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Guaranteed Proofs Using Interval Arithmetic

*
17th IEEE Symposium on Computer Arithmetic (ARITH'05)
*

This paper presents a set of tools for mechanical reasoning of numerical bounds

doi:10.1109/arith.2005.25
dblp:conf/arith/DaumasMM05
fatcat:obkunjnxhfgo7ihyuspmoluuvq
*using**interval**arithmetic*. ... This paper presents a set of tools for mechanical reasoning of numerical bounds*using**interval**arithmetic*. ... Acknowledgment*Proofs*of Sections 4 and 5 were checked on high performance clusters set up and maintained by the Reso research project of the LIP computer science laboratory. ...##
###
Enclosing Solutions of Linear Equations

1998
*
SIAM Journal on Numerical Analysis
*

*using*Brouwer's fixed-point theorem. ... It is shown that Rump's method for enclosing solutions of linear equations can be reformulated in an

*interval*-free form and that the underlying inclusion result can be proved by elementary means without ... When

*using*the

*interval*

*arithmetic*operations, downwardly and upwardly oriented rounding must be

*used*to

*guarantee*that Y ⊂ Int(X) holds; then Y is a verified enclosure of the solution x * . ...

##
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Verified Real Number Calculations: A Library for Interval Arithmetic
[article]

2007
*
arXiv
*
pre-print

In order to reduce the dependency effect of

arXiv:0708.3721v1
fatcat:5a6672hxfzhlrnavvchbzlc7aq
*interval**arithmetic*, we integrate two techniques:*interval*splitting and taylor series expansions. ... Our ultimate goal is to provide*guaranteed**proofs*of numerical properties with minimal human theorem-prover interaction. ... It*guarantees*that evaluations of an expression*using**interval**arithmetic*bound its exact real value. ...##
###
Verified Real Number Calculations: A Library for Interval Arithmetic

2009
*
IEEE transactions on computers
*

In order to reduce the dependency effect of

doi:10.1109/tc.2008.213
fatcat:4aadux52sncpjan3yjo7fkauja
*interval**arithmetic*, we integrate two techniques:*interval*splitting and taylor series expansions. ... Our ultimate goal is to provide*guaranteed**proofs*of numerical properties with minimal human theorem-prover interaction. ... It*guarantees*that evaluations of an expression*using**interval**arithmetic*bound its exact real value. ...##
###
Real Number Calculations and Theorem Proving
[chapter]

2008
*
Lecture Notes in Computer Science
*

Then, based on these bounds, we develop a rational

doi:10.1007/978-3-540-71067-7_19
fatcat:lccfqatnmrg5fal3zvorrlktuu
*interval**arithmetic*where real number calculations can be performed in an algebraic setting. ... Wouldn't it be nice to be able to conveniently*use*ordinary real number expressions within*proof*assistants? In this paper we outline how this can be done within a theorem proving framework. ... Acknowledgment The authors would like to thank Marc Daumas for his early interest on this work, and his key suggestion to*use**interval**arithmetic*to automate our initial lower/upper bound technique. ...##
###
Real Number Calculations and Theorem Proving
[chapter]

2005
*
Lecture Notes in Computer Science
*

Then, based on these bounds, we develop a rational

doi:10.1007/11541868_13
fatcat:isfmzar76bh3rhuq6c6npolepm
*interval**arithmetic*where real number calculations can be performed in an algebraic setting. ... Wouldn't it be nice to be able to conveniently*use*ordinary real number expressions within*proof*assistants? In this paper we outline how this can be done within a theorem proving framework. ... Acknowledgment The authors would like to thank Marc Daumas for his early interest on this work, and his key suggestion to*use**interval**arithmetic*to automate our initial lower/upper bound technique. ...##
###
Interval methods that are guaranteed to underestimate (and the resulting new justification of Kaucher arithmetic)

1996
*
Reliable Computing
*

Proposition 1 gives a mathematical

doi:10.1007/bf02425913
fatcat:qqcwxasvvjhbrhynmb3b4ne5ti
*proof*of these estimates being the best; this*proof*is not so easy as the*proofs*of many algebraic results about*interval**arithmetic*because we have to consider all possible ... with traditional*interval**arithmetic*; operations between improper*intervals*; these operations are equivalent to operations of traditional*interval**arithmetic*; operations between proper and improper*intervals*...##
###
Interval methods for verifying structural optimality of circle packing configurations in the unit square

