Hend Dawood
Senior Lecturer of Computational Mathematics
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt. (email)
Parametric Intervals: More Reliable or Foundationally Problematic?
in
- Cardinality
- Categoricity
- Classical interval arithmetic
- Computational Mathematics
- Computer Science
- Consistency
- Constrained Interval Arithmetic
- Dawood's InCLosure
- Dependency Predicate
- Functional Dependence
- Fuzzy Mathematics
- InCL
- InCLosure
- Interval Analysis
- Interval arithmetic
- Interval computations
- Interval dependency
- Interval Enclosures
- Interval Functions
- Interval Mathematics
- Mathematical Analysis
- Mathematical Logic
- Mathematical Software
- Mathematics
- Metamathematics
- Numerical Analysis
- Order theory
- Overestimation-Free Interval Arithmetic
- Parametric Interval Arithmetic
- Quantifiable Uncertainty
- Quantitative Knowledge
- Real Analysis
- Real Functions
- Reliability
- Reliable computing
- Rounding Error
- S-Semiring
- Scientific Knowledge
- Set-Valued Algebras
- Set-valued functions
- Skolemization
- Subdistributive Semiring
- Uncertainty
- Uncertainty Analysis
- Uncertainty Modeling
- Uncertainty Quantification
- Universal intervals
- Citation:
- Dawood, Hend, and Yasser Dawood. "Parametric Intervals: More Reliable or Foundationally Problematic?" Online Mathematics Journal 1, no. 3 (2019): 37-54.
Abstract:
Interval arithmetic has been proved to be very subtle, reliable, and most fundamental in addressing uncertainty and imprecision. However, the theory of classical interval arithmetic and all its alternates suffer from algebraic anomalies, and all have difficulties with interval dependency. A theory of interval arithmetic that seems promising is the theory of parametric intervals. The theory of parametric intervals is presented in the literature with the zealous claim that it provides a radical solution to the long-standing dependency problem in the classical interval theory, along with the claim that parametric interval arithmetic, unlike Moore's classical interval arithmetic, has additive and multiplicative inverse elements, and satisfies the distributive law. So, does the theory of parametric intervals accomplish these very desirable objectives? Here it is argued that it does not.
Keywords: Interval mathematics, Classical interval arithmetic, Parametric interval arithmetic, Constrained interval arithmetic, Overestimation-free interval arithmetic, Interval dependency, Functional dependence, Dependency predicate, Interval enclosures, S-semiring, Uncertainty, Reliability.