Interval Contractor¶
A box-to-box operator that narrows an interval search region without