## Formalising exact arithmetic in type theory (2005)

Venue: | New Computational Paradigms: First Conference on Computability in Europe, CiE 2005 |

Citations: | 3 - 0 self |

### BibTeX

@INPROCEEDINGS{Niqui05formalisingexact,

author = {Milad Niqui},

title = {Formalising exact arithmetic in type theory},

booktitle = {New Computational Paradigms: First Conference on Computability in Europe, CiE 2005},

year = {2005},

pages = {368--377},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. In this work we focus on a formalisation of the algorithms of lazy exact arithmetic à la Potts and Edalat [1]. We choose the constructive type theory as our formal verification tool. We discuss an extension of the constructive type theory with coinductive types that enables one to formalise and reason about the infinite objects. We show examples of how infinite objects such as streams and expression trees can be formalised as coinductive types. We study the type theoretic notion of productivity which ensures the infiniteness of the outcome of the algorithms on infinite objects. Syntactical methods are not always strong enough to ensure the productivity. However, if some information about the complexity of a function is provided, one may be able to show the productivity of that function. In the case of the normalisation algorithm we show that such information can be obtained from the choice of real number representation that is used to represent the input and the output. 1

### Citations

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(Show Context)
Citation Context ...verification of the NAlg algorithm. 2 Coinductive Types Coinductive types were added to the type theory in order to make it capable of dealing with infinite objects. This extension was done by Hagino =-=[6]-=- using the categorical semantics. The idea is to consider an ambient category for the type theory, and interpret the final coalgebras of this category as coinductive types. The standard way to conside... |

39 |
Categorical Logic and Type Theory, Volume 141
- Jacobs
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(Show Context)
Citation Context ...d interpret the final coalgebras of this category as coinductive types. The standard way to consider a categorical model for type theory is to interpret types as objects and typing rules as morphisms =-=[7]-=-. But one can also base the presentation of coinductive types on Martin-Löf’s constructive type theory. That means that in order to present the rules for a type, one should present the formation, intr... |

20 |
Inductively defined types (preliminary version
- Coquand, Paulin
- 1990
(Show Context)
Citation Context ...hat correspond to strictly positive type operators. Strictly positive type operators and their functorial extensions are used in defining inductive types in CIC; their definition can be found e.g. in =-=[9]-=-. Below we shall present the rules of coinductive types. The method of construction will be given as a type constructor symbol ν, with a total decision procedure is-νF. This method of construction is ... |

17 | Un Calcul de Constructions Infinies et son application a la vérification de systèmes communicants
- Giménez
- 1996
(Show Context)
Citation Context ... order to formalise these algorithms we choose a type theory extended with coinductive types. More specifically we choose the Calculus of Inductive Constructions (CIC) extended with coinductive types =-=[4]-=-. There are two reasons for this choice: 1. CIC distinguishes between propositions (computationally irrelevant proofs) and sets (computational content of proofs); 2. CIC is implemented as the Coq proo... |

14 | Exact Real Computer Arithmetic
- Potts, Edalat
- 1997
(Show Context)
Citation Context ...d Information Sciences, Radboud University Nijmegen, The Netherlands milad@cs.ru.nl Abstract. In this work we focus on a formalisation of the algorithms of lazy exact arithmetic à la Potts and Edalat =-=[1]-=-. We choose the constructive type theory as our formal verification tool. We discuss an extension of the constructive type theory with coinductive types that enables one to formalise and reason about ... |

8 | Lazy computation with exact real numbers
- Edalat, Potts, et al.
- 1998
(Show Context)
Citation Context ...ormalisation. This is especially the case if one uses the approaches to the exact arithmetic whose formalisation require a complex type system, such as the lazy exact arithmetic à la Potts and Edalat =-=[1, 2]-=-. Formalising Potts and Edalat’s algorithms requires a type theory that is enriched with infinite objects. This is because the Normalisation Algorithm (NAlg) — which is the core of Potts and Edalat’s ... |

5 | Formalising Exact Arithmetic: Representations, Algorithms and Proofs - Niqui - 2004 |

2 |
Intuitionistic Type Theory. Biblioplois
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- 1984
(Show Context)
Citation Context ...of coinductive types on Martin-Löf’s constructive type theory. That means that in order to present the rules for a type, one should present the formation, introduction, elimination and equality rules =-=[8]-=-. Here we use this approach to define coinductive types for polynomial functors. Such coinductive types are all that is needed for formalising NAlg. In the Martin-Löf’s setting, the formation rule is ... |

1 |
http://coq.inria.fr/doc/main.html, [cited 31st
- INRIA
- 2004
(Show Context)
Citation Context ...of coinductive types on Martin-Löf’s constructive type theory. That means that in order to present the rules for a type, one should present the formation, introduction, elimination and equality rules =-=[8]-=-. Here we use this approach to define coinductive types for polynomial functors. Such coinductive types are all that is needed for formalising NAlg. In the Martin-Löf’s setting, the formation rule is ... |