Some results in probability and theoretical computer science

As typical examples for nonlinear dynamical systems, the logistic maps mapping x to cx(1 - x) with x is in [0,1] and c is a constant in [0,4] have been extensively studied. Bhattacharya and Rao (1993) studied the case that c is a random variable rather than a constant. In this case, each of the logistic maps above defines a Markov Chain on [0,1]. In this dissertation, we give some sufficient conditions for the existence of an invariant probability on (0,1) and some sufficient conditions for the nonexistence of invariant probability measures on (0,1) as well. When there exists an invariant probability on (0,1), we study the problem of the uniqueness of invariant probability measure on (0,1). We give some sufficient conditions for the invariant probability measure to be unique. We also provide an example where c takes only two values such that there exist two distinct invariant probability distributions supported by the open interval (0,1). This settles a question raised by R. N. Bhattacharya. In this dissertation, we also study the resource bounded measure that was introduced by Jack Lutz in 1992. It is shown that under Jack Lutz\u27s Strong Hypothesis, for any integer k that is at least 2, there is a sequence of k languages that is sequentially complete for NP, but no nontrivial permutation of this sequence is sequentially complete for NP. We also prove a stronger version of Resource-Bounded Kolmogorov Zero-One Law. We prove that if a class X of languages is a tail set, and has outer-measure less than 1, then it is measurable and has resource-bounded measure 0

Sub-computable Boundedness Randomness

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen's theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness

Constructive Dimension and Turing Degrees

This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dim_H(S) and constructive packing dimension dim_P(S) is Turing equivalent to a sequence R with dim_H(R) <= (dim_H(S) / dim_P(S)) - epsilon, for arbitrary epsilon > 0. Furthermore, if dim_P(S) > 0, then dim_P(R) >= 1 - epsilon. The reduction thus serves as a *randomness extractor* that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dim_H(S) / dim_P(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dim_H(S) = dim_P(S)) such that dim_H(S) > 0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension.Comment: The version of this paper appearing in Theory of Computing Systems, 45(4):740-755, 2009, had an error in the proof of Theorem 2.4, due to insufficient care with the choice of delta. This version modifies that proof to fix the error

Is there a physically universal cellular automaton or Hamiltonian?

It is known that both quantum and classical cellular automata (CA) exist that are computationally universal in the sense that they can simulate, after appropriate initialization, any quantum or classical computation, respectively. Here we introduce a different notion of universality: a CA is called physically universal if every transformation on any finite region can be (approximately) implemented by the autonomous time evolution of the system after the complement of the region has been initialized in an appropriate way. We pose the question of whether physically universal CAs exist. Such CAs would provide a model of the world where the boundary between a physical system and its controller can be consistently shifted, in analogy to the Heisenberg cut for the quantum measurement problem. We propose to study the thermodynamic cost of computation and control within such a model because implementing a cyclic process on a microsystem may require a non-cyclic process for its controller, whereas implementing a cyclic process on system and controller may require the implementation of a non-cyclic process on a "meta"-controller, and so on. Physically universal CAs avoid this infinite hierarchy of controllers and the cost of implementing cycles on a subsystem can be described by mixing properties of the CA dynamics. We define a physical prior on the CA configurations by applying the dynamics to an initial state where half of the CA is in the maximum entropy state and half of it is in the all-zero state (thus reflecting the fact that life requires non-equilibrium states like the boundary between a hold and a cold reservoir). As opposed to Solomonoff's prior, our prior does not only account for the Kolmogorov complexity but also for the cost of isolating the system during the state preparation if the preparation process is not robust.Comment: 27 pages, 1 figur

Derandomization in Game-theoretic Probability

We give a general method for constructing a deterministic strategy of Reality from a randomized strategy in game-theoretic probability. The construction can be seen as derandomization in game-theoretic probability.Comment: 19 page