25,577 research outputs found
Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability
Previously referred to as `miraculous' in the scientific literature because
of its powerful properties and its wide application as optimal solution to the
problem of induction/inference, (approximations to) Algorithmic Probability
(AP) and the associated Universal Distribution are (or should be) of the
greatest importance in science. Here we investigate the emergence, the rates of
emergence and convergence, and the Coding-theorem like behaviour of AP in
Turing-subuniversal models of computation. We investigate empirical
distributions of computing models in the Chomsky hierarchy. We introduce
measures of algorithmic probability and algorithmic complexity based upon
resource-bounded computation, in contrast to previously thoroughly investigated
distributions produced from the output distribution of Turing machines. This
approach allows for numerical approximations to algorithmic
(Kolmogorov-Chaitin) complexity-based estimations at each of the levels of a
computational hierarchy. We demonstrate that all these estimations are
correlated in rank and that they converge both in rank and values as a function
of computational power, despite fundamental differences between computational
models. In the context of natural processes that operate below the Turing
universal level because of finite resources and physical degradation, the
investigation of natural biases stemming from algorithmic rules may shed light
on the distribution of outcomes. We show that up to 60\% of the
simplicity/complexity bias in distributions produced even by the weakest of the
computational models can be accounted for by Algorithmic Probability in its
approximation to the Universal Distribution.Comment: 27 pages main text, 39 pages including supplement. Online complexity
calculator: http://complexitycalculator.com
Improved bounds for testing Dyck languages
In this paper we consider the problem of deciding membership in Dyck
languages, a fundamental family of context-free languages, comprised of
well-balanced strings of parentheses. In this problem we are given a string of
length in the alphabet of parentheses of types and must decide if it is
well-balanced. We consider this problem in the property testing setting, where
one would like to make the decision while querying as few characters of the
input as possible.
Property testing of strings for Dyck language membership for , with a
number of queries independent of the input size , was provided in [Alon,
Krivelevich, Newman and Szegedy, SICOMP 2001]. Property testing of strings for
Dyck language membership for was first investigated in [Parnas, Ron
and Rubinfeld, RSA 2003]. They showed an upper bound and a lower bound for
distinguishing strings belonging to the language from strings that are far (in
terms of the Hamming distance) from the language, which are respectively (up to
polylogarithmic factors) the power and the power of the input size
.
Here we improve the power of in both bounds. For the upper bound, we
introduce a recursion technique, that together with a refinement of the methods
in the original work provides a test for any power of larger than .
For the lower bound, we introduce a new problem called Truestring Equivalence,
which is easily reducible to the -type Dyck language property testing
problem. For this new problem, we show a lower bound of to the power of
Finite-State Complexity and the Size of Transducers
Finite-state complexity is a variant of algorithmic information theory
obtained by replacing Turing machines with finite transducers. We consider the
state-size of transducers needed for minimal descriptions of arbitrary strings
and, as our main result, we show that the state-size hierarchy with respect to
a standard encoding is infinite. We consider also hierarchies yielded by more
general computable encodings.Comment: In Proceedings DCFS 2010, arXiv:1008.127
The descriptive complexity approach to LOGCFL
Building upon the known generalized-quantifier-based first-order
characterization of LOGCFL, we lay the groundwork for a deeper investigation.
Specifically, we examine subclasses of LOGCFL arising from varying the arity
and nesting of groupoidal quantifiers. Our work extends the elaborate theory
relating monoidal quantifiers to NC1 and its subclasses. In the absence of the
BIT predicate, we resolve the main issues: we show in particular that no single
outermost unary groupoidal quantifier with FO can capture all the context-free
languages, and we obtain the surprising result that a variant of Greibach's
``hardest context-free language'' is LOGCFL-complete under quantifier-free
BIT-free projections. We then prove that FO with unary groupoidal quantifiers
is strictly more expressive with the BIT predicate than without. Considering a
particular groupoidal quantifier, we prove that first-order logic with majority
of pairs is strictly more expressive than first-order with majority of
individuals. As a technical tool of independent interest, we define the notion
of an aperiodic nondeterministic finite automaton and prove that FO
translations are precisely the mappings computed by single-valued aperiodic
nondeterministic finite transducers.Comment: 10 pages, 1 figur
Input-Driven Tissue P Automata
We introduce several variants of input-driven tissue P automata where the
rules to be applied only depend on the input symbol. Both strings and multisets are
considered as input objects; the strings are either read from an input tape or defined
by the sequence of symbols taken in, and the multisets are given in an input cell at the
beginning of a computation, enclosed in a vesicle. Additional symbols generated during a
computation are stored in this vesicle, too. An input is accepted when the vesicle reaches a
final cell and it is empty. The computational power of some variants of input-driven tissue
P automata is illustrated by examples and compared with the power of the input-driven
variants of other automata as register machines and counter automata
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