Computability of Probability Distributions and Distribution Functions

We define the computability of probability distributions on the real line as well as that of distribution functions. Mutual relationships between the computability notion of a probability distribution and that of the corresponding distribution function are discussed. It is carried out through attempts to effectivize some classical fundamental theorems concerning probability distributions. We then define the effective convergence of probability distributions as an effectivization of the classical vague convergence. For distribution functions, computability and effective convergence are naturally defined as real functions. A weaker effective convergence is also defined as an effectivization of pointwise convergence

Fine Computability of Probability Distribution Functions and Computability of Probability Distributions on the Real Line

　We continue our work in [9] on an effective relationship between the sequence of probability distributions and the corresponding sequence of probability distribution functions. In order to deal with discontinuous distribution functions, we define the notion of Fine topology on the whole real line, and show that, when a probability distribution is associated with a Fine continuous distribution function, the computability of the former and the sequential computability of the latter can be effectively mutually translatable under a certain condition. The effectivity of the translations is secured by the treatment of the sequences of the objects in concern. The equivalences of effective convergences will also be proved

Computable de Finetti measures

We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically rewritten as procedures that do not modify non-local state. Along the way, we prove that a distribution on the unit interval is computable if and only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor corrections. To appear in Annals of Pure and Applied Logic. Extended abstract appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23

Quantum Algorithm for Hilbert's Tenth Problem

We explore in the framework of Quantum Computation the notion of {\em Computability}, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle--that is, if certain hamiltonian and its ground state can be physically constructed according to the proposal--quantum computability would surpass classical computability as delimited by the Church-Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles

Computability of entropy and information in classical Hamiltonian systems

We consider the computability of entropy and information in classical Hamiltonian systems. We define the information part and total information capacity part of entropy in classical Hamiltonian systems using relative information under a computable discrete partition. Using a recursively enumerable nonrecursive set it is shown that even though the initial probability distribution, entropy, Hamiltonian and its partial derivatives are computable under a computable partition, the time evolution of its information capacity under the original partition can grow faster than any recursive function. This implies that even though the probability measure and information are conserved in classical Hamiltonian time evolution we might not actually compute the information with respect to the original computable partition

Complexity vs Energy: Theory of Computation and Theoretical Physics

This paper is a survey dedicated to the analogy between the notions of {\it complexity} in theoretical computer science and {\it energy} in physics. This analogy is not metaphorical: I describe three precise mathematical contexts, suggested recently, in which mathematics related to (un)computability is inspired by and to a degree reproduces formalisms of statistical physics and quantum field theory.Comment: 23 pages. Talk at the satellite conference to ECM 2012, "QQQ Algebra, Geometry, Information", Tallinn, July 9-12, 201