62 research outputs found
Very Simple Chaitin Machines for Concrete AIT
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 reason the bits should be the way they are, if we define
``reason'' to be a computable explanation smaller than the data itself. Since
that time, only {\em two} explicit universal Chaitin machines have been
proposed, both by Chaitin himself.
Concrete algorithmic information theory involves the study of particular
universal Turing machines, about which one can state theorems with specific
numerical bounds, rather than include terms like O(1). We present several new
tiny Chaitin machines (those with a prefix-free domain) suitable for the study
of concrete algorithmic information theory. One of the machines, which we call
Keraia, is a binary encoding of lambda calculus based on a curried lambda
operator. Source code is included in the appendices.
We also give an algorithm for restricting the domain of blank-endmarker
machines to a prefix-free domain over an alphabet that does not include the
endmarker; this allows one to take many universal Turing machines and construct
universal Chaitin machines from them
Algorithmic Randomness as Foundation of Inductive Reasoning and Artificial Intelligence
This article is a brief personal account of the past, present, and future of
algorithmic randomness, emphasizing its role in inductive inference and
artificial intelligence. It is written for a general audience interested in
science and philosophy. Intuitively, randomness is a lack of order or
predictability. If randomness is the opposite of determinism, then algorithmic
randomness is the opposite of computability. Besides many other things, these
concepts have been used to quantify Ockham's razor, solve the induction
problem, and define intelligence.Comment: 9 LaTeX 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
Alan Turing y los orígenes de la complejidad
The 75th anniversary of Turing’s seminal paper and his centennial anniversary occur in 2011 and 2012, respectively. It is natural to review and assess Turing’s contributions in diverse fields in the light of new developments that his thought has triggered in many scientific communities. Here, the main idea is to discuss how the work of Turing allows us to change our views on the foundations of Mathematics, much as quantum mechanics changed our conception of the world of Physics. Basic notions like computability and universality are discussed in a broad context, placing special emphasis on how the notion of complexity can be given a precise meaning after Turing, i.e., not just qualitatively but also quantitatively Turing’s work is given some historical perspective with respect to some of his precursors, contemporaries and mathematicians who took his ideas further.El 75 aniversario del artículo seminal de Turing y el centenario de su nacimiento ocurren en 2011 y 2012, respectivamente. Es natural revisar y valorar las contribuciones que hizo Turing en campos muy diversos a la luz de los desarrollos que sus pensamientos han producido en muchas comunidades científicas. Aquí, la idea principal es discutir como el trabajo de Turing nos permite cambiar nuestra visión sobre los fundamentos de las Matemáticas, de forma similar a como la mecánica cuántica cambió nuestra concepción de la Física. Nociones básicas como compatibilidad y universalidad se discuten en un contexto amplio, haciendo énfasis especial en como a la noción de complejidad se le puede dar un significado preciso después de Turing, es decir, no solo cualitativo sino cuantitativo. Al trabajo de Turing se le da una perspectiva histórica en relación a algunos de sus precursores, contemporáneos y matemáticos que tomaron y llevaron sus ideas aún más allá
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