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Up-to-date Interval Arithmetic: From Closed Intervals to Connected Sets of Real Numbers
[chapter]
2016
Lecture Notes in Computer Science
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To a great deal this can be seen as an extension of arithmetic for closed real intervals to open and halfopen real intervals. ...
Here interval arithmetic just deals with connected sets of real numbers. These can be closed, open, half-open, bounded or unbounded. ...
He gratefully acknowledges frequent e-mail exchange with John Gustafson on the contents of the paper. ...
doi:10.1007/978-3-319-32152-3_39
fatcat:7efbekt3h5dkhiwoomd2phn62m
Real number computability and domain theory
[chapter]
1993
Lecture Notes in Computer Science
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Domain theory is a useful tool to study higher order computability on real numbers. ...
An important result in this paper is the proof that every computable functional on real numbers is continuous w.r.t. the compact open topology on the function space. ...
In this representation a real number is defined as the limit of a computable sequence of rational intervals. ...
doi:10.1007/3-540-57182-5_33
fatcat:zvjqtdqh5vgpld2rrdpedx7wie
Real Number Computability and Domain Theory
1996
Information and Computation
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Domain theory is a useful tool to study higher order computability on real numbers. ...
An important result in this paper is the proof that every computable functional on real numbers is continuous w.r.t. the compact open topology on the function space. ] ...
In this representation a real number is defined as the limit of a computable sequence of rational intervals. ...
doi:10.1006/inco.1996.0046
fatcat:cb73dq2qfnacrmmlhetmw5rmou
Interval Representations
2004
TEMA
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The concept is often applied in interval mathematics and captures the essence of that theory; namely: Interval analysis is a language that designates computations with real numbers. ...
The idea of interval objects as representation of real objects is defined and its relation with some aspects of interval analysis is showed. ...
Introduction One of the ideas behind interval analysis is that it is a kind of language which designates real computations, in the sense that an interval [a, b] represents any real number r ∈ [a, b] 3 ...
doi:10.5540/tema.2004.05.02.0317
fatcat:nqk5w3nirjhjbekue5od5c63wa
An induction principle over real numbers
2016
Archive for Mathematical Logic
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We give a constructive proof of the open induction principle on real numbers, using bar induction and enumerative open sets. We comment the algorithmic content of this result. ...
Real numbers and enumerative open sets. The set R of real numbers is axiomatically defined as a Cauchy complete Archimedean ordered field. ...
We call open induction over real numbers an elementary lemma of real analysis which states that an open set of the closed interval [0, 1] satisfying an inductive property in fact covers entirely the space ...
doi:10.1007/s00153-016-0513-8
fatcat:7n33mbyqqbflpa2vappslmulka
Computability and the morphological complexity of some dynamics on continuous domains
2008
Theoretical Computer Science
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The partially ordered set of compact intervals provides a convenient embedding space for the analysis of some Dynamical Systems. ...
Crucial dynamical properties are transferred to it, while allowing an investigation of stability and chaoticity, in terms of computability, in particular in the presence of singularities. ...
(Preliminary or revised versions of Longo's papers are downloadable from: http://www.di.ens.fr/users/longo or Search: Google: Giuseppe Longo). ...
doi:10.1016/j.tcs.2008.01.048
fatcat:pbesijwghngjzioblranl242ty
A computable approach to measure and integration theory
2007
22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)
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This leads us to the interval versions of the main results in classical measure theory. ...
We introduce a computable framework for Lebesgue's measure and integration theory in the spirit of domain theory. ...
μ on the space if the μ-measure of the inverse image of each open dyadic rational interval is a computable real number. ...
doi:10.1109/lics.2007.5
dblp:conf/lics/Edalat07
fatcat:s6w4uc3hhvde7mnx3aihs2wxfq
A computable approach to measure and integration theory
2009
Information and Computation
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This leads us to the interval versions of the main results in classical measure theory. ...
We introduce a computable framework for Lebesgue's measure and integration theory in the spirit of domain theory. ...
μ on the space if the μ-measure of the inverse image of each open dyadic rational interval is a computable real number. ...
doi:10.1016/j.ic.2008.05.003
fatcat:poxmj243evb2xmndlzuog3o53i
Linear Models of Computation and Program Learning
[article]
2015
arXiv
pre-print
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We look at the recent advances in the "sampling the samplers" paradigm in higher-order probabilistic programming. ...
We consider two classes of computations which admit taking linear combinations of execution runs: probabilistic sampling and generalized animation. ...
The open sets of the upper topology on the reals is a particular representation of real numbers extended with ±∞, the same is true about the open sets of the lower topology on the reals. ...
arXiv:1512.04639v1
fatcat:b4oap7ko2bhvzmvr6b6lgamqbm
Characterizing Computable Analysis with Differential Equations
2008
Electronical Notes in Theoretical Computer Science
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The functions of Computable Analysis are defined by enhancing the capacities of normal Turing Machines to deal with real number inputs. ...
In a similar model, Shannon's General Purpose Analog Computer, Bournez et. al. 2007 [3] also characterize the functions of Computable Analysis. ...
For an open interval G of G, we consider its closureḠ, which is in the domain of g. The minimum, m, and the maximum, M , of g onḠ are computable. ...
doi:10.1016/j.entcs.2008.12.004
fatcat:bvmaynycpnetbi45zvb6fsapjq
Interval Domains and Computable Sequences: A Case Study of Domain Reductions
2012
Computer journal
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The interval domain as a model of approximations of real numbers is not unique, in fact, there are many variations of the interval domain. We study these variations with respect to domain reductions. ...
A (directed) complete partial order, abbreviated CPO, is a partial order, D = (D; , ⊥), such that ⊥ is the least element in D and any directed set A ⊆ D has a supremum, Downloaded from Definition 2.5. ...
It has been used for semantics of computable reals, interval arithmetic and constraint satisfaction over the reals. ...
doi:10.1093/comjnl/bxs121
fatcat:hx2g6bxfjrevvgpvrufrszikcq
Using Constraint Programming for Reconfiguration of Electrical Power Distribution Networks
[chapter]
2005
Lecture Notes in Computer Science
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We analyze the performance of our system in canonical distribution networks of up to 60 nodes. ...
A variety of approximation computational models have recently been proposed. We describe a constraint programming model for this problem, using the MOzArt system. ...
Future Work We implemented in CRE2 a network of around 600 nodes corresponding to six electric circuits of the power system for the city of Buenaventura in Colombia. ...
doi:10.1007/978-3-540-31845-3_22
fatcat:awrwh2bjkregfejwkktm6mli2a
Interval Interpolation By Newtons Divided Differences
2014
Journal of Mathematics and Computer Science
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In this paper, we present applied of interval algebra operation in interpolation, when the support points are intervals. We compute interpolation polynomial that coefficients are interval. ...
We compute interpolation polynomial by Newton's divided difference formula. ...
Conclusion We present applied of interval algebra operation in interpolation, when the support points are intervals. We compute interpolation polynomial with Newton`s divided differences. ...
doi:10.22436/jmcs.013.03.05
fatcat:mddsp5oilzh4jhq72kcxjj5fbi
Foundation of a computable solid modelling
2002
Theoretical Computer Science
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In practice, correctness of algorithms in computational geometry is usually proved using the unrealistic Real RAM machine model of computation, which allows comparison of real numbers, with the undesirable ...
This framework is equipped with a well deÿned and realistic notion of computability which re ects the observable properties of real solids. ...
Acknowledgements The ÿrst author has been supported by EPSRC and would like to thank Reinhold Heckmann for constructing a regular open set, on the real line, with a boundary of non-zero Lebesgue measure ...
doi:10.1016/s0304-3975(01)00091-3
fatcat:ixti3liujzg7hne4w4tzsza6pm
GiNaCRA: A C++ Library for Real Algebraic Computations
[chapter]
2011
Lecture Notes in Computer Science
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In contrast to other similar tools, our open source library aids exact, real algebraic computations based on an appropriate data type representing real zeros of polynomials. ...
It is based on the C++ library GiNaC, supporting the symbolic representation and manipulation of polynomials. ...
It supports different representations of real algebraic numbers (order, sign, and interval representation [14, p. 327] ). ...
doi:10.1007/978-3-642-20398-5_41
fatcat:fgtq4xf7jvalbppj3ggw4ozqki
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