Abstract not found

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. In this paper we will try to understand how oracles and advice functions, which are mathematical abstractions in the theory of computability and complexity, can be seen as physical measurements in Classical Physics. First, we consider how physical measurements are a natural external source of information to an algorithmic computation. We argue that oracles and advice functions can help us to understand how the structure of space and time has information content that can be processed by Turing machines (after Cooper and Odifreddi [10] and Copeland and Proudfoot [11, 12]). We show that non-uniform complexity is an adequate framework for classifying feasible computations by Turing machines interacting with an oracle in Nature. By classifying the information content of such an oracle using Kolmogorov complexity, we obtain a hierarchical structure for advice classes. 1

Abstract. We argue that there is currently no satisfactory general framework for comparing the extensional computational power of arbitrary computational models operating over arbitrary domains. We propose a conceptual framework for comparison, by linking computational models to hypothetical physical devices. Accordingly, we deduce a mathematical notion of relative computational power, allowing the comparison of arbitrary models over arbitrary domains. In addition, we claim that the method commonly used in the literature for “strictly more powerful” is problematic, as it allows for a model to be more powerful than itself. On the positive side, we prove that Turing machines and the recursive functions are “complete ” models, in the sense that they are not susceptible to this anomaly, justifying the standard means of showing that a model is “hypercomputational.” 1

We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations. A representation reduces to another if its representing map factors through the other representation. Reductions form a pre-order on representations. A spectrum is a class of representations divided by the equivalence relation induced by reductions. We establish some basic properties of spectra, such as, non-triviality. Equivalent representations represent the same set of functions on the represented space. Within a class of representations, a representation is universal if all representations in the class reduce to it. We show that notions of admissibility, considered both for domains and within Weihrauch’s TTE, are universality concepts in the appropriate spectra. Viewing TTE representations as domain representations, the reduction notion here is a natural generalisation of the one from TTE. To illustrate the framework, we consider some domain representations of real numbers and show that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. On the other hand, however, we show that a substructure of the interval domain more suitable for efficient computation of operations is equivalent to the usual interval domain with respect to reducibility. 1.

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 not found

Is there a physical constant with the value of the halting function? An answer to this question, as holds true for other discussions of hypercomputation, assumes a fixed interpretation of nature by mathematical entities. We discuss the subjectiveness of viewing the mathematical properties of nature, and the possibility of comparing computational models having different views of the world. For that purpose, we propose a conceptual framework for power comparison, by linking computational models to hypothetical physical devices. Accordingly, we deduce a mathematical notion of relative computational power, allowing for the comparison of arbitrary models over arbitrary domains. In addition, we claim that the method commonly used in the literature for “strictly more powerful ” is problematic, as it allows for a model to be more powerful than itself. On the positive side, we prove that Turing machines and the recursive functions are “complete ” models, in the sense that they are not susceptible to this anomaly, justifying the standard means of showing that a model is more powerful than Turing machines.

Abstract. We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0−1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting system?

Abstract not found