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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 2-dimensional 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.

this paper, we show how the use of quantum computing can speed up some computations related to interval and probabilistic uncertainty. We end the paper with speculations on whether (and how) "hypothetic" physical devices can compute NP-hard problems faster than in exponential time

Abstract. Let (α, β) ⊆ R denote the maximal interval of existence of solution for the initial-value problem { dx = f(t, x) dt x(t0) = x0, where E is an open subset of R m+1, f is continuous in E and (t0, x0) ∈ E. We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable f and (t0, x0) whose maximal interval of existence (α, β) is noncomputable. The fact that f may be taken to be analytic shows that this is not a lack of regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that (α, β) is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable. 1.

Up to Turing power, all computations are describable by suitable programs, which correspond to the prescription by finite means of some rational parameters of the system or some computable reals. ¿From Turing power up we have computations that are not describable by finite means: computation without a program. When we observe natural phenomena and endow them with computational significance, it is not the algorithm we are observing but the process. Some objects near us may be performing hypercomputation: we observe them, but we will never be able to simulate their behaviour on a computer. What is then the profit of such a theory of computation to Science? The theory of analog computation, where the internal states of a computer are continuous rather than discrete, has enjoyed a recent resurgence of interest. This stems partly from a wider program of exploring alternative approaches to computation, such as neural and quantum computation; partly as an abstraction

Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with super-Turing power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or super-Turing. 1

Consider the initial-value problem with computable parameters � dx = p(t, x) dt x(t0) = x0, where p: R n+1 → R n is a vector of polynomials and (t0, x0) ∈ R n+1. We show that the problem of determining whether the maximal interval of definition of this initial-value problem is bounded or not is in general undecidable. 1

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.

We present an informal overview of a number of approaches to differential equations which are popular in computer algebra. This includes symmetry and completion theory, local analysis, differential ideal and Galois theory, dynamical systems and numerical analysis. A large bibliography is provided. 1 Introduction Differential equations represent one of the largest fields within mathematics. Besides being an interesting subject of their own right one can hardly overestimate their importance for applications. They appear in natural and engineering sciences and increasingly often in economics and social sciences. Whenever a continuous process is modeled mathematically, chances are high that differential equations are used. Thus it is not surprising that differential equations also play an important role in computer algebra and most general purpose computer algebra systems provide some kind of solve command. Many casual users believe that designing and improving such procedures is a centra...

We overview different approaches to the study of hypercomputation and other investigations on the plausibility of the physical Church–Turing thesis. We propose five thesis to classify investigation in this area. Sly does it. Tiptoe catspaws. Slide and creep.