2007
*
Journal of Computational and Applied Mathematics
*

The required statements are verified with mathematical rigor

doi:10.1016/j.cam.2005.08.039
fatcat:ycxy53vjpjeuhowtcxr6ykkf4y
*using**interval**arithmetic*tools. ... Recently, a global optimization method based exclusively on*interval**arithmetic*calculations has been designed for this problem. ... In our studies the necessary numerical computations are performed*using**interval**arithmetic*, which enables*us*to compute reliable bounds on the results of floating point calculations, and thus to prove ...##
###
Guaranteed state estimation by zonotopes

2005
*
Automatica
*

*Interval*

*arithmetic*is

*used*to calculate a

*guaranteed*trajectory of the state process. Two examples have been provided for clarifying the algorithm. ... This paper presents a new approach to

*guaranteed*state estimation for non linear discrete-time systems with a bounded description of noise and parameters. ... Each sample time, a

*guaranteed*bound of the uncertain trajectory of the system is calculated

*using*

*interval*

*arithmetic*. ...

##
###
An overview of the upcoming IEEE P-1788 working group document: Standard for interval arithmetic

2013
*
2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS)
*

We provide a summary of the goals, underlying philosophy, work, decisions, product, and completion schedule of the IEEE P-1788 working group on

doi:10.1109/ifsa-nafips.2013.6608444
dblp:conf/ifsa/Kearfott13
fatcat:f4govcc2qzb3rj4kwb3zzj4she
*interval**arithmetic*. ... This leads to the ability to*use*floating point computations to provide mathematical*proofs*of appropriately posed assertions, to provide*guarantees*that a system subject to manufacturing uncertainties ... Such*proofs*are largely effected by computing mathematically rigorous upper and lower bounds on the range of a function over hyper-rectangles,*using*the fundamental theorem of*interval**arithmetic*: Theorem ...##
###
Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

2015
*
Journal of automated reasoning
*

It is based on a formalization of floating-point and

doi:10.1007/s10817-015-9350-4
fatcat:4ges4bruovhqbphmqstyfhqkz4
*interval**arithmetic*, associated with an on-the-fly computation of Taylor expansions. ... In fact, the inherent difficulty of computing such bounds often mandates a formal*proof*of them. ... Computations are performed*using**interval**arithmetic*. For multivariate polynomial expressions, the PVS*proof*system*uses*an approach based on Bernstein polynomials [23] . ...##
###
Interval computation as deduction in chip

1993
*
The Journal of Logic Programming
*

To reduce

doi:10.1016/0743-1066(93)90045-i
fatcat:wva2qfziurfsfddc2fglybk2zm
*arithmetic*primitives with*interval*domains, we*use*our*interval*narrowing technique as an implementation of the looking-ahead inference rule. ... Cleary proposed a relational form of*interval**arithmetic*that was incorporated in BNR Prolog in such a way that variables already bound can be bound again. ...*Interval**arithmetic*contributes methods*guaranteeing*correctness at the level of a simple*arithmetic*expression. ...##
###
An intervalal gorithm for solving systems of linear equations to prespecified accuracy

1985
*
Computing
*

We describe an

doi:10.1007/bf02259840
fatcat:5rzcjs7vg5clvciywd354ow26m
*interval**arithmetic*algorithm for solving a special class of simultaneous linear equations. ... The algorithm*uses*fixed point*arithmetic*, and has two properties which distinguish it from earlier algorithms: given the absolute accuracy a desired, the algorithm*uses*only as much precision as needed ... Kriickeberg's students who implemented the fixed point*arithmetic*and assisted in programming the algorithm. ...##
###
Nonstandard methods and the Erdős-Turán conjecture
[chapter]

2007
*
The Strength of Nonstandard Analysis
*

Here we recall that a < t, d, w > cell is merely a collection of t

doi:10.1007/978-3-211-49905-4_9
fatcat:unkvmc3xcfas3e2fqekq2pv6ye
*intervals*in*arithmetic*progression on which A is nonempty.*Proof*. ... A complete*proof*of this theorem appears in [8] , but we will outline the*proof*here, as it provides the clearest illustration of how the*use*of the nonstandard model provides*us*with a new set of tools ...##
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Large Subsets of Euclidean Space Avoiding Infinite Arithmetic Progressions
[article]

2022
*
arXiv
*
pre-print

We prove that this result does not extend to infinite

arXiv:2205.04786v1
fatcat:64ihdtgs3nghxngoixsyu7n4ga
*arithmetic*progressions in the following sense: for each λ in [0,1), we construct a subset of ℝ that intersects every*interval*of unit length in a ... set of measure at least λ, but that does not contain any infinite*arithmetic*progression. ... The set does not contain any*arithmetic*progression with integer gap size.*Proof*. ...
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