## Coinduction for Exact Real Number Computation (2007)

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Citations: | 4 - 3 self |

### BibTeX

@MISC{Berger07coinductionfor,

author = {Ulrich Berger and Tie Hou},

title = {Coinduction for Exact Real Number Computation},

year = {2007}

}

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### Abstract

This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the correctness of various corecursive exact real number algorithms. The basic framework is the classical theory of coinductive sets as greatest fixed points of monotone operators and hence is different from (though related to) the type theoretic approach by Ciaffaglione and Gianantonio. Key words: Exact real number computation, coinduction, corecursion, signed digit streams. 1

### Citations

298 | Universal coalgebra: a theory of systems
- Rutten
(Show Context)
Citation Context ...on Martin-Löf Type theory. In the literature on coinduction in general, coinduction is usually defined as the principle expressing that equality is the largest bisimulation on streams (see e.g. [10], =-=[17]-=-). This form of coinduction is too restricted to express equality of real numbers given as infinite streams, due to the –for computability reasons inevitable– redundancy of real number representations... |

228 | A tutorial on (co)algebras and (co)induction
- Jacobs, Rutten
- 1997
(Show Context)
Citation Context ...based on Martin-Löf Type theory. In the literature on coinduction in general, coinduction is usually defined as the principle expressing that equality is the largest bisimulation on streams (see e.g. =-=[10]-=-, [17]). This form of coinduction is too restricted to express equality of real numbers given as infinite streams, due to the –for computability reasons inevitable– redundancy of real number represent... |

162 |
Complexity theory of real functions
- Ko
- 1991
(Show Context)
Citation Context ...putability on infinite streams can be explained by means of ‘Oracle Turing machines’ (Turing [21]). More recent accounts of the computability and complexity of stream functions are studied by e.g. Ko =-=[12]-=- and Weihrauch [22]. All the stream functions defined in this paper will be defined either explicitly from previously defined functions or by the corecursion scheme discussed in Section 2.3. Hence the... |

152 |
Computable Analysis, An Introduction
- Weihrauch
- 2000
(Show Context)
Citation Context ...ite streams can be explained by means of ‘Oracle Turing machines’ (Turing [21]). More recent accounts of the computability and complexity of stream functions are studied by e.g. Ko [12] and Weihrauch =-=[22]-=-. All the stream functions defined in this paper will be defined either explicitly from previously defined functions or by the corecursion scheme discussed in Section 2.3. Hence they are clearly compu... |

123 |
Systems of Logic Based on Ordinals
- Turing
- 1939
(Show Context)
Citation Context ...ere are computable back-and-forth translations between it and the Cauchy representation. The concept of computability on infinite streams can be explained by means of ‘Oracle Turing machines’ (Turing =-=[21]-=-). More recent accounts of the computability and complexity of stream functions are studied by e.g. Ko [12] and Weihrauch [22]. All the stream functions defined in this paper will be defined either ex... |

42 | A new representation for exact real numbers
- Edalat, Potts
- 1997
(Show Context)
Citation Context ..., Cauchy sequences [18], continued fractions [9], golden-ratio based systems with binary digits, as well as continued fractions and their generalisation, linear fractional (or Möbius) transformations =-=[6, 7]-=-. Many algorithms have been proposed 1sfor real computations using these representations, however their correctness is rarely proved formally. Plume [16] developed algorithms for the basic arithmetic ... |

16 | R.: Computing with real numbers: I. The LFT approach to real number computation; II. A domain framework for computational geometry
- Edalat, Heckmann
- 2002
(Show Context)
Citation Context ..., Cauchy sequences [18], continued fractions [9], golden-ratio based systems with binary digits, as well as continued fractions and their generalisation, linear fractional (or Möbius) transformations =-=[6, 7]-=-. Many algorithms have been proposed 1sfor real computations using these representations, however their correctness is rarely proved formally. Plume [16] developed algorithms for the basic arithmetic ... |

15 | From set-theoretic coinduction to coalgebraic coinduction: some results, some problems. ENTCS
- Lenisa
- 1999
(Show Context)
Citation Context ... real numbers given as infinite streams, due to the –for computability reasons inevitable– redundancy of real number representations. A more general form of coinduction, studied for example by Lenisa =-=[13]-=-, is based on the fact that every monotone set operator has a greatest fixed point. In this paper, we follow the same approach: We define relations of the form “stream a represents real number r” as t... |

13 |
A term calculus for (co-)recursive definitions on streamlike data structures
- Buchholz
- 2005
(Show Context)
Citation Context ...ons of stream producing functions in this paper will be instances of the simple corecursion scheme (∗). More general schemes have been discussed, for example, by Telford and Turner [20], and Buchholz =-=[3]-=-. Important examples of recursive definitions of stream functions related to exact 4sreal number computation that do not fit into the recursion scheme (∗) can be found in [6, 7, 14]. 3 Coinductive rep... |

12 |
Affine functions and series with co-inductive real numbers
- Bertot
- 2007
(Show Context)
Citation Context ... the Cauchy sequence representation in the systems Nuprl and Minlog, respectively. Approaches to exact real arithmetic based on coinduction were undertaken by Ciaffaglione and Gianantonio [4], Bertot =-=[1, 2]-=- as well as Niqui [14], who formalised and implemented real numbers as a coinductive type of streams in the Coq proof assistant which is based on Martin-Löf Type theory. In the literature on coinducti... |

12 | Gianantonio, P.: A certified, corecursive implementation of exact real numbers
- Ciaffaglione, Di
- 2006
(Show Context)
Citation Context ...hms based on the Cauchy sequence representation in the systems Nuprl and Minlog, respectively. Approaches to exact real arithmetic based on coinduction were undertaken by Ciaffaglione and Gianantonio =-=[4]-=-, Bertot [1, 2] as well as Niqui [14], who formalised and implemented real numbers as a coinductive type of streams in the Coq proof assistant which is based on Martin-Löf Type theory. In the literatu... |

12 | Implementing constructive real analysis: a preliminary report - Chirimar, Howe - 1991 |

10 | A universal characterization of the closed euclidean interval (extended abstract - Escardo, Simpson - 2001 |

5 | Completing the rationals and metric spaces in LEGO - Jones - 1992 |

4 |
Realizability of monotone coinductive definitions and its application to program synthesis
- Tatsuta
- 1998
(Show Context)
Citation Context ... further work we intend to use coinduction not only to verify algorithms, but also to synthesise them from proofs using a realisability interpretation of coinductive definitions as studied by Tatsuta =-=[19]-=-. Work in a similar direction is currently being done by Bertot [2] who uses Coq’s proof search engine to construct correct real number algorithms. The expected advantages of this approach are at leas... |

3 | Coinduction in Coq
- Bertot
(Show Context)
Citation Context ... the Cauchy sequence representation in the systems Nuprl and Minlog, respectively. Approaches to exact real arithmetic based on coinduction were undertaken by Ciaffaglione and Gianantonio [4], Bertot =-=[1, 2]-=- as well as Niqui [14], who formalised and implemented real numbers as a coinductive type of streams in the Coq proof assistant which is based on Martin-Löf Type theory. In the literature on coinducti... |

3 | Streaming Representation-Changers
- Gibbons
- 2004
(Show Context)
Citation Context ...exact real arithmetic, such as integral base systems with positive and negative digits, base 2/3 with binary digits, nested sequences of rational intervals, Cauchy sequences [18], continued fractions =-=[9]-=-, golden-ratio based systems with binary digits, as well as continued fractions and their generalisation, linear fractional (or Möbius) transformations [6, 7]. Many algorithms have been proposed 1sfor... |

3 | Formalising exact arithmetic in type theory
- Niqui
- 2005
(Show Context)
Citation Context ...resentation in the systems Nuprl and Minlog, respectively. Approaches to exact real arithmetic based on coinduction were undertaken by Ciaffaglione and Gianantonio [4], Bertot [1, 2] as well as Niqui =-=[14]-=-, who formalised and implemented real numbers as a coinductive type of streams in the Coq proof assistant which is based on Martin-Löf Type theory. In the literature on coinduction in general, coinduc... |

3 | Ensuring the Productivity of Infinite Structures
- Telford, Turner
- 1997
(Show Context)
Citation Context ... recursive definitions of stream producing functions in this paper will be instances of the simple corecursion scheme (∗). More general schemes have been discussed, for example, by Telford and Turner =-=[20]-=-, and Buchholz [3]. Important examples of recursive definitions of stream functions related to exact 4sreal number computation that do not fit into the recursion scheme (∗) can be found in [6, 7, 14].... |

2 | Coinductive correctness of homographic and quadratic algorithms for exact real numbers - Niqui |

2 |
A Calculator for Exact Real Number Computation. 4th year project
- Plume
- 1998
(Show Context)
Citation Context ...inear fractional (or Möbius) transformations [6, 7]. Many algorithms have been proposed 1sfor real computations using these representations, however their correctness is rarely proved formally. Plume =-=[16]-=- developed algorithms for the basic arithmetic operations, transcendental functions and integration, for various representation, but gave only informal proofs of the correctness for some of the algori... |

2 |
Inverting monotone continuous functions in constructive analysis
- Schwichtenberg
- 2006
(Show Context)
Citation Context ... representations used for exact real arithmetic, such as integral base systems with positive and negative digits, base 2/3 with binary digits, nested sequences of rational intervals, Cauchy sequences =-=[18]-=-, continued fractions [9], golden-ratio based systems with binary digits, as well as continued fractions and their generalisation, linear fractional (or Möbius) transformations [6, 7]. Many algorithms... |