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A zoo of computable binary normal sequences.

ABSTRACT: Historically there has been a virtual absence of constructive methods to produce broad classes of "certifiably random" infinite sequences, despite considerable interest in this endeavor. Previously, we proved a theorem that yielded explicit algorithms to produce diverse sets of normal numbers, reasonable candidates for random sequences, given their limiting equidistribution of subblocks of all lengths. Herein, we develop this algorithmic approach much further, systematizing the normal number generation process in several ways. We construct delineated, distinct sets of normal numbers (classified by the extent to which initial segments deviate from maximal irregularity), with virtually any allowable specified rate of convergence to 0 of this deviation, encompassing arbitrarily fast and slow rates, and accommodating asymmetric behavior above or below a centered median. As a corollary, we provide an explicit construction of a normal number that satisfies the Law of the Iterated Logarithm. We also produce distinct families of "biased" normal numbers, with virtually any specified rate of convergence of the bias (to 0). This latter theory is in part motivated by the remarkable observation that the binary version of Champernowne's number, which is also normal, is biased-any initial segment has more 1s than 0s. Finally, we construct an interesting normal sequence with arbitrarily fast convergence to equidistribution of singleton blocks, yet arbitrarily slow convergence of pairs, which has profound implications both for probability theory, and for metrics to evaluate the "near-randomness" of sequences.

SUBMITTER: Pincus S

PROVIDER: S-EPMC3511067 | biostudies-literature | 2012 Nov

REPOSITORIES: biostudies-literature

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Publications

A zoo of computable binary normal sequences.

Pincus Steve S Singer Burton H BH

Proceedings of the National Academy of Sciences of the United States of America 20121102 47

Historically there has been a virtual absence of constructive methods to produce broad classes of "certifiably random" infinite sequences, despite considerable interest in this endeavor. Previously, we proved a theorem that yielded explicit algorithms to produce diverse sets of normal numbers, reasonable candidates for random sequences, given their limiting equidistribution of subblocks of all lengths. Herein, we develop this algorithmic approach much further, systematizing the normal number gen ...[more]

PMID: 23125196

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Project description:BackgroundThe ability to search for and precisely compare similar phenotypic appearances within and across species has vast potential in plant science and genetic research. The difficulty in doing so lies in the fact that many visual phenotypic data, especially visually observed phenotypes that often times cannot be directly measured quantitatively, are in the form of text annotations, and these descriptions are plagued by semantic ambiguity, heterogeneity, and low granularity. Though several bio-ontologies have been developed to standardize phenotypic (and genotypic) information and permit comparisons across species, these semantic issues persist and prevent precise analysis and retrieval of information. A framework suitable for the modeling and analysis of precise computable representations of such phenotypic appearances is needed.ResultsWe have developed a new framework called the Computable Visually Observed Phenotype Ontological Framework for plants. This work provides a novel quantitative view of descriptions of plant phenotypes that leverages existing bio-ontologies and utilizes a computational approach to capture and represent domain knowledge in a machine-interpretable form. This is accomplished by means of a robust and accurate semantic mapping module that automatically maps high-level semantics to low-level measurements computed from phenotype imagery. The framework was applied to two different plant species with semantic rules mined and an ontology constructed. Rule quality was evaluated and showed high quality rules for most semantics. This framework also facilitates automatic annotation of phenotype images and can be adopted by different plant communities to aid in their research.ConclusionsThe Computable Visually Observed Phenotype Ontological Framework for plants has been developed for more efficient and accurate management of visually observed phenotypes, which play a significant role in plant genomics research. The uniqueness of this framework is its ability to bridge the knowledge of informaticians and plant science researchers by translating descriptions of visually observed phenotypes into standardized, machine-understandable representations, thus enabling the development of advanced information retrieval and phenotype annotation analysis tools for the plant science community.

Dataset Information

A zoo of computable binary normal sequences.

Publications

A zoo of computable binary normal sequences.

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