1,526 research outputs found
On the Size Complexity of Non-Returning Context-Free PC Grammar Systems
Improving the previously known best bound, we show that any recursively
enumerable language can be generated with a non-returning parallel
communicating (PC) grammar system having six context-free components. We also
present a non-returning universal PC grammar system generating unary languages,
that is, a system where not only the number of components, but also the number
of productions and the number of nonterminals are limited by certain constants,
and these size parameters do not depend on the generated language
Minimum Description Length Induction, Bayesianism, and Kolmogorov Complexity
The relationship between the Bayesian approach and the minimum description
length approach is established. We sharpen and clarify the general modeling
principles MDL and MML, abstracted as the ideal MDL principle and defined from
Bayes's rule by means of Kolmogorov complexity. The basic condition under which
the ideal principle should be applied is encapsulated as the Fundamental
Inequality, which in broad terms states that the principle is valid when the
data are random, relative to every contemplated hypothesis and also these
hypotheses are random relative to the (universal) prior. Basically, the ideal
principle states that the prior probability associated with the hypothesis
should be given by the algorithmic universal probability, and the sum of the
log universal probability of the model plus the log of the probability of the
data given the model should be minimized. If we restrict the model class to the
finite sets then application of the ideal principle turns into Kolmogorov's
minimal sufficient statistic. In general we show that data compression is
almost always the best strategy, both in hypothesis identification and
prediction.Comment: 35 pages, Latex. Submitted IEEE Trans. Inform. Theor
Computability and human symbolic output
This paper concerns “human symbolic output,” or strings of characters produced by humans in our various symbolic systems; e.g., sentences in a natural language, mathematical propositions, and so on. One can form a set that consists of all of the strings of characters that have been produced by at least one human up to any given moment in human history. We argue that at any particular moment in human history, even at moments in the distant future, this set is finite. But then, given fundamental results in recursion theory, the set will also be recursive, recursively enumerable, axiomatizable, and could be the output of a Turing machine. We then argue that it is impossible to produce a string of symbols that humans could possibly produce but no Turing machine could. Moreover, we show that any given string of symbols that we could produce could also be the output of a Turing machine. Our arguments have implications for Hilbert’s sixth problem and the possibility of axiomatizing particular sciences, they undermine at least two distinct arguments against the possibility of Artificial Intelligence, and they entail that expert systems that are the equals of human experts are possible, and so at least one of the goals of Artificial Intelligence can be realized, at least in principle
Generic algorithms for halting problem and optimal machines revisited
The halting problem is undecidable --- but can it be solved for "most"
inputs? This natural question was considered in a number of papers, in
different settings. We revisit their results and show that most of them can be
easily proven in a natural framework of optimal machines (considered in
algorithmic information theory) using the notion of Kolmogorov complexity. We
also consider some related questions about this framework and about asymptotic
properties of the halting problem. In particular, we show that the fraction of
terminating programs cannot have a limit, and all limit points are Martin-L\"of
random reals. We then consider mass problems of finding an approximate solution
of halting problem and probabilistic algorithms for them, proving both positive
and negative results. We consider the fraction of terminating programs that
require a long time for termination, and describe this fraction using the busy
beaver function. We also consider approximate versions of separation problems,
and revisit Schnorr's results about optimal numberings showing how they can be
generalized.Comment: a preliminary version was presented at the ICALP 2015 conferenc
On Time-Bounded Incompressibility of Compressible Strings and Sequences
For every total recursive time bound , a constant fraction of all
compressible (low Kolmogorov complexity) strings is -bounded incompressible
(high time-bounded Kolmogorov complexity); there are uncountably many infinite
sequences of which every initial segment of length is compressible to yet -bounded incompressible below ; and there are
countable infinitely many recursive infinite sequence of which every initial
segment is similarly -bounded incompressible. These results are related to,
but different from, Barzdins's lemma.Comment: 9 pages, LaTeX, no figures, submitted to Information Processing
Letters. Changed and added a Barzdins-like lemma for infinite sequences with
different quantification oreder, a fixed constant, and uncountably many
sequence
On Measuring Non-Recursive Trade-Offs
We investigate the phenomenon of non-recursive trade-offs between
descriptional systems in an abstract fashion. We aim at categorizing
non-recursive trade-offs by bounds on their growth rate, and show how to deduce
such bounds in general. We also identify criteria which, in the spirit of
abstract language theory, allow us to deduce non-recursive tradeoffs from
effective closure properties of language families on the one hand, and
differences in the decidability status of basic decision problems on the other.
We develop a qualitative classification of non-recursive trade-offs in order to
obtain a better understanding of this very fundamental behaviour of
descriptional systems
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