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60
Computational depth and reducibility
 Theoretical Computer Science
, 1994
"... This paper reviews and investigates Bennett's notions of strong and weak computational depth (also called logical depth) for in nite binary sequences. Roughly, an in nite binary sequence x is de ned to be weakly useful if every element of a nonnegligible set of decidable sequences is reducible to x ..."
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Cited by 12 (2 self)
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This paper reviews and investigates Bennett's notions of strong and weak computational depth (also called logical depth) for in nite binary sequences. Roughly, an in nite binary sequence x is de ned to be weakly useful if every element of a nonnegligible set of decidable sequences is reducible to x in recursively bounded time. It is shown that every weakly useful sequence is strongly deep. This result (which generalizes Bennett's observation that the halting problem is strongly deep) implies that every high Turing degree contains strongly deep sequences. It is also shown that, in the sense of Baire category, almost
SpaceTime Foam Dense Singularities and de Rham Cohomology
, 2004
"... In an earlier paper of the authors it was shown that the sheaf theoretically based recently developed abstract differential geometry of the first author can in an easy and natural manner incorporate singularities on arbitrary closed nowhere dense sets in Euclidean spaces, singularities which therefo ..."
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Cited by 10 (4 self)
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In an earlier paper of the authors it was shown that the sheaf theoretically based recently developed abstract differential geometry of the first author can in an easy and natural manner incorporate singularities on arbitrary closed nowhere dense sets in Euclidean spaces, singularities which therefore can have arbitrary large positive Lebesgue measure. As also shown, one can construct in such a singular context a de Rham cohomology, as well as a short exponential sequence, both of which are fundamental in differential geometry. In this paper, these results are significantly strengthened, motivated by the so called spacetime foam structures in general relativity, where singularities can be dense. In fact, this time one can deal with singularities on arbitrary sets, provided that their complementaries are dense, as well. In particular, the cardinal of the set of singularities can be larger than that of the nonsingular points. Note: This paper is an augmented version of the paper with the same title, published in Acta Applicandae Mathematicae 67(1):5989,2001, and it is posted here with the kind permission of Kluwer Academic Publishers. ’We do not possess any method at all to derive systematically solutions that are free of singularities... ’ 1
A ZeroOne Law for Dynamical Properties
 In Topological dynamics and applications (Minneapolis, MN
, 1995
"... . For any countable group \Gamma satisfying the "weak Rohlin property", and for each dynamical property, the set of \Gammaactions with that property is either residual or meager. The class of groups with the weak Rohlin property includes each lattice Z \Thetad ; indeed, all countable discrete a ..."
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Cited by 9 (2 self)
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. For any countable group \Gamma satisfying the "weak Rohlin property", and for each dynamical property, the set of \Gammaactions with that property is either residual or meager. The class of groups with the weak Rohlin property includes each lattice Z \Thetad ; indeed, all countable discrete amenable groups. For \Gamma an arbitrary countable group, let A be the set of \Gammaactions on the unit circle Y . We establish an Equivalence theorem by showing that a dynamical property is Baire/meager/residual in A if and only if it is Baire/meager/residual in the set of shiftinvariant measures on the product space Y \Theta\Gamma . x1 Introduction Halmos's book Ergodic Theory introduced many of us to the study of determining which dynamical properties are generic (i.e, topologically residual) in the socalled "coarse topology" on transformations. For instance, "weakmixing" is generic, whereas "mixing" is not, [Hal, pp. 77,78]. The exploration of this notion of genericity became a...
Axiomatising Various Classes of Relation and Cylindric Algebras
 Logic Journal of the IGPL
, 1997
"... We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras. 1 Introduction Relation algebras are to bina ..."
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Cited by 8 (5 self)
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We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras. 1 Introduction Relation algebras are to binary relations what boolean algebras are to unary ones. They are used in artificial intelligence, where, for example, the AllenKoomen temporal planning system checks the consistency of given relations between time intervals. In mathematics, they form a part of algebraic logic. The history of this goes back to the nineteenth century, the early workers including Boole, de Morgan, Peirce, and Schroder; it was studied intensively by Tarski's group (including, at various times, Chin, Givant, Henkin, J'onsson, Lyndon, Maddux, Monk, N'emeti) from around the 1950s, and currently we know of active groups in Amsterdam, Budapest, Rio de Janeiro, South Africa, and the U.S., among other places. Abstract...
ResourceBounded Baire Category: A Stronger Approach
 Proceedings of the Tenth Annual IEEE Conference on Structure in Complexity Theory
, 1996
"... This paper introduces a new definition of resourcebounded Baire category in the style of Lutz. This definition gives an almostall/almostnone theory of various complexity classes. The meagerness/comeagerness of many more classes can be resolved in the new definition than in previous definitions. ..."
