intervalleilla

Intervalleilla is a theoretical construct used in interval analysis and related fields to denote a minimal interval determined by a constraint. The term blends intervalle, the form of the word for interval in several languages, with the diminutive suffix -illa, signaling a specific or reduced instance of an interval.

Definition: Let A be a finite or infinite subset of the real numbers and let P be a predicate defined on closed intervals I = [a,b]. The intervalleilla of A under P is the unique smallest interval I = [a,b] such that A ⊆ I and P(I) holds, provided such an interval exists and P is monotone with respect to interval inclusion.

Properties: If P is monotone, the intervalleilla exists and is the intersection of all intervals containing A that satisfy P. It generalizes the convex hull concept to include side constraints on endpoints or width.

Relation and use: In interval arithmetic, constraint programming, and static analysis, intervalleilla serves as a bound that respects additional requirements on the interval. It helps bound a quantity when only partial information about endpoints is available, and it supports pruning in search algorithms and verification of numerical properties.

Example: Let A = {3.2, 4.7} and P(I) require that endpoints are integers. The intervalleilla is [3,5], the smallest integer-ended interval containing A.

See also: The concept is closely related to the convex hull, but enriches it with predicate-driven constraints and end-point requirements.