1,892 research outputs found
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
Testing data types implementations from algebraic specifications
Algebraic specifications of data types provide a natural basis for testing
data types implementations. In this framework, the conformance relation is
based on the satisfaction of axioms. This makes it possible to formally state
the fundamental concepts of testing: exhaustive test set, testability
hypotheses, oracle. Various criteria for selecting finite test sets have been
proposed. They depend on the form of the axioms, and on the possibilities of
observation of the implementation under test. This last point is related to the
well-known oracle problem. As the main interest of algebraic specifications is
data type abstraction, testing a concrete implementation raises the issue of
the gap between the abstract description and the concrete representation. The
observational semantics of algebraic specifications bring solutions on the
basis of the so-called observable contexts. After a description of testing
methods based on algebraic specifications, the chapter gives a brief
presentation of some tools and case studies, and presents some applications to
other formal methods involving datatypes
Polynomial tuning of multiparametric combinatorial samplers
Boltzmann samplers and the recursive method are prominent algorithmic
frameworks for the approximate-size and exact-size random generation of large
combinatorial structures, such as maps, tilings, RNA sequences or various
tree-like structures. In their multiparametric variants, these samplers allow
to control the profile of expected values corresponding to multiple
combinatorial parameters. One can control, for instance, the number of leaves,
profile of node degrees in trees or the number of certain subpatterns in
strings. However, such a flexible control requires an additional non-trivial
tuning procedure. In this paper, we propose an efficient polynomial-time, with
respect to the number of tuned parameters, tuning algorithm based on convex
optimisation techniques. Finally, we illustrate the efficiency of our approach
using several applications of rational, algebraic and P\'olya structures
including polyomino tilings with prescribed tile frequencies, planar trees with
a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures,
colours. Implementation and examples are available at [1]
https://github.com/maciej-bendkowski/boltzmann-brain [2]
https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler
Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers
The KuÄeraâGĂĄcs theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Löf random real. If the computation of the first n bits of a sequence requires n+h(n) bits of the random oracle, then h is the redundancy of the computation. KuÄera implicitly achieved redundancy nlogâĄn while GĂĄcs used a more elaborate coding procedure which achieves redundancy View the MathML source. A similar bound is implicit in the later proof by Merkle and MihailoviÄ. In this paper we obtain optimal strict lower bounds on the redundancy in computations from Martin-Löf random oracles. We show that any nondecreasing computable function g such that ân2âg(n)=â is not a general upper bound on the redundancy in computations from Martin-Löf random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a Martin-Löf random oracle satisfies ân2âg(n)<â. Moreover, the class of such reals is comeager and includes a View the MathML source real as well as all weakly 2-generic reals. On the other hand, it has been recently shown that any real is computable from a Martin-Löf random oracle with redundancy g, provided that g is a computable nondecreasing function such that ân2âg(n)<â. Hence our lower bound is optimal, and excludes many slow growing functions such as logâĄn from bounding the redundancy in computations from random oracles for a large class of reals. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop
Kolmogorov complexity and computably enumerable sets
We study the computably enumerable sets in terms of the: (a) Kolmogorov
complexity of their initial segments; (b) Kolmogorov complexity of finite
programs when they are used as oracles. We present an extended discussion of
the existing research on this topic, along with recent developments and open
problems. Besides this survey, our main original result is the following
characterization of the computably enumerable sets with trivial initial segment
prefix-free complexity. A computably enumerable set is -trivial if and
only if the family of sets with complexity bounded by the complexity of is
uniformly computable from the halting problem
Noncomputable Functions in the Blub-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
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