6,254 research outputs found
Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals
We describe the basic theory of infinite time Turing machines and some recent
developments, including the infinite time degree theory, infinite time
complexity theory, and infinite time computable model theory. We focus
particularly on the application of infinite time Turing machines to the
analysis of the hierarchy of equivalence relations on the reals, in analogy
with the theory arising from Borel reducibility. We define a notion of infinite
time reducibility, which lifts much of the Borel theory into the class
in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference,
200
Game interpretation of Kolmogorov complexity
The Kolmogorov complexity function K can be relativized using any oracle A,
and most properties of K remain true for relativized versions. In section 1 we
provide an explanation for this observation by giving a game-theoretic
interpretation and showing that all "natural" properties are either true for
all sufficiently powerful oracles or false for all sufficiently powerful
oracles. This result is a simple consequence of Martin's determinacy theorem,
but its proof is instructive: it shows how one can prove statements about
Kolmogorov complexity by constructing a special game and a winning strategy in
this game. This technique is illustrated by several examples (total conditional
complexity, bijection complexity, randomness extraction, contrasting plain and
prefix complexities).Comment: 11 pages. Presented in 2009 at the conference on randomness in
Madison
Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes
We survey diverse approaches to the notion of information: from Shannon
entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov
complexity are presented: randomness and classification. The survey is divided
in two parts in the same volume. Part I is dedicated to information theory and
the mathematical formalization of randomness based on Kolmogorov complexity.
This last application goes back to the 60's and 70's with the work of
Martin-L\"of, Schnorr, Chaitin, Levin, and has gained new impetus in the last
years.Comment: 40 page
Noncomputable functions in the Blum-Shub-Smale model
Working in the Blum-Shub-Smale model of computation on the real numbers, we
answer several questions of Meer and Ziegler. First, we show that, for each
natural number d, an oracle for the set of algebraic real numbers of degree at
most d is insufficient to allow an oracle BSS-machine to decide membership in
the set of algebraic numbers of degree d + 1. We add a number of further
results on relative computability of these sets and their unions. Then we show
that the halting problem for BSS-computation is not decidable below any
countable oracle set, and give a more specific condition, related to the
cardinalities of the sets, necessary for relative BSS-computability. Most of
our results involve the technique of using as input a tuple of real numbers
which is algebraically independent over both the parameters and the oracle of
the machine
On approximate decidability of minimal programs
An index in a numbering of partial-recursive functions is called minimal
if every lesser index computes a different function from . Since the 1960's
it has been known that, in any reasonable programming language, no effective
procedure determines whether or not a given index is minimal. We investigate
whether the task of determining minimal indices can be solved in an approximate
sense. Our first question, regarding the set of minimal indices, is whether
there exists an algorithm which can correctly label 1 out of indices as
either minimal or non-minimal. Our second question, regarding the function
which computes minimal indices, is whether one can compute a short list of
candidate indices which includes a minimal index for a given program. We give
some negative results and leave the possibility of positive results as open
questions
On the necessity of complexity
Wolfram's Principle of Computational Equivalence (PCE) implies that universal
complexity abounds in nature. This paper comprises three sections. In the first
section we consider the question why there are so many universal phenomena
around. So, in a sense, we week a driving force behind the PCE if any. We
postulate a principle GNS that we call the Generalized Natural Selection
Principle that together with the Church-Turing Thesis is seen to be equivalent
to a weak version of PCE. In the second section we ask the question why we do
not observe any phenomena that are complex but not-universal. We choose a
cognitive setting to embark on this question and make some analogies with
formal logic. In the third and final section we report on a case study where we
see rich structures arise everywhere.Comment: 17 pages, 3 figure
Instruction sequence processing operators
Instruction sequence is a key concept in practice, but it has as yet not come
prominently into the picture in theoretical circles. This paper concerns
instruction sequences, the behaviours produced by them under execution, the
interaction between these behaviours and components of the execution
environment, and two issues relating to computability theory. Positioning
Turing's result regarding the undecidability of the halting problem as a result
about programs rather than machines, and taking instruction sequences as
programs, we analyse the autosolvability requirement that a program of a
certain kind must solve the halting problem for all programs of that kind. We
present novel results concerning this autosolvability requirement. The analysis
is streamlined by using the notion of a functional unit, which is an abstract
state-based model of a machine. In the case where the behaviours exhibited by a
component of an execution environment can be viewed as the behaviours of a
machine in its different states, the behaviours concerned are completely
determined by a functional unit. The above-mentioned analysis involves
functional units whose possible states represent the possible contents of the
tapes of Turing machines with a particular tape alphabet. We also investigate
functional units whose possible states are the natural numbers. This
investigation yields a novel computability result, viz. the existence of a
universal computable functional unit for natural numbers.Comment: 37 pages; missing equations in table 3 added; combined with
arXiv:0911.1851 [cs.PL] and arXiv:0911.5018 [cs.LO]; introduction and
concluding remarks rewritten; remarks and examples added; minor error in
proof of theorem 4 correcte
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