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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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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.
Computability properties of the Gauian function
, 1999
"... We speculate on the Gauian function [x] as an example of a noncontinuous function which is nevertheless possessed of some properties of computability. An algorithm how to compute [x] for a single computable real number is first described, followed by a remark that [x] does not necessarily preserve s ..."
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We speculate on the Gauian function [x] as an example of a noncontinuous function which is nevertheless possessed of some properties of computability. An algorithm how to compute [x] for a single computable real number is first described, followed by a remark that [x] does not necessarily preserve sequential computability. Second, [x] is studied in the light of the notion of upper semicomputability. Then two Fr'echet spaces, R Z and L 1 loc (R) , in which the Gauian function is computable, will be taken up. 1 Introduction The Gauian function [x] is a real function which assigns to x the greatest integer not exceeding x. It is a good example of a function which is not continuous but is easy to understand. In the naive sense of computability, that is, in the sense of Chapter 0 of [7], it is not computable for the obvious reason that it is not continuous. On the other hand, in the sense of BlumShubSmale, it is a simplest example of a computable function (Example 5 in [1]). The r...
Computability of discontinuous functions Functional space approach  A gerenal report at Dagstuhl Seminar
"... Contents 1 Our standpoint We are to discuss how to view notions of computability for discontinuous functions. We confine ourselves to realvalued functions from some spaces. Our standpoint in studying computability problems in mathematics is doing mathematics. That is, we would like to talk abou ..."
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Contents 1 Our standpoint We are to discuss how to view notions of computability for discontinuous functions. We confine ourselves to realvalued functions from some spaces. Our standpoint in studying computability problems in mathematics is doing mathematics. That is, we would like to talk about computable functions and other mathematical objects just as one talks about continuous functions, integrable functions, etc. In any naive notion of computability of a function (on a compact set), uniform continuity is inherent. On the other hand, one often approximates discontinuous functions, for instance in numerical computations and drawing graphs. Very often such approximations are successful and satisfactory. It is therefore meaningful and important to speculate on computability of discontinuous functions. According to our standpoint, it is a mathematical investigation to formulate computability o