## Guaranteed proofs using interval arithmetic (2005)

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Venue: | Proceedings of the 17th Symposium on Computer Arithmetic, Cape Cod |

Citations: | 22 - 11 self |

### BibTeX

@INPROCEEDINGS{Daumas05guaranteedproofs,

author = {Marc Daumas and Guillaume Melquiond and César Muñoz},

title = {Guaranteed proofs using interval arithmetic},

booktitle = {Proceedings of the 17th Symposium on Computer Arithmetic, Cape Cod},

year = {2005}

}

### Years of Citing Articles

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### Abstract

This paper presents a set of tools for mechanical reasoning of numerical bounds using interval arithmetic. The tools implement two techniques for reducing decorrelation: interval splitting and Taylor’s series expansions. Although the tools are designed for the proof assistant system PVS, expertise on PVS is not required. The ultimate goal of the tools is to provide guaranteed proofs of numerical properties with a minimal human-theorem prover interaction. 1

### Citations

550 | PVS: A Prototype Verification System
- Owre, Rushby, et al.
(Show Context)
Citation Context ...echnique is based on interval splitting. The second technique is based on Taylor’s series expansions. The tools are designed for the verification system PVS, 2 which is developed by SRI International =-=[16]-=-. However, minimal PVS expertise is required to use our tools as most of the technical burden of proving properties in a proof assistant system is hidden from the user. A C++ library generates proof o... |

519 |
Interval methods for systems of equations
- Neumaier
(Show Context)
Citation Context ... to provide guaranteed formal proofs of numerical properties with a minimum effort. Interval arithmetic has been used for decades as a standard tool for numerical analysis on engineering applications =-=[8,15]-=-. In interval arithmetic, operations are evaluated on range of numbers rather than on real numbers. Formally, an interval x = [a, b] is the set of real numbers between a and b, i.e., [a, b] = {x | a ≤... |

277 |
Applied Interval Analysis
- Jaulin, Kieffer, et al.
- 2001
(Show Context)
Citation Context ... to provide guaranteed formal proofs of numerical properties with a minimum effort. Interval arithmetic has been used for decades as a standard tool for numerical analysis on engineering applications =-=[8,15]-=-. In interval arithmetic, operations are evaluated on range of numbers rather than on real numbers. Formally, an interval x = [a, b] is the set of real numbers between a and b, i.e., [a, b] = {x | a ≤... |

262 | The art of computer programming, seminumerical algorithms - KNUTH - 1969 |

134 |
The Chebyshev Polynomials
- Rivlin
- 1974
(Show Context)
Citation Context ...ries expansion, e.g., the second order expansion yields e(φ) ∈ e(a) + (Φ − a)e ′ (a) + (Φ − a)2 e 2 ′′ (Φ). However, note that ˆr(φ) is a least square approximation of r(φ) on Chebyshev’s polynomials =-=[18]-=- and r(φ) is a relatively smooth function. Therefore, the first order expansion, along with interval splitting, is sufficient in this case to show the required accuracy. Listing 6 illustrates the PVS ... |

82 | How to write a proof
- Lamport
- 1993
(Show Context)
Citation Context ...1 Introduction Deadly and disastrous failures [4, 7, 12] confirm the shared belief that the traditional peer-review process is not sufficient to guarantee correctness of published proofs and software =-=[11]-=-. Despite this belief, mechanical theorem provers and proof assistants are not widely used in the applied mathematics community. Part of the problem is the lack of user friendly interfaces that result... |

81 |
The GNU Multiple Precision Arithmetic Library. TMG Datakonsult
- Granlund
- 1996
(Show Context)
Citation Context ...can efficiently explore many possibilities and generate the lemmas for a local optimal solution. • Publicly available C++ libraries, such as Boost [1] and GMP’s multiple precision rational arithmetic =-=[5]-=-, can be used. Although the C++ library checks that the reported intervals are sufficiently accurate compared to the one that are produced using exact rational arithmetic, the C++ library does not for... |

36 | Interval computations: Introduction, Uses and Resources
- Kearfott
- 1996
(Show Context)
Citation Context ..., respectively. Furthermore, the proposition X > x states that all values in X are strictly greater than x; similarly for X ≥ x, X < x, and X ≤ x. The four basic interval operations are defined as in =-=[9]-=-: x + y = [x + y, x + y], x − y = [x − y, x − y], x × y = [min{xy, xy, xy, xy}, max{xy, xy, xy, xy}], [ ] 1 1 x ÷ y = x × , , if yy > 0. y y We also define negative, absolute value, square, and power ... |

31 | Floating point verification in HOL light: the exponential function
- Harrison
- 1997
(Show Context)
Citation Context ...(phi) / r(phi) ## [|-136/1000000000,136/1000000000|] 6 Conclusion and Perspective The examples presented in Sections 3 and 4 could have been handled in HOL-light 6 using one of the tools presented in =-=[6]-=-. According to Sturm’s theorem [10, p. 434] that development is more efficient on these specific examples but it is limited to problems that can ultimately be approximated by polynomial functions. On ... |

