Steven M. Kautz

Relative to a Random Oracle, NP Is Not Small

Resource-boundedmeasure as originated by Lutz is an extension of classical measure theory which provides a probabilistic means of describing the relative sizes of complexity classes. Lutz has proposed the hypothesis that NP does not havep-measure zero, meaning loosely that NP contains a non-negligible subset of exponential time. This hypothesis implies a strong separation of P from NP and is supported by a growing body of plausible consequences which are not known to follow from the weaker assertion P?NP. It is shown in this paper that relative to a random oracle, NP does not havep-measure zero. The proof exploits the followingindependenceproperty of algorithmically random sequences: ifAis an algorithmically random sequence and a subsequenceA0is chosen by means of abounded Kolmogorov? Loveland place selection, then the sequenceA1of unselected bits is random relative toA0, i.e.,A0andA1are independent. A bounded Kolmogorov?Loveland place selection is a very general type of recursive selection rule which may be interpreted as the sequence of oracle queries of a time-bounded Turing machine, so the methods used may be applicable to other questions involving random oracles.

Relative to a random oracle, NP is not small

The resource-bounded measure (J. Lutz, 1992) is an extension of classical measure theory which provides a probabilistic means of describing the relative sizes of complexity classes. Lutz proposed the hypothesis that NP does not have measure zero in the class E/sub 2/=DTIME(2/sup polynomial/), meaning loosely that NP contains a non-negligible subset of exponential time. This hypothesis implies a strong separation of P from NP and is supported by a growing body of plausible consequences which are not known to follow from the weaker assertion P/spl ne/NP. It is shown that relative to a random oracle, NP does not have measure zero in E/sub 2/, improving the result of Bennett and Gill (1981) that P/spl ne/NP relative to a random oracle. Several new techniques are introduced; in particular the proof exploits the independence properties of algorithmically random sequences, and a strong independence result is shown: if A is an algorithmically random sequence and a subsequence A/sub 0/ is chosen by means of a bounded Kolmogorov-Loveland place selection, then the sequence A/sub 1/ of unselected bits is random relative to A/sub 0/, i.e. A/sub 0/ and A/sub 1/ are independent. A bounded Kolmogorov-Loveland place selection is a very general type of recursive selection rule which may be interpreted as the sequence of oracle queries of a time-bounded Turing machine, so the methods used may be applicable to other questions involving random oracles.<<ETX>>

Self-assembly of the Discrete Sierpinski Carpet and Related Fractals

It is well known that the discrete Sierpinski triangle can be defined as the nonzero residues modulo 2 of Pascals triangle, and that from this definition one can construct a tileset with which the discrete Sierpinski triangle self-assembles in Winfrees tile assembly model. In this paper we introduce an infinite class of discrete self-similar fractals (a class that includes both the Sierpinski triangle and the Sierpinski carpet) that are defined by the residues modulo a prime p of the entries in a two-dimensional matrix obtained from a simple recursive equation. We prove that every fractal in this class self-assembles and that there is a uniform procedure that generates the corresponding tilesets. As a special case we show that the discrete Sierpinski carpet self-assembles using a set of 30 tiles.

Self-Assembling Rulers for Approximating Generalized Sierpinski Carpets

Discrete self-similar fractals have been used as test cases for self-assembly, both in the laboratory and in mathematical models, ever since Winfree exhibited a tile assembly system in which the Sierpinski triangle self-assembles. For strict self-assembly, where tiles are not allowed to be placed outside the target structure, it is an open question whether any self-similar fractal can self-assemble. This has motivated the development of techniques to approximate fractals with strict self-assembly. Ideally, such an approximation would produce a structure with the same fractal dimension as the intended fractal, but with specially labeled tiles at positions corresponding to points in the fractal. We show that the Sierpinski carpet, along with an infinite class of related fractals, can approximately self-assemble in this manner. Our construction takes a set of parameters specifying a target fractal and creates a tile assembly system in which the fractal approximately self-assembles. This construction introduces rulers and readers to control the self-assembly of a fractal structure without distorting it. To verify the fractal dimension of the resulting assemblies, we prove a result on the dimension of sets embedded into discrete fractals. We also give a conjecture on the limitations of approximating self-similar fractals.

Resource-Bounded Randomness and Compressibility with Respect to Nonuniform Measures

Most research on resource-bounded measure and randomness has focused on the uniform probability density, or Lebesgue measure, on {0,1}∞; the study of resource-bounded measure theory with respect to a nonuniform underlying measure was recently initiated by Breutzmann and Lutz [1]. In this paper we prove a series of fundamental results on the role of nonuniform measures in resource-bounded measure theory. These results provide new tools for analyzing and constructing martingales and, in particular, offer new insight into the compressibility characterization of randomness given recently by Buhrman and Longpre [2]. We give several new characterizations of resource-bounded randomness with respect to an underlying measure μ: the first identifies those martingales whose rate of success is asymptotically optimal on the given sequence; the second identifies martingales which induce a maximal compression of the sequence; the third is a (nontrivial) extension of the compressibility characterization to the nonuniform case. In addition we prove several technical results of independent interest, including an extension to resource-bounded measure of the classical theorem of Kakutani on the equivalence of product measures; this answers an open question in [1].

Self-assembling rulers for approximating generalized sierpinski carpets

Discrete self-similar fractals have been studied as test cases for self-assembly ever since Winfree exhibited a tile assembly system in which the Sierpinski triangle self-assembles. For strict self-assembly, where tiles are not allowed to be placed outside the target structure, it is an open question whether any self-similar fractal can self-assemble. This has motivated the development of techniques to approximate fractals with strict self-assembly. Ideally, such an approximation would produce a structure with the same fractal dimension as the intended fractal and with specially labeled tiles at positions corresponding to points in the fractal. We show that the Sierpinski carpet, along with an infinite class of related fractals, can approximately self-assemble in this manner. Our construction takes a set of parameters specifying a target fractal and creates a tile assembly system in which the fractal approximately selfassembles. This construction introduces rulers and readers to control the self-assembly of a fractal structure without distorting it. To verify the fractal dimension of the resulting assemblies, we prove a result on the dimension of sets embedded into discrete fractals. We also give a conjecture on the limitations of approximating self-similar fractals.

Almost free concurrency! (using GOF patterns)

We present a framework that provides concurrency-enhanced versions of the GOF object-oriented design patterns. The main benefit of our work is that if programmers improve program modularity by applying standard GOF design patterns while using the reusable pattern implementations from our framework, they receive implicit concurrency for free.

An improved zero-one law for algorithmically random sequences

Abstract Results on random oracles typically involve showing that a class {ifX:P(X)} has Lebesgue measure one, i.e., that some property P ( X ) holds for “almost every X ”. A potentially more informative approach is to show that P ( X ) is true for every X in some explicitly defined class of random sequences or languages. In this note we consider the algorithmically random sequences originally defined by Martin-Lof and their generalizations, the n -random sequences. Our result is an effective form of the classical zero-one law: for each n ≥ 1, if a class {ifX:P(X)} is closed under finite variation and has arithmetical complexity σ 0 n +1 or Π 0 n +1 (roughly, the property P can be expressed with n +1 alternations of quantifiers), then either P holds for every n -random sequence or else holds for none of them. This result has been used by Book and Mayordomo to give new characterizations of complexity classes of the form ALMOST-oR, the languages which can be ≤ oR -reduced to almost every oracle, where oR is a reducibility.