31,119 research outputs found

    Uniform test of algorithmic randomness over a general space

    Get PDF
    The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend the theory to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary distributions (by Levin). We recall the main ideas and problems of Levin's theory, and report further progress in the same framework. - We allow non-compact spaces (like the space of continuous functions, underlying the Brownian motion). - The uniform test (deficiency of randomness) d_P(x) (depending both on the outcome x and the measure P should be defined in a general and natural way. - We see which of the old results survive: existence of universal tests, conservation of randomness, expression of tests in terms of description complexity, existence of a universal measure, expression of mutual information as "deficiency of independence. - The negative of the new randomness test is shown to be a generalization of complexity in continuous spaces; we show that the addition theorem survives. The paper's main contribution is introducing an appropriate framework for studying these questions and related ones (like statistics for a general family of distributions).Comment: 40 pages. Journal reference and a slight correction in the proof of Theorem 7 adde

    Computability of probability measures and Martin-Lof randomness over metric spaces

    Get PDF
    In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption).Comment: 29 page

    A Manifestly Gauge-Invariant Approach to Quantum Theories of Gauge Fields

    Full text link
    In gauge theories, physical histories are represented by space-time connections modulo gauge transformations. The space of histories is thus intrinsically non-linear. The standard framework of constructive quantum field theory has to be extended to face these {\it kinematical} non-linearities squarely. We first present a pedagogical account of this problem and then suggest an avenue for its resolution.Comment: 27 pages, CGPG-94/8-2, latex, contribution to the Cambridge meeting proceeding

    Bohrification of operator algebras and quantum logic

    Get PDF
    Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n-by-n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the "Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of functors from C(A) to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns-Lakser completions. Finally, we establish a connection between probability measure on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A for A = B(H).Comment: 31 page
    • …
    corecore