3,001 research outputs found

    Limit complexities revisited [once more]

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    The main goal of this article is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result of Vereshchagin saying that lim sup⁥nC(x∣n)\limsup_n C(x|n) equals C0â€Č(x)C^{0'}(x). Then we use the same argument to prove similar results for prefix complexity, a priori probability on binary tree, to prove Conidis' theorem about limits of effectively open sets, and also to improve the results of Muchnik about limit frequencies. As a by-product, we get a criterion of 2-randomness proved by Miller: a sequence XX is 2-random if and only if there exists cc such that any prefix xx of XX is a prefix of some string yy such that C(y)â‰„âˆŁy∣−cC(y)\ge |y|-c. (In the 1960ies this property was suggested in Kolmogorov as one of possible randomness definitions.) We also get another 2-randomness criterion by Miller and Nies: XX is 2-random if and only if C(x)â‰„âˆŁx∣−cC(x)\ge |x|-c for some cc and infinitely many prefixes xx of XX. This is a modified version of our old paper that contained a weaker (and cumbersome) version of Conidis' result, and the proof used low basis theorem (in quite a strange way). The full version was formulated there as a conjecture. This conjecture was later proved by Conidis. Bruno Bauwens (personal communication) noted that the proof can be obtained also by a simple modification of our original argument, and we reproduce Bauwens' argument with his permission.Comment: See http://arxiv.org/abs/0802.2833 for the old pape

    A Casual Tour Around a Circuit Complexity Bound

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    I will discuss the recent proof that the complexity class NEXP (nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial size. The proof will be described from the perspective of someone trying to discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News, September 201

    On approximate decidability of minimal programs

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    An index ee in a numbering of partial-recursive functions is called minimal if every lesser index computes a different function from ee. Since the 1960's it has been known that, in any reasonable programming language, no effective procedure determines whether or not a given index is minimal. We investigate whether the task of determining minimal indices can be solved in an approximate sense. Our first question, regarding the set of minimal indices, is whether there exists an algorithm which can correctly label 1 out of kk indices as either minimal or non-minimal. Our second question, regarding the function which computes minimal indices, is whether one can compute a short list of candidate indices which includes a minimal index for a given program. We give some negative results and leave the possibility of positive results as open questions

    Causality - Complexity - Consistency: Can Space-Time Be Based on Logic and Computation?

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    The difficulty of explaining non-local correlations in a fixed causal structure sheds new light on the old debate on whether space and time are to be seen as fundamental. Refraining from assuming space-time as given a priori has a number of consequences. First, the usual definitions of randomness depend on a causal structure and turn meaningless. So motivated, we propose an intrinsic, physically motivated measure for the randomness of a string of bits: its length minus its normalized work value, a quantity we closely relate to its Kolmogorov complexity (the length of the shortest program making a universal Turing machine output this string). We test this alternative concept of randomness for the example of non-local correlations, and we end up with a reasoning that leads to similar conclusions as in, but is conceptually more direct than, the probabilistic view since only the outcomes of measurements that can actually all be carried out together are put into relation to each other. In the same context-free spirit, we connect the logical reversibility of an evolution to the second law of thermodynamics and the arrow of time. Refining this, we end up with a speculation on the emergence of a space-time structure on bit strings in terms of data-compressibility relations. Finally, we show that logical consistency, by which we replace the abandoned causality, it strictly weaker a constraint than the latter in the multi-party case.Comment: 17 pages, 16 figures, small correction

    Quantum Associative Memory

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    This paper combines quantum computation with classical neural network theory to produce a quantum computational learning algorithm. Quantum computation uses microscopic quantum level effects to perform computational tasks and has produced results that in some cases are exponentially faster than their classical counterparts. The unique characteristics of quantum theory may also be used to create a quantum associative memory with a capacity exponential in the number of neurons. This paper combines two quantum computational algorithms to produce such a quantum associative memory. The result is an exponential increase in the capacity of the memory when compared to traditional associative memories such as the Hopfield network. The paper covers necessary high-level quantum mechanical and quantum computational ideas and introduces a quantum associative memory. Theoretical analysis proves the utility of the memory, and it is noted that a small version should be physically realizable in the near future

    Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

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    The Kučera–GĂĄcs 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. Kučera implicitly achieved redundancy nlog⁥n 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 Mihailović. 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 ∑n2−g(n)=∞ 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 ∑n2−g(n)<∞. 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 ∑n2−g(n)<∞. Hence our lower bound is optimal, and excludes many slow growing functions such as log⁥n 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

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    In dieser Dissertation werden neue Ergebnisse ĂŒber KolmogorovkomplexitĂ€t diskutiert. Ihr erster Teil konzentriert sich auf das Studium von KolmogorovkomplexitĂ€t ohne Zeitschranken. Hier beschĂ€ftigen wir uns mit dem Konzept nicht-monotoner ZufĂ€lligkeit, d.h. ZufĂ€lligkeit, die von Martingalen charakterisiert wird, die in nicht-monotoner Reihenfolge wetten dĂŒrfen. Wir werden in diesem Zusammenhang eine Reihe von ZufĂ€lligkeitsklassen einfĂŒhren, und diese dann von einander separieren. Wir prĂ€sentieren auß erdem einen systematischen ĂŒberblick ĂŒber verschiedene Traceability-Begriffe und charakterisieren diese durch (Auto-)KomplexitĂ€tsbegriffe. Traceabilities sind eine Gruppe von Begriffen, die ausdrĂŒcken, dass eine Menge beinahe berechenbar ist. Der zweite Teil dieses Dokuments beschĂ€ftigt sich mit dem Thema zeitbeschrĂ€nkter KolmogorovkomplexitĂ€t. ZunĂ€chst untersuchen wir den Unterschied zwischen zwei Arten, ein Wort zu beschreiben: Die KomplexitĂ€t, es genau genug zu beschreiben, damit es von anderen Wörter unterschieden werden kann; sowie die KomplexitĂ€t, es genau genug zu beschreiben, damit das Wort aus der Beschreibung tatsĂ€chlich generiert werden kann. Diese Unterscheidung ist im Falle zeitunbeschrĂ€nkter KolmogorovkomplexitĂ€t nicht von Bedeutung; sobald wir jedoch Zeitschranken einfĂŒhren, wird sie essentiell. Als nĂ€chstes fĂŒhren wir den Begriff der Tiefe ein und beweisen ein ihn betreffendes Dichotomieresultat, das in seiner Struktur an Kummers bekanntes Gap-Theorem erinnert. Zu guter Letzt betrachten wir den wichtigen Begriff der Solovayfunktionen. Hierbei handelt es sich um berechenbare obere Schranken der KolmogorovkomplexitĂ€t, die unendlich oft scharf sind. Wir benutzen sie, um in einem gewissen Zusammenhang Martin-Löf-ZufĂ€lligkeit zu charakterisieren, und um eine Charakterisierung von Jump-Traceability anzugeben
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