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20
Abstract versus concrete computation on metric partial algebras
 ACM Transactions on Computational Logic
, 2004
"... Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological manysorted algebras. In the theory of computation on topological algebras there is a considerable gap between socalled abstract and concrete models of computation. We pr ..."
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Cited by 28 (17 self)
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Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological manysorted algebras. In the theory of computation on topological algebras there is a considerable gap between socalled 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
On the algebraic complexity of some families of coloured Tutte polynomials
 Advances in Applied Mathematics
, 2004
"... Abstract. We investigate the coloured Tutte polynomial in Valiant’s algebraic framework of NPcompleteness. Generalising the well known relationship between the Tutte polynomial and the partition function from the Ising model, we establish a reduction from the permanent to the coloured Tutte polynom ..."
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Cited by 10 (4 self)
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Abstract. We investigate the coloured Tutte polynomial in Valiant’s algebraic framework of NPcompleteness. Generalising the well known relationship between the Tutte polynomial and the partition function from the Ising model, we establish a reduction from the permanent to the coloured Tutte polynomial, thus showing that its evaluation is a VNP−complete problem.
Reducibility of Domain Representations and CantorWeihrauch Domain Representations
, 2006
"... 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 preorder on representatio ..."
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Cited by 8 (4 self)
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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 preorder 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, nontriviality. 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.
Some Properties of the Effectively Uniform Topological Space
 Computability and Complexity in Analysis, LNCS 2064
, 2000
"... Our main objective is to investigate algorithmic features latent in some discontinuous functions in terms of uniform topological spaces. We develop the theory of computability structure and computable functions on a uniform topological space, and will apply the results to some real functions which a ..."
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Cited by 6 (3 self)
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Our main objective is to investigate algorithmic features latent in some discontinuous functions in terms of uniform topological spaces. We develop the theory of computability structure and computable functions on a uniform topological space, and will apply the results to some real functions which are discontinuous in the Euclidean space. Keywords: Effectively uniform topological space; Computability structure; Computable function; Binary tree structure, Cylinder function, Amalgamated space of reals, Function with jump points 1 Introduction We have recently been speculating on computability problems of some discontinuous real functions such as the Gauian function ([?]), Rademacher functions ([?]) and Walsh functions ([?]). These are functions either on the whole real line or on the interval [0; 1], jumping at integer points or at some binary rationals, and assuming computable values. Throughout this article, knowledge of the notion of computability of a real number or a sequence of...
Oracles and Advice as Measurements
"... 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 ..."
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Cited by 5 (4 self)
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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 nonuniform 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
RZ: A tool for bringing constructive and computable mathematics closer to programming practice
 CiE 2007: Computation and Logic in the Real World, volume 4497 of LNCS
, 2007
"... Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Obje ..."
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Cited by 5 (2 self)
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Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. 1
Abstract computability and algebraic specification
 ACM Transactions on Computational Logic
, 2002
"... Abstract computable functions are defined by abstract finite deterministic algorithms on manysorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable functions on any manysorted algebra; (ii) all functi ..."
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Cited by 5 (3 self)
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Abstract computable functions are defined by abstract finite deterministic algorithms on manysorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable functions on any manysorted algebra; (ii) all functions effectively approximable by abstract computable functions on any metric algebra. We show that there exist universal algebraic specifications for all the classically computable functions on the set R of real numbers. The algebraic specifications used are mainly bounded universal equations and conditional equations. We investigate the initial algebra semantics of these specifications, and derive situations where algebraic specifications precisely define the computable functions.
Computable total functions on metric algebras, universal algebraic specifications and dynamical systems
 THE JOURNAL OF LOGIC AND ALGEBRAIC PROGRAMMING
, 2005
"... ..."
M.: Uncomputability Below the Real Halting Problem
 CiE 2006. LNCS
, 2006
"... Abstract. Most of the existing work in real number computation theory concentrates on complexity issues rather than computability aspects. Though some natural problems like deciding membership in the Mandelbrot set or in the set of rational numbers are known to be undecidable in the BlumShubSmale ..."
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Cited by 2 (1 self)
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Abstract. Most of the existing work in real number computation theory concentrates on complexity issues rather than computability aspects. Though some natural problems like deciding membership in the Mandelbrot set or in the set of rational numbers are known to be undecidable in the BlumShubSmale (BSS) model of computation over the reals, there has not been much work on different degrees of undecidability. A typical question into this direction is the real version of Post’s classical problem: Are there some explicit undecidable problems below the real Halting Problem? In this paper we study three different topics related to such questions: First an extension of a positive answer to Post’s problem to the linear setting. We then analyze how additional real constants increase the power of a BSS machine. And finally a real variant of the classical word problem for groups is presented which we establish reducible to and from (that is, complete for) the BSS Halting problem. 1