655 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
Bicategorical Semantics for Nondeterministic Computation
We outline a bicategorical syntax for the interaction between public and
private information in classical information theory. We use this to give
high-level graphical definitions of encrypted communication and secret sharing
protocols, including a characterization of their security properties.
Remarkably, this makes it clear that the protocols have an identical abstract
form to the quantum teleportation and dense coding procedures, yielding
evidence of a deep connection between classical and quantum information
processing. We also formulate public-key cryptography using our scheme.
Specific implementations of these protocols as nondeterministic classical
procedures are recovered by applying our formalism in a symmetric monoidal
bicategory of matrices of relations.Comment: 21 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
Algorithmic Thermodynamics
Algorithmic entropy can be seen as a special case of entropy as studied in
statistical mechanics. This viewpoint allows us to apply many techniques
developed for use in thermodynamics to the subject of algorithmic information
theory. In particular, suppose we fix a universal prefix-free Turing machine
and let X be the set of programs that halt for this machine. Then we can regard
X as a set of 'microstates', and treat any function on X as an 'observable'.
For any collection of observables, we can study the Gibbs ensemble that
maximizes entropy subject to constraints on expected values of these
observables. We illustrate this by taking the log runtime, length, and output
of a program as observables analogous to the energy E, volume V and number of
molecules N in a container of gas. The conjugate variables of these observables
allow us to define quantities which we call the 'algorithmic temperature' T,
'algorithmic pressure' P and algorithmic potential' mu, since they are
analogous to the temperature, pressure and chemical potential. We derive an
analogue of the fundamental thermodynamic relation dE = T dS - P d V + mu dN,
and use it to study thermodynamic cycles analogous to those for heat engines.
We also investigate the values of T, P and mu for which the partition function
converges. At some points on the boundary of this domain of convergence, the
partition function becomes uncomputable. Indeed, at these points the partition
function itself has nontrivial algorithmic entropy.Comment: 20 pages, one encapsulated postscript figur
Most Programs Stop Quickly or Never Halt
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 growing interest, not only academically, in understanding
the problem better and in providing alternative solutions. Halting computations
can be recognised by simply running them; the main difficulty is to detect
non-halting programs. Our approach is to have the probability space extend over
both space and time and to consider the probability that a random -bit
program has halted by a random time. We postulate an a priori computable
probability distribution on all possible runtimes and we prove that given an
integer k>0, we can effectively compute a time bound T such that the
probability that an N-bit program will eventually halt given that it has not
halted by T is smaller than 2^{-k}. We also show that the set of halting
programs (which is computably enumerable, but not computable) can be written as
a disjoint union of a computable set and a set of effectively vanishing
probability. Finally, we show that ``long'' runtimes are effectively rare. More
formally, the set of times at which an N-bit program can stop after the time
2^{N+constant} has effectively zero density.Comment: Shortened abstract and changed format of references to match Adv.
Appl. Math guideline
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