354,226 research outputs found

    Resource-Bounded Category and Measure in Exponential Complexity Classes

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    This thesis presents resource-bounded category and resource- bounded measure - two new tools for computational complexity theory - and some applications of these tools to the structure theory of exponential complexity classes. Resource-bounded category, a complexity-theoretic version of the classical Baire category method, identifies certain subsets of PSPACE, E, ESPACE, and other complexity classes as meager. These meager sets are shown to form a nontrivial ideal of "small" subsets of the complexity class. The meager sets are also (almost) characterized in terms of curtain two-person infinite games called resource-bounded Banach-Maxur games. Similarly, resource-bounded measure, a complexity-theoretic version of Lebesgue measure theory, identifies the measure 0 subsets of E, ESPACE, and other complexity classes, and these too are shown to form nontrivial ideals of "small" subsets. A resource-bounded extension of the classical Kolmogorov zero-one law is also proven. This shows that measurable sets of complexity-theoretic interest either have measure 0 or are the complements of sets of measure 0. Resource-bounded category and measure are then applied to the investigation of uniform versus nonuniform complexity. In particular, Kannan's theorem that ESPACE P/Poly is extended by showing that P/Poly fl ESPACE is only a meager, measure 0 subset of ESPACE. A theorem of Huynh is extended similarly by showing that all but a meager, measure 0 subset of the languages i n ESPACE have high space-bounded Kolmogorov complexity

    Baire categories on small complexity classes and meager–comeager laws

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    We introduce two resource-bounded Baire category notions on small complexity classes such as P, QUASIPOLY, SUBEXP and PSPACE and on probabilistic classes such as BPP, which differ on how the corresponding finite extension strategies are computed. We give an alternative characterization of small sets via resource-bounded Banach-Mazur games. As an application of the first notion, we show that for almost every language A (i.e. all except a meager class) computable in subexponential time, PA = BPPA. We also show that almost all languages in PSPACE do not have small nonuniform complexity. We then switch to the second Baire category notion (called locally-computable), and show that the class SPARSE is meager in P. We show that in contrast to the resource-bounded measure case, meager–comeager laws can be obtained for many standard complexity classes, relative to locally-computable Baire category on BPP and PSPACE. Another topic where locally-computable Baire categories differ from resource-bounded measure is regarding weak-completeness: we show that there is no weak-completeness notion in P based on locally-computable Baire categories, i.e. every P-weakly-complete set is complete for P. We also prove that the class of complete sets for P under Turing-logspace reductions is meager in P, if P is not equal to DSPACE (log n), and that the same holds unconditionally for QUASIPOLY. Finally we observe that locally-computable Baire categories are incomparable with all existing resource-bounded measure notions on small complexity classes, which might explain why those two settings seem to differ so fundamentally

    The Use of Fuzzy Set Theory in Remote Sensing Pattern Recognition

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    Satellite images increasingly become a major data source for monitoring changes in the natural environment. A main task in the analysis of satellite images is concerned with the modelling of land use classes by reducing uncertainty during a classification process. In the approach presented in this paper uncertainty is perceived to be due to the vagueness of geographical categories caused by either the complexity of the category (like 'urban area') or by the use of the category in several application contexts. Two circumstances of use of an extended set theoretical concept (fuzzy sets) are discussed. First, two algorithms from the ISODATA class are used to determine the grades of membership to three a priori defined classes (woodland, suburban area, urban area) of a LANDSAT TM satellite image of Vienna, Austria. The results are visualized by a RGB image of the grades of membership to each category. Second, a measure of fuzziness is employed on the results. The measure relies on the concept of distance between a seUcategory and its complement. The so determined vagueness provide more information on the uncertainty of the different categories and may improve further information processing tasks. (authors' abstract)Series: Discussion Papers of the Institute for Economic Geography and GIScienc

    Relativized topological size of sets of partial recursive functions

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    AbstractIn [1], a recursive topology on the set of unary partial recursive functions was introduced and recursive variants of Baire topological notions of nowhere dense and meagre sets were defined. These tools were used to measure the size of some classes of partial recursive (p.r.) functions. Thus, for example, it was proved that measured sets or complexity classes are recursively meagre in contrast with the sets of all p.r. functions or recursive functions, which are sets of recursively second Baire category. In this paper we measure the size of sets of p.r. functions using the above Baire notions relativized to the topological spaces induced by these sets. In this way we strengthen, in a uniform way, most results of [4, 5, 6, 3, 2], and we also obtain new results. For many sets of p.r. functions, strong differences between “local” and “global” topological size are established

    The Neural Representation Benchmark and its Evaluation on Brain and Machine

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    A key requirement for the development of effective learning representations is their evaluation and comparison to representations we know to be effective. In natural sensory domains, the community has viewed the brain as a source of inspiration and as an implicit benchmark for success. However, it has not been possible to directly test representational learning algorithms directly against the representations contained in neural systems. Here, we propose a new benchmark for visual representations on which we have directly tested the neural representation in multiple visual cortical areas in macaque (utilizing data from [Majaj et al., 2012]), and on which any computer vision algorithm that produces a feature space can be tested. The benchmark measures the effectiveness of the neural or machine representation by computing the classification loss on the ordered eigendecomposition of a kernel matrix [Montavon et al., 2011]. In our analysis we find that the neural representation in visual area IT is superior to visual area V4. In our analysis of representational learning algorithms, we find that three-layer models approach the representational performance of V4 and the algorithm in [Le et al., 2012] surpasses the performance of V4. Impressively, we find that a recent supervised algorithm [Krizhevsky et al., 2012] achieves performance comparable to that of IT for an intermediate level of image variation difficulty, and surpasses IT at a higher difficulty level. We believe this result represents a major milestone: it is the first learning algorithm we have found that exceeds our current estimate of IT representation performance. We hope that this benchmark will assist the community in matching the representational performance of visual cortex and will serve as an initial rallying point for further correspondence between representations derived in brains and machines.Comment: The v1 version contained incorrectly computed kernel analysis curves and KA-AUC values for V4, IT, and the HT-L3 models. They have been corrected in this versio

    Topological complexity of motion planning and Massey products

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    We employ Massey products to find sharper lower bounds for the Schwarz genus of a fibration than those previously known. In particular we give examples of non-formal spaces XX for which the topological complexity \TC(X) (defined to be the genus of the free path fibration on XX) is greater than the zero-divisors cup-length plus one.Comment: 11 pages; minor revisions and 1 added reference; to appear in the Proceedings of the M. M. Postnikov Memorial Conferenc

    Complexity over Uncertainty in Generalized Representational\ud Information Theory (GRIT): A Structure-Sensitive General\ud Theory of Information

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    What is information? Although researchers have used the construct of information liberally to refer to pertinent forms of domain-specific knowledge, relatively few have attempted to generalize and standardize the construct. Shannon and Weaver(1949)offered the best known attempt at a quantitative generalization in terms of the number of discriminable symbols required to communicate the state of an uncertain event. This idea, although useful, does not capture the role that structural context and complexity play in the process of understanding an event as being informative. In what follows, we discuss the limitations and futility of any generalization (and particularly, Shannon’s) that is not based on the way that agents extract patterns from their environment. More specifically, we shall argue that agent concept acquisition, and not the communication of\ud states of uncertainty, lie at the heart of generalized information, and that the best way of characterizing information is via the relative gain or loss in concept complexity that is experienced when a set of known entities (regardless of their nature or domain of origin) changes. We show that Representational Information Theory perfectly captures this crucial aspect of information and conclude with the first generalization of Representational Information Theory (RIT) to continuous domains
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