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This paper introduces a new definition of resourcebounded Baire category in the style of Lutz. This definition gives an almostall/almostnone theory of various complexity classes. The meagerness/comeagerness of many more classes can be resolved in the new definition than in previous definitions. For example, almost no sets in EXP are EXPcomplete, and NP is PFmeager unless NP = EXP. It is also seen under the new definition that no recrandom set can be (recursively) ttreducible to any PFgeneric set. We weaken our definition by putting arbitrary bounds on the length of extension strategies, obtaining a spectrum of different theories of Baire Category that includes Lutz's original definition. 1
Survey on dissipative KAM theory including quasiperiodic bifurcation theory based on lectures by Henk Broer
 Peyresq Lectures in Geometric Mechanics and Symmetry
, 2005
"... Based on lectures by Henk Broer KolmogorovArnol’dMoser Theory classically was mainly developed for conservative systems, establishing persistence results for quasiperiodic invariant tori in nearly integrable systems. In this survey we focus on dissipative systems, where similar results hold. In n ..."
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Cited by 8 (5 self)
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Based on lectures by Henk Broer KolmogorovArnol’dMoser Theory classically was mainly developed for conservative systems, establishing persistence results for quasiperiodic invariant tori in nearly integrable systems. In this survey we focus on dissipative systems, where similar results hold. In nonconservative settings often parameters are needed for the persistence of invariant tori. When considering families of such dynamical systems bifurcations of quasiperiodic tori may occur. As an example we discuss the quasiperiodic Hopf bifurcation. 1.1 Motivation 1
On Nonconvex Caustics of Convex Billiards
, 1996
"... There are billiard tables of constant width, for which the billiard map has invariant curves in the phase space which belong to continuous but nowhere differentiable caustics. We apply this to construct ruled surfaces which have a nowhere differentiable lines of striction. We use it also to get Riem ..."
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Cited by 6 (0 self)
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There are billiard tables of constant width, for which the billiard map has invariant curves in the phase space which belong to continuous but nowhere differentiable caustics. We apply this to construct ruled surfaces which have a nowhere differentiable lines of striction. We use it also to get Riemannian metrics on the sphere such that the caustic belonging at least one point on the sphere is nowhere differentiable. For three dimensional billiards, we find three dimensional billiard surfaces with nonconvex rough caustics. 1 Convex billiards Describing the long time behavior of a path of a light ray or billiard ball in a convex domain is an interesting mathematical problem. Good starting points in the literature are [21, 31, 6, 33, 27]. In the case, when the domain is a strictly convex planar region, the return map to the boundary T defines an area preserving map OE of the annulus A = T \Theta [0; 1], where the circle T parametrizes the table with arc length s: the impact point s at t...
A Topological Characterization of Random Sequences
, 2003
"... The set of random sequences is large in the sense of measure, but small in the sense of category. This is the case when we regard the set of infinite sequences over a finite alphabet as a subset of the usual Cantor space. In this note we will show that the above result depends on the topology chosen ..."
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Cited by 3 (0 self)
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The set of random sequences is large in the sense of measure, but small in the sense of category. This is the case when we regard the set of infinite sequences over a finite alphabet as a subset of the usual Cantor space. In this note we will show that the above result depends on the topology chosen. To this end we will use a relativisation of the Cantor topology, the U topology introduced in Staiger (1987). This topology is also metric, but the distance between two sequences does not depend on their longest common prefix (Cantor metric), but on the number of their common prefixes in a given language U . The resulting space is complete, but not always compact. We will show how to derive a computable set U from a universal MartinLf test such that the set of nonrandom sequences is nowhere dense in the U topology. As a byproduct we obtain a topological characterization of the set of random sequences. We also show that the Law of Large Numbers, which fails with respect to the usual topology, is true for the U topology. Keywords: MartinLf test, random sequence, theory of computation 1
Baire Category and Nowhere Differentiability for Feasible Real Functions ⋆
"... Abstract. A notion of resourcebounded Baire category is developed for the class PC[0,1] of all polynomialtime computable realvalued functions on the unit interval. The meager subsets of PC[0,1] are characterized in terms of resourcebounded BanachMazur games. This characterization is used to pro ..."
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Abstract. A notion of resourcebounded Baire category is developed for the class PC[0,1] of all polynomialtime computable realvalued functions on the unit interval. The meager subsets of PC[0,1] are characterized in terms of resourcebounded BanachMazur games. This characterization is used to prove that, in the sense of Baire category, almost every function in PC[0,1] is nowhere differentiable. This is a complexitytheoretic extension of the analogous classical result that Banach proved for the class C[0, 1] in 1931. 1
Probability as typicality
, 2006
"... The concept of typicality refers to properties holding for the “vast majority” of cases and is a fundamental idea of the qualitative approach to dynamical problems. We argue that measuretheoretical typicality would be the adequate viewpoint of the role of probability in classical statistical mechan ..."
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The concept of typicality refers to properties holding for the “vast majority” of cases and is a fundamental idea of the qualitative approach to dynamical problems. We argue that measuretheoretical typicality would be the adequate viewpoint of the role of probability in classical statistical mechanics, particularly in understanding the micro to macroscopic change of levels of description. Keywords: Statistical mechanics; Typicality; Probability.