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? ACM Reference Format: Bergstra, J. A. and Tucker, J. V. 2007. The rational numbers as an abstract data type

We discuss combining physical experiments with machine computations and introduce a form of analogue-digital (AD) Turing machine. We examine in detail a case study where an experimental procedure based on Newtonian kinematics is combined with a class of Turing machines. Three forms of AD machine are studied, in which physical parameters can be set exactly and approximately. Using non-uniform complexity theory, and some probability, we prove theorems that show that these machines can compute more than classical Turing machines.

Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological many-sorted algebras. In the theory of computation on topological algebras there is a considerable gap between so-called abstract and concrete models of computation. We prove theorems that bridge the gap in the case of metric algebras with partial operations.With an abstract model of computation on an algebra, the computations are invariant under isomorphisms and do not depend on any representation of the algebra. Examples of such models are the ' while' programming language and the BCSS model. With a concrete model of computation, the computations depend on the choice of a representation of the algebra and are not invariant under isomorphisms. Usually, the representations are made from the set N of natural numbers, and computability is reduced to classical computability on N. Examples of such models are computability via effective metric spaces, effective domain representations, and type two enumerability.The theory of abstract models is stable: there are many models of computation, and conditions under which they are equivalent are largely known. The theory of concrete models is not yet stable, though it seems to be converging: several interesting models are known to be equivalent over special types of topological algebra. We investigate the problem of comparing the two types of models and, hence, establishing a unified and stable theory of computation for topological algebras.First, we show that to compute functions on topological algebras using an abstract model, it is necessary that one must use algebras with partial operations and computable functions that are continuous and multivalued. This multivaluedness is needed even to compute singlevalued functions, and so abstract models must be nondeterministic even to compute * The research of the second author was supported by a grant from the Natural Sciences and Engineering Research Council (Canada) and by a Visiting Fellowship from the Engineering and Physical Sciences Research Council (U.K.) 1 2 deterministic problems. Then we choose the ' while'-array programming language as an abstract model for computing on any data type, and extend it with a nondeterministic assignment of "countable choice". This is the new WhileCC * model. Finally, we introduce the notion of approximable multivalued computation on metric algebras. As a concrete model, we choose effective metric spaces. Among a number of results we prove the following.For any metric algebra A with an effective representation, any function WhileCC * approximable over A is computable in the effective representation of the metric algebra A. Conversely, we show that, under certain reasonable conditions on the effective metric algebra A, any function that is effective is also WhileCC * approximable. We give an equivalence theorem, and examples of algebras where equivalence holds.

The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of equational specifications of data types cannot use field theory in applications to number systems based upon rational, real and complex numbers. We study a new axiomatic concept for number systems with division that uses only equations: a meadow is a commutative ring with a total inverse operator satisfying two equations which imply 0 −1 = 0. All fields and products of fields can be viewed as meadows. After reviewing alternate axioms for inverse, we start the development of a theory of meadows. We give a general representation theorem for meadows and find, as a corollary, that the conditional equational theory of meadows coincides with the conditional equational theory of zero totalized fields. We also prove representation results for meadows of finite characteristic.

We pose the following question: If a physical experiment were to be completely controlled by an algorithm, what effect would the algorithm have on the physical measurements made possible by the experiment?In a programme to study the nature of computation possible by physical systems, and by algorithms coupled with physical systems, we have begun to analyse (i) the algorithmic nature of experimental procedures, and (ii) the idea of using a physical experiment as an oracle to Turing Machines. To answer the question, we will extend our theory of experimental oracles in order to use Turing machines to model the experimental procedures that govern the conduct of physical experiments. First, we specify an experiment that measures mass via collisions in Newtonian Dynamics; we examine its properties in preparation for its use as an oracle. We start to classify the computational power of polynomial time Turing machines with this experimental oracle using non-uniform complexity classes. Second, we show that modelling an experimenter and experimental procedure algorithmically imposes a limit on what can be measured with equipment. Indeed, the theorems suggest a new form of uncertainty principle for our knowledge of physical quantities measured in simple physical experiments. We argue that the results established here are representative of a huge class of experiments.

Earlier, to explore the idea of combining physical experiments with algorithms, we introduced a new form of analogue-digital (AD) Turing machine. We examined in detail a case study where an experimental procedure, based on Newtonian kinematics, is used as an oracle with classes of Turing machines. The physical cost of oracle calls was counted and three forms of AD queries were studied, in which physical parameters can be set exactly and approximately. Here, in this sequel, we complete the classification of the computational power of these AD Turing machines and determine precisely what they can compute, using non-uniform complexity classes and probabilities.

We present an algebraic framework, called Constructive Volume Geometry (CVG), for modelling complex spatial objects using combinational operations. By utilising scalar fields as fundamental building blocks, CVG provides high-level algebraic representations of objects that are defined mathematically or built upon sampled or simulated datasets. It models amorphous phenomena as well as solid objects, and describes the interior as well as the exterior of objects. We also describe a hierarchical representation scheme for CVG, and a direct rendering method with a new approach for consistent sampling. The work has demonstrated the feasibility of combining a variety of graphics data types in a coherent modelling scheme.

Following a methodology we have proposed for analysing the nature of experimental computation, we prove that there is a three-dimensional Newtonian machine which given any point x ∈[0, 1] can generate an infinite sequence [ p n , q n ], for n =1, 2, …, of rational number interval approximations, that converges to x as n →∞. The machine is a system for scattering and collecting particles. The theorem implies that every point x∈[0, 1] is computable by a simple Newtonian kinematic system that is bounded in space and mass and for which the calculation of the nth approximation of x takes place in O ( n ) time with O ( n ) energy. We describe variants of the scatter machine which explain why our machine is non-deterministic.