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Localic completion of generalized metric spaces II: Powerlocales
, 2009
"... The work investigates the powerlocales (lower, upper, Vietoris) of localic completions of generalized metric spaces. The main result is that all three are localic completions of generalized metric powerspaces, on the Kuratowski finite powerset. This is a constructive, localic version of spatial resu ..."
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Cited by 11 (2 self)
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The work investigates the powerlocales (lower, upper, Vietoris) of localic completions of generalized metric spaces. The main result is that all three are localic completions of generalized metric powerspaces, on the Kuratowski finite powerset. This is a constructive, localic version of spatial results of Bonsangue et al. and of Edalat and Heckmann. As applications, a localic completion is always overt, and is compact iff its generalized metric space is totally bounded. The representation is used to discuss closed intervals of the reals, with the localic Heine–Borel Theorem as a consequence. The work is constructive in the toposvalid sense.
Constructing NonComputable Julia Sets
 Proc. of STOC 2007
"... While most polynomial Julia sets are computable, it has been recently shown [12] that there exist noncomputable Julia sets. The proof was nonconstructive, and indeed there were doubts as to whether specific examples of parameters with noncomputable Julia sets could be constructed. It was also unk ..."
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Cited by 4 (0 self)
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While most polynomial Julia sets are computable, it has been recently shown [12] that there exist noncomputable Julia sets. The proof was nonconstructive, and indeed there were doubts as to whether specific examples of parameters with noncomputable Julia sets could be constructed. It was also unknown whether the noncomputability proof can be extended to the filled Julia sets. In this paper we give an answer to both of these questions, which were the main open problems concerning the computability of polynomial Julia sets. We show how to construct a specific polynomial with a noncomputable Julia set. In fact, in the case of Julia sets of quadratic polynomials we give a precise characterization of Julia sets with computable parameters. Moreover, assuming a widely believed conjecture in Complex Dynamics, we give a polytime algorithm for computing a number c such that the Julia set J z 2 +cz is noncomputable. In contrast with these results, we show that the filled Julia set of a polynomial is always computable.
FOUNDATIONS OF A DISCRETE PHYSICS*
, 1989
"... Starting from the principles of finiteness, discreteness, finite computability and absolute nonuniqueness, we develop the ordering operator calculus, a strictly constructive mathematical system having the empirical properties required by quantum mechanical and special relativistic phenomena. We sho ..."
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Cited by 1 (1 self)
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Starting from the principles of finiteness, discreteness, finite computability and absolute nonuniqueness, we develop the ordering operator calculus, a strictly constructive mathematical system having the empirical properties required by quantum mechanical and special relativistic phenomena. We show how to construct discrete distance functions, and both rectangular and spherical coordinate systems (with a discrete version of ‘Y). The richest discrete space constructible without a preferred axis and preserving translational and rotational invariance is shown to be a discrete 3space with the usual symmetries. We introduce a local ordering parameter with local (proper) timelike properties and universal ordering parameters with global (cosmological) timelike properties. Constructed “attribute velocities ” connect ensembles with attributes that are invariant as the appropriate timelike parameter increases. For each such attribute, we show how to construct attribute velocities which must satisfy the “relativistic Doppler shift ” and the “relativistic velocity composition law,” as well as the Lorentz transformations. By construction, these velocities have finite maximum and minimum values. In the space of all attributes, the minimum of these maximum velocities will predominate in all multiple attribute computations, and hence can be identified as a fundamental limiting velocity. General commutation relations are constructed which under the physical interpretation are shown to reduce to the usual quantum mechanical commutation relations. 1.
On the calculating power of Laplace’s demon (Part I)
, 2006
"... We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a s ..."
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We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a start on our programme by proposing a general logical framework for describing physical theories, and analysing several possible formulations of a simple Newtonian theory from the point of view of determinism. Our emphasis throughout is on clarifying the precise physical and metaphysical assumptions that typically underlie a claim that some physical theory is ‘deterministic’. A sequel paper is planned, in which we shall apply similar methods to the analysis of other physical theories. Along the way, we discuss some possible repercussions of this kind of investigation for both physics and logic. 1
On the calculating power of Laplace’s demon
"... Abstract. We discuss some of the choices that arise when one tries to make the idea of physical determinism more precise. Broadly speaking, ‘ontological ’ notions of determinism are parameterized by one’s choice of mathematical ideology, whilst ‘epistemological ’ notions of determinism are parameter ..."
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Abstract. We discuss some of the choices that arise when one tries to make the idea of physical determinism more precise. Broadly speaking, ‘ontological ’ notions of determinism are parameterized by one’s choice of mathematical ideology, whilst ‘epistemological ’ notions of determinism are parameterized by the choice of an appropriate notion of computability. We present some simple examples to show that these choices can indeed make a difference to whether a given physical theory is ‘deterministic’ or not. Keywords: Laplace’s demon, physical determinism, philosophy of mathematics, notions of computability. 1
Received (Day Month Year)
"... Accepted (Day Month Year) Communicated by (xxxxxxxxxx) In this paper, we explain why, in our opinion, logic and constructive mathematics are playing – and should play – an important role in the design, understanding, and analysis of unconventional computation. ..."
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Accepted (Day Month Year) Communicated by (xxxxxxxxxx) In this paper, we explain why, in our opinion, logic and constructive mathematics are playing – and should play – an important role in the design, understanding, and analysis of unconventional computation.