition'', a key ingredient of quantum algorithm, is--in this specific case--classically available.Comment: 8 page

Natural Halting Probabilities, Partial Randomness, and Zeta Functions

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 according to their zeta numbers is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness--which cannot be naturally characterised in terms of plain complexity--asymptotic randomness admits such a characterisation.Comment: Accepted for publication in Information and Computin

Is Complexity a Source of Incompleteness?

In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is invariant under the change of the G\"odel numbering. For this measure, the theorems of a finitely-specified, sound, consistent theory strong enough to formalize arithmetic which is arithmetically sound (like Zermelo-Fraenkel set theory with choice or Peano Arithmetic) have bounded complexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. Previous results showing that incompleteness is not accidental, but ubiquitous are here reinforced in probabilistic terms: the probability that a true sentence of length $n$ is provable in the theory tends to zero when $n$ tends to infinity, while the probability that a sentence of length $n$ is true is strictly positive.Comment: 15 pages, improved versio

Quantum randomness and value indefiniteness

As computability implies value definiteness, certain sequences of quantum outcomes cannot be computable.Comment: 13 pages, revise

Spurious, Emergent Laws in Number Worlds

We study some aspects of the emergence of logos from chaos on a basal model of the universe using methods and techniques from algorithmic information and Ramsey theories. Thereby an intrinsic and unusual mixture of meaningful and spurious, emerging laws surfaces. The spurious, emergent laws abound, they can be found almost everywhere. In accord with the ancient Greek theogony one could say that logos, the Gods and the laws of the universe, originate from "the void," or from chaos, a picture which supports the unresolvable/irreducible lawless hypothesis. The analysis presented in this paper suggests that the "laws" discovered in science correspond merely to syntactical correlations, are local and not universal.Comment: 24 pages, invited contribution to "Contemporary Natural Philosophy and Philosophies - Part 2" - Special Issue of the journal Philosophie