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19
Computations via experiments with kinematic systems
, 2004
"... Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be des ..."
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Cited by 15 (4 self)
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Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be designed to operate under (a) Newtonian mechanics or (b) Relativistic mechanics. The theorem proves that valid models of mechanical systems can compute all possible functions on discrete data. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on this undesirable situation and argue that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems. We conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms. We set these theorems and ideas in the context of the literature on the general problem “Is physical behaviour computable? ” and state some open problems.
NONCOMPUTABLE CONDITIONAL DISTRIBUTIONS
"... Abstract. We study the computability of conditional probability, a fundamental notion in probability theory and Bayesian statistics. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In more general settings, conditional probability is defined axiomaticall ..."
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Abstract. We study the computability of conditional probability, a fundamental notion in probability theory and Bayesian statistics. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In more general settings, conditional probability is defined axiomatically, and the search for more constructive definitions is the subject of a rich literature in probability theory and statistics. However, we show that in general one cannot compute conditional probabilities. Specifically, we construct a pair of computable random variables (X, Y) in the unit interval whose conditional distribution P[YX] encodes the halting problem. Nevertheless, probabilistic inference has proven remarkably successful in practice, even in infinitedimensional continuous settings. We prove several results giving general conditions under which conditional distributions are computable. In the discrete or dominated setting, under suitable computability hypotheses, conditional distributions are computable. Likewise, conditioning is a computable operation in the presence of certain additional structure, such as independent absolutely continuous noise.
Computable Invariance
 THEORETICAL COMPUTER SCIENCE
, 1996
"... In Computable Analysis each computable function is continuous and computably invariant, i.e. it maps computable points to computable points. On the other hand, discontinuity is a sufficient condition for noncomputability, but a discontinuous function might still be computably invariant. We investig ..."
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Cited by 8 (3 self)
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In Computable Analysis each computable function is continuous and computably invariant, i.e. it maps computable points to computable points. On the other hand, discontinuity is a sufficient condition for noncomputability, but a discontinuous function might still be computably invariant. We investigate algebraic conditions which guarantee that a discontinuous function is sufficiently discontinuous and sufficiently effective such that it is not computably invariant. Our main theorem generalizes the First Main Theorem ouf PourEl & Richards (cf. [20]). We apply our theorem to prove that several setvalued operators are not computably invariant.
ON THE COMPUTABILITY OF CONDITIONAL PROBABILITY
"... Abstract. We study the problem of computing conditional probabilities, a fundamental operation in statistics and machine learning. In the elementary discrete setting, conditional probability is defined axiomatically and the search for more constructive definitions is the subject of a rich literature ..."
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Cited by 5 (4 self)
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Abstract. We study the problem of computing conditional probabilities, a fundamental operation in statistics and machine learning. In the elementary discrete setting, conditional probability is defined axiomatically and the search for more constructive definitions is the subject of a rich literature in probability theory and statistics. In the discrete or dominated setting, under suitable computability hypotheses, conditional probabilities are computable. However, we show that in general one cannot compute conditional probabilities. We do this by constructing a pair of computable random variables in the unit interval whose conditional distribution encodes the halting problem at almost every point. We show that this result is tight, in the sense that given an oracle for the halting problem, one can compute this conditional distribution. On the other hand, we show that conditioning in abstract settings is computable in the presence of certain additional structure, such as independent absolutely continuous noise. 1.
Baire Category and Nowhere Differentiability for Feasible Real Functions ⋆
"... Abstract. A notion of resourcebounded Baire category is developed for the class PC[0,1] of all polynomialtime computable realvalued functions on the unit interval. The meager subsets of PC[0,1] are characterized in terms of resourcebounded BanachMazur games. This characterization is used to pro ..."
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Cited by 3 (0 self)
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Abstract. A notion of resourcebounded Baire category is developed for the class PC[0,1] of all polynomialtime computable realvalued functions on the unit interval. The meager subsets of PC[0,1] are characterized in terms of resourcebounded BanachMazur games. This characterization is used to prove that, in the sense of Baire category, almost every function in PC[0,1] is nowhere differentiable. This is a complexitytheoretic extension of the analogous classical result that Banach proved for the class C[0, 1] in 1931. 1
Index Sets for Computable Differential Equations
 Mathematical Logic Quarterly
, 2004
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Computation over arbitrary models of time  A unified model of discrete, analog, quantum and hybrid computation
, 2001
"... This technical report is an updated record of a talk given by the author at the 1998 "Xmachines Day" hosted by Sheffield University Computer Science Department. Where appropriate the introductory section has been streamlined, but the underlying content is otherwise unchanged. ..."
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Cited by 2 (0 self)
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This technical report is an updated record of a talk given by the author at the 1998 "Xmachines Day" hosted by Sheffield University Computer Science Department. Where appropriate the introductory section has been streamlined, but the underlying content is otherwise unchanged.
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
DAVID PINES
"... All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. ..."
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All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. This volume was typeset using TEXtures on a Macintosh computer. Cameraready output from an