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### Up-to-date Interval Arithmetic: From Closed Intervals to Connected Sets of Real Numbers [chapter]

Ulrich Kulisch
2016 Lecture Notes in Computer Science
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.  ...

### Real number computability and domain theory [chapter]

Pietro Gianantonio
1993 Lecture Notes in Computer Science
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.  ...

### Real Number Computability and Domain Theory

Pietro Di Gianantonio
1996 Information and Computation
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.  ...

### Interval Representations

R.H.N. Santiago, B.R.C Bedregal, B.M. Acióly
2004 TEMA
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  ...

### An induction principle over real numbers

Assia Mahboubi
2016 Archive for Mathematical Logic
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  ...

### Computability and the morphological complexity of some dynamics on continuous domains

Mathieu Hoyrup, Arda Kolçak, Giuseppe Longo
2008 Theoretical Computer Science
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).  ...

### A computable approach to measure and integration theory

Abbas Edalat
2007 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)
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.  ...

### A computable approach to measure and integration theory

Abbas Edalat
2009 Information and Computation
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.  ...

### Linear Models of Computation and Program Learning [article]

Michael Bukatin, Steve Matthews
2015 arXiv   pre-print
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.  ...

### Characterizing Computable Analysis with Differential Equations

Manuel L. Campagnolo, Kerry Ojakian
2008 Electronical Notes in Theoretical Computer Science
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.  ...

### Interval Domains and Computable Sequences: A Case Study of Domain Reductions

J. Blanck
2012 Computer journal
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.  ...

### Using Constraint Programming for Reconfiguration of Electrical Power Distribution Networks [chapter]

Juan Francisco Díaz, Gustavo Gutierrez, Carlos Alberto Olarte, Camilo Rueda
2005 Lecture Notes in Computer Science
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.  ...

### Interval Interpolation By Newtons Divided Differences

Ali Salimi Shamloo, Parisa Hajagharezalou
2014 Journal of Mathematics and Computer Science
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.  ...

### Foundation of a computable solid modelling

Abbas Edalat, André Lieutier
2002 Theoretical Computer Science
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  ...

### GiNaCRA: A C++ Library for Real Algebraic Computations [chapter]

Ulrich Loup, Erika Ábrahám
2011 Lecture Notes in Computer Science
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] ).  ...
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