John V. Tucker

Constructive Volume Geometry

We present an algebraic framework, called Constructive Volume Geometryn (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.

Algebraic specifications of computable and semicomputable data types

Abstract An extensive survey is given of the properties of various specification mechanisms based on initial algebra semantics.

Initial and final algebra semantics for data type specifications, two characterization theorems

We prove that those data types which may be defined by conditional equation specifications and final algebra semantics are exactly the cosemicomputable data types-those data types which are effectively computable, but whose inequality relations are recursively enumerable. And we characterize the computable data types as those data types which may be specified by conditional equation specifications using both initial algebra semantics and final algebra semantics. Numerical bounds for the number of auxiliary functions and conditional equations required are included in both theorems.

Concrete models of computation for topological algebras

A concrete model of computation for a topological algebra is based on a representation of the algebra made from functions on the natural numbers. The functions computable in a concrete model are computable in the representation in the classical sense of the Chruch-Turing Thesis. Moreover, the functions turn out to be continuous in the topology of the algebra. In this paper we consider different concrete models for computing in topological algebras and prove their mutual equivalence in certain commonly occurring circumstances. For topological algebras, the concrete models we use are: effective representation by algebraic domains (Stoltenberg-Hansen and Tucker); effective representation by continuous domains (Edelat); effective representation by type two recursion on Baire space (Weihrauch). And for metric and normed algebras we use: effective metric spaces (Moschovakis) and computability structures (Pour-El and Richards). The result are evidence that computability theory for topological algebras is a stable theory independent of the specific models of computation, just as classical computability theory for discrete algebras is stable.

Equational specifications, complete term rewriting systems, and computable and semicomputable algebras

We classify the computable and semicomputable algebras in terms of finite equational initial algebra specifications and their properties as term term rewriting systems, such as completeness. Further results on properties of these specifications, such as on their size and orthogonality, are provided which show that our main results are the best possible.

On the Power of Algebraic Specifications

We study the expressive power of different algebraic specification methods. In contrast to (nonhierarchical) initial and terminal algebra specifications which correspond to semicomputable and cosemicomputable algebras, hierarchical specifications — as e.g. in the specification language CLEAR — allow to specify hyperarithmetical algebras and are characterized by them. For partial abstract types we prove that every computable partial algebra has an equational hidden enrichment specification and discuss the power of hierarchical partial algebras. Finally we give an example of the specification of a simple nondeterministic programming language.

Computational complexity with experiments as oracles

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.

Expressiveness and the completeness of Hoare's logic☆

Three theorems are proven which reconsider the completeness of Hoares logic for the partial correctness of while-programs equipped with a first-order assertion language. The results are about the expressiveness of the assertion language and the role of specifications in completeness concerns for the logic: (1) expressiveness is not a necessary condition on a structure for its Hoare logic to be complete, (2) complete number theory is the only extension of Peano Arithmetic which yields a logically complete Hoare logic and (3) a computable structure with enumeration is expressive if and only if its Hoare logic is complete.

Coupled map lattices as computational systems

The coupled map lattice (CML) as a mathematical model for a computer is considered. Using the theory of synchronous concurrent algorithms, it is shown that the CML is a valid new model for a parallel deterministic analog machine, but that, in principle, such a CML computer does not generate computations that cannot be reproduced by the standard mathematical models for computing on real numbers. The analysis is based on new general mathematical definitions of CMLs, and an axiomatic approach to determining which models of computation can be used to simulate CMLs.

Can excitable media be considered as computational systems

Abstract Actual excitable media, and discrete models of excitable media, may be used to process images. Discrete time, discrete space models of excitable media are shown to be examples of synchronous concurrent algorithms and so may be considered as formal computational systems.