175 research outputs found

### 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

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