27 |
Taylor models and other validated functional inclusion methods
- Makino, Berz
(Show Context)
Citation Context ...eatures: • Floating point arithmetic rather than rational arithmetic as developed in [2]. • Use of high speed multiple precision techniques. • Implementation of latest developments on Taylor’s models =-=[13]-=-, and mix Taylor’s models and floating point arithmetic [17]. The tools are currently being used to check numerical properties of aircraft navigation algorithms developed at the National Institute of ... |

20 |
Tactical conflict detection and resolution in a 3-D airspace
- Dowek, Geser, et al.
- 2001
(Show Context)
Citation Context ... is used to translate aircraft geodesic coordinates, as calculated by global positioning systems, to Cartesian coordinates used, for example, by geometric conflict detection and resolution algorithms =-=[3]-=-. For efficiency reasons, one may want to approximate the function r(φ) by polynomial ˆr(φ) = 4439091 + (φ 4 2 m − φ 2 )× ( 9023647 + (φ 4 2 m − φ 2 )× ( 13868737 + (φ 64 2 m − φ 2 )× ( 13233647 2048 ... |

16 |
Generating formally certified bounds on values and round-off errors
- Daumas, Melquiond
- 2004
(Show Context)
Citation Context ...most cases. Research is conducted to study the feasibility of enhancing the prototypes with some of the following features: • Floating point arithmetic rather than rational arithmetic as developed in =-=[2]-=-. • Use of high speed multiple precision techniques. • Implementation of latest developments on Taylor’s models [13], and mix Taylor’s models and floating point arithmetic [17]. The tools are currentl... |

15 | Invisible formal methods for embedded control systems
- Tiwari, Shankar, et al.
(Show Context)
Citation Context ...ocessed by PVS in batch mode and a summary of the status of the proofs is printed. User interaction with the theorem prover is minimized. This approach is usually referred as invisible formal methods =-=[21]-=-. The rest of this paper is organized as follows. Section 2 presents an overview of the PVS interval arithmetic library. The usage of the library is illustrated with the toy example x × (1 − x) ∈ [0, ... |

14 |
et al., “Ariane 5 flight 501 failure report by the inquiry board,” European Space Agency
- Lions
- 1996
(Show Context)
Citation Context ... is not required. The ultimate goal of the tools is to provide guaranteed proofs of numerical properties with a minimal human-theorem prover interaction. 1 Introduction Deadly and disastrous failures =-=[4, 7, 12]-=- confirm the shared belief that the traditional peer-review process is not sufficient to guarantee correctness of published proofs and software [11]. Despite this belief, mechanical theorem provers an... |

13 | Formal verification of conflict detection algorithms
- Muñoz, Carreño, et al.
(Show Context)
Citation Context ... functions are implemented by approximation series. A PVS library of approximations was originally developed by one of the authors for the verification of an algorithm for aircraft conflict detection =-=[14]-=-. It was completed and extended with logarithm, exponential and arc tangent functions by David Lester. The approximation library is part of the NASA Langley PVS libraries 3 . The basic idea is to prov... |

11 | Taylor models and floating-point arithmetic: proof that arithmetic operations are validated
- Revol, Makino, et al.
(Show Context)
Citation Context ...ithmetic as developed in [2]. • Use of high speed multiple precision techniques. • Implementation of latest developments on Taylor’s models [13], and mix Taylor’s models and floating point arithmetic =-=[17]-=-. The tools are currently being used to check numerical properties of aircraft navigation algorithms developed at the National Institute of Aerospace (NIA). 7 Acknowledgment Proofs of Sections 4 and 5... |

8 | Formal verification of divide and square root algorithms using series calculation
- Sawada
- 2002
(Show Context)
Citation Context ...st to real operations, interval operations are defined everywhere. The empty interval acts as an exceptional value in cases where the real function is undefined. Contrary to the approach described in =-=[19]-=-, we do not have to generate a new Taylor approximation for each subrange. By using an interval-based Taylor expansion, the same expression can be reused for all the subranges. We do not suffer from t... |

5 |
The boost interval arithmetic library
- Bronnimann, Melquiond
- 2003
(Show Context)
Citation Context ...ic is sub-distributive, i.e., x×(y +z) ⊆ x×y+x×z. In the general case, the inclusion is strict. This may have surprising effects, for instance x−x is, in general, different from the interval 0, e.g., =-=[0, 1]-=- − [0, 1] = [−1, 1]. This effect is also called decorrelation and it is due to the fact that interval identity is lost in interval arithmetic. Consider the function f(x) = x × (1 − x). A simple analys... |

3 |
We did nothing wrong
- Gage, McCormick
- 2004
(Show Context)
Citation Context ... is not required. The ultimate goal of the tools is to provide guaranteed proofs of numerical properties with a minimal human-theorem prover interaction. 1 Introduction Deadly and disastrous failures =-=[4, 7, 12]-=- confirm the shared belief that the traditional peer-review process is not sufficient to guarantee correctness of published proofs and software [11]. Despite this belief, mechanical theorem provers an... |