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Hierarchies Of Generalized Kolmogorov Complexities And Nonenumerable Universal Measures Computable In The Limit
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 2000
"... The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin measure. Here we introduce more general, nonenumerable but cumulatively enumerable m ..."
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Cited by 38 (20 self)
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The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the "true" information content of some (possibly in nite) bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. Among other things we show that there are objects computable in the limit yet more random than Chaitin's "number of wisdom" Omega, that any approximable measure of x is small for any x lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of x is small for any x lacking a short enumerating program. We briey mention consequences for universes sampled from such priors.
Algorithmic Theories Of Everything
, 2000
"... The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lac ..."
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Cited by 32 (15 self)
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The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lacking a short description, and study the spectrum of TOEs spanned by two Ps, one reflecting the most compact constructive descriptions, the other the fastest way of computing everything. The former derives from generalizations of traditional computability, Solomonoff’s algorithmic probability, Kolmogorov complexity, and objects more random than Chaitin’s Omega, the latter from Levin’s universal search and a natural resourceoriented postulate: the cumulative prior probability of all x incomputable within time t by this optimal algorithm should be 1/t. Between both Ps we find a universal cumulatively enumerable measure that dominates traditional enumerable measures; any such CEM must assign low probability to any universe lacking a short enumerating program. We derive Pspecific consequences for evolving observers, inductive reasoning, quantum physics, philosophy, and the expected duration of our universe.
REPRESENTATIONS OF THE REAL NUMBERS AND OF THE OPEN SUBSETS OF THE SET OF REAL NUMBERS
, 1987
"... In previous papers we have presented a unified Type 2 theory of computability and continuity and a theory of representations. In this paper the concepts developed so far are used for the foundation of a new kind of constructive analysis. Different standard representations of the real numbers are com ..."
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Cited by 9 (1 self)
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In previous papers we have presented a unified Type 2 theory of computability and continuity and a theory of representations. In this paper the concepts developed so far are used for the foundation of a new kind of constructive analysis. Different standard representations of the real numbers are compared. It turns out that the crucial differences are of topological nature and that most of the representations (e.g., the decimal representation) are not reasonable for topological reasons. In the second part some effective representations of the open subsets of the real numbers are introduced and compared.
Some notes on Fine computability
 JUCS
, 2002
"... Abstract: A metric defined by Fine induces a topology on the unit interval which is strictly stronger than the ordinary Euclidean topology and which has some interesting applications in Walsh analysis. We investigate computability properties of a corresponding Fine representation of the real numbers ..."
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Cited by 4 (0 self)
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Abstract: A metric defined by Fine induces a topology on the unit interval which is strictly stronger than the ordinary Euclidean topology and which has some interesting applications in Walsh analysis. We investigate computability properties of a corresponding Fine representation of the real numbers and we construct a structure which characterizes this representation. Moreover, we introduce a general class of Fine computable functions and we compare this class with the class of locally uniformly Fine computable functions defined by Mori. Both classes of functions include all ordinary computable functions and, additionally, some important functions which are discontinuous with respect to the usual Euclidean metric. Finally, we prove that the integration operator on the space of Fine continuous functions is lower semicomputable.
Inferability of Recursive RealValued Functions
 Moscow Math. Soc
, 1997
"... This paper presents a method of inductive inference of realvalued functions from given pairs of observed data of (x; h(x)), where h is a target function to be inferred. Each of such observed data inevitably involves some ranges of errors, and hence it is usually represented by a pair of rational nu ..."
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Cited by 1 (1 self)
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This paper presents a method of inductive inference of realvalued functions from given pairs of observed data of (x; h(x)), where h is a target function to be inferred. Each of such observed data inevitably involves some ranges of errors, and hence it is usually represented by a pair of rational numbers which show the approximate value and the error bound, respectively. On the other hand, a real number called a recursive real number can be represented by a pair of two sequences of rational numbers, which converges to the real number and converges to zero, respectively. These sequences show an approximate value of the real number and an error bound at each point. Such a real number can also be represented by a sequence of closed intervals with rational end points which converges to a singleton interval with the real number as both end points. In this paper, we propose a notion of recursive realvalued functions that can enjoy the merits of the both representations of the recursive real...
Computable padic Numbers
, 1999
"... : In the present work the notion of the computable (primitive recursive, polynomially time computable) padic number is introduced and studied. Basic properties of these numbers and the set of indices representing them are established and it is proved that the above defined fields are padically c ..."
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: In the present work the notion of the computable (primitive recursive, polynomially time computable) padic number is introduced and studied. Basic properties of these numbers and the set of indices representing them are established and it is proved that the above defined fields are padically closed. Using the notion of a notation system introduced by Y. Moschovakis an abstract characterization of the indices representing the field of computable padic numbers is established. Keywords: Computable numbers, computable (primitive recursive, polynomially time computable padic numbers, padically closed fields, notation systems. 1 Introduction The present work brings together two ideas, namely type two computability, and padic fields. We start with a brief review of the notion a computable number and of a padic field as a background. The basic idea for computable real numbers is contained in Turing's fundamental paper [20], where he introduced the notion of a computable (or...
doi:10.1093/comjnl/bxs121 Interval Domains and Computable Sequences: A Case Study of Domain Reductions
, 2012
"... The interval domain as a model of approximations of real numbers is not unique, in fact, there are many variations of the interval domain. We study these variations with respect to domain reductions. ..."
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The interval domain as a model of approximations of real numbers is not unique, in fact, there are many variations of the interval domain. We study these variations with respect to domain reductions.
Computability and analysis: the legacy of Alan Turing
, 2012
"... For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a par ..."
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For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a particular geometric