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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.

Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the input-output behaviour of the network with system parameters from A is modelled by a function Φ: C[T, A] p ×A r → C[T, A] q (p, q> 0, r ≥ 0), where C[T, A] is the set of all continuous streams equipped with the compact-open topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C[T, A] using a contraction principle. We analyse a case study involving a mechanical system. Finally, we introduce a custom-made concrete computation theory over C[T, A] and show that if the modules are concretely computable then so is the function Φ. 1

Abstract. We give a quantifier elimination procedure for term algebras over suitably expanded finite first-order languages. Our expansion is purely functional. Our method works by substituting finitely many parametric test terms. This allows us to obtain in addition sample solutions for an outermost existential quantifier block. The existence of our method implies that the considered quantifier elimination problem and as well the corresponding decision problem are in the fourth Grzegorcyk complexity class. For prenex input formulas with a bounded number of quantifiers our quantifier elimination procedure is elementary recursive. The same applies to arbitrary input formulas in case the language has only constants and unary function symbols. As a corollary we get corresponding upper bounds for the decision problem for term algebras. 1

Mathematics can provide precise formulations of relatively vague concepts and problems from the real world, and bring out underlying structure common to diverse scientific areas. Sometimes very natural mathematical concepts lie neglected and not widely understood for many years, before their fundamental relevance is recognised and their explanatory power is fully exploited. The notion of definability in a structure is such a concept, and Turing’s [77] 1939 model of interactive computation provides a fruitful context in which to exercise the usefulness of definability as a powerful and widely applicable source of understanding. In this article we set out to relate this simple idea to one of the oldest and apparently least scientifically approachable of problems — that of realistically modelling how mental properties supervene on physical ones. Mathematics can provide precise formulations of relatively vague concepts and problems from the real world, and bring out underlying structure common to diverse scientific areas. Sometimes very natural mathematical concepts lie neglected and not widely understood for many years, before their fundamental relevance is recognised and their explanatory power is fully exploited. Previously we have argued that the notion of definability in a structure is such a concept, and pointed to Turing’s [77] 1939 model of interactive computation as providing a fruitful context in which to exercise the usefulness of definability as a powerful and widely applicable source of understanding. Below, we relate this simple idea to one of the oldest and apparently least scientifically approachable of problems — that of realistically modelling how mental properties supervene on physical ones. We will first briefly review the origins with René Descartes of mind-body dualism, and the problem of mental causation. We will then summarise the subsequent difficulties encountered, and their current persistence, and the more recent usefulness of the concept of supervenience in

We study computational properties of linear cellular automata on configurations that differ from spatially periodic ones in only finitely many places. It is shown that the degree structure of the orbits of cellular automata is the same on these configurations as on the space of finite configurations. We also