A new definition of program-size complexity is made. H(A;B=C;D) is defined to be the size in bits of the shortest self-delimiting program for calculating strings A and B if one is given a minimal-size selfdelimiting program for calculating strings C and D. This differs from previous definitions: (1) programs are required to be self-delimiting, i.e. no program is a prefix of another, and (2) instead of being given C and D directly, one is given a program for calculating them that is minimal in size. Unlike previous definitions, this one has precisely the formal 2 G. J. Chaitin properties of the entropy concept of information theory. For example, H(A;B) = H(A) + H(B=A) + O(1). Also, if a program of length k is assigned measure 2 \Gammak , then H(A) = \Gamma log 2 (the probability that the standard universal computer will calculate A) +O(1). Key Words and Phrases: computational complexity, entropy, information theory, instantaneous code, Kraft inequality, minimal program, probab...

The existence of probability in the sense of the frequency interpretation, i.e. probability as “long term relative frequency, ” is shown to follow from the dynamics and the interpretational rules of Everett quantum mechanics in the Heisenberg picture. This proof is free of the difficulties encountered in applying to the Everett interpretation previous results regarding relative frequency and probability in quantum mechanics. The ontology of the Everett interpretation in the Heisenberg picture is also discussed.

The main purpose of this paper is to promote the study of computational aspects, primarily the convergence rate, of nonlinear dynamical systems from a combinatorial perspective. We identify the class of symmetric quadratic systems. Such systems have been widely used to model phenomena in the natural sciences, and also provide an appropriate framework for the study of genetic algorithms in combinatorial optimisation. We prove several fundamental general properties of these systems, notably that every trajectory converges to a fixed point. We go on to give a detailed analysis of a quadratic system defined in a natural way on probability distributions over the set of matchings in a graph. In particular, we prove that convergence to the limit requires only polynomial time when the graph is a tree. This result demonstrates that such systems, though nonlinear, are amenable to quantitative analysis. 1 Introduction 1.1 Dynamical systems Many natural phenomena can be described by dynamical sy...

Abstract. Document fields, such as the title or the headings of a document, offer a way to consider the structure of documents for retrieval. Most of the proposed approaches in the literature employ either a linear combination of scores assigned to different fields, or a linear combination of frequencies in the term frequency normalisation component. In the context of the Divergence From Randomness framework, we have a sound opportunity to integrate document fields in the probabilistic randomness model. This paper introduces novel probabilistic models for incorporating fields in the retrieval process using a multinomial randomness model and its information theoretic approximation. The evaluation results from experiments conducted with a standard TREC Web test collection show that the proposed models perform as well as a state-of-the-art field-based weighting model, while at the same time, they are theoretically founded and more extensible than current field-based models. 1

We suggest an attack on a symmetric non-ideal quantum coin-tossing protocol suggested by Mayers Salvail and Chiba-Kohno. The analysis of the attack shows that the protocol is insecure. 1

We study the behavior of infinite systems of coupled harmonic oscillators as the time t ! 1, and generalize the Central Limit Theorem (CLT) to show that their reduced Wigner distributions become Gaussian under quite general conditions. This shows that generalized coherent states tend to be produced naturally. A sufficient condition for this to happen is shown to be that the spectral function is analytic and nonlinear. For a chain of coupled oscillators, the nonlinearity requirement means that waves must be dispersive, so that localized wave-packets become suppressed. Virtually all harmonic heat-bath models in the literature satisfy this constraint, and we have good reason to believe that coherent states and their generalizations are not merely a useful analytical tool, but that nature is indeed full of them. Standard proofs of the CLT rely heavily on the fact that probability densities are non-negative. Although the CLT is generally not applicable if the densities are allowed to take negative values, we show that a CLT does indeed hold for a special class of such functions. We find that, intriguingly, nature has arranged things so that all Wigner functions belong to this class. PACS Codes: 5.30.-d, 5.30.ch, 2.50.+s, 3.65.-w y Published in Phys. Rev. E, 50, 2538 (1994) 1 I.

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