An improved zero-one law for algorithmically random sequences

An Improved Zero-One Law for Algorithmically Random Sequences

Results on random oracles typically involve showing that a class {X : 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 Ma...

Independence Properties of Algorithmically Random Sequences

A bounded Kolmogorov-Loveland selection rule is an adaptive strategy for recursively selecting a subsequence of an infinite binary sequence; such a subsequence may be interpreted as the query sequence of a time-bounded Turing machine. In this paper we show that if A is an algorithmically random sequence, A0 is selected from A via a bounded Kolmogorov-Loveland selection rule, and A1 denotes the ...

Very Weak Zero One Law for Random Graphs with Order and Random Binary Functions Sh548

Let G<(n, p) denote the usual random graph G(n, p) on a totally ordered set of n vertices. (We naturally think of the vertex set as 1, . . . , n with the usual <). We will fix p = 12 for definiteness. Let L < denote the first order language with predicates equality (x = y), adjacency (x ∼ y) and less than (x < y). For any sentence A in L let f(n) = fA(n) denote the probability that the random G...

Very weak zero one law for random graphs with order and random binary functions

On the zero-one law for connectivity in one-dimensional geometric random graphs

We consider the geometric random graph where n points are distributed uniformly and independently on the unit interval [0, 1]. Using the method of first and second moments, we provide a simple proof of the “zero-one” law for the property of graph connectivity under the asymptotic regime created by having n become large and the transmission range scaled appropriately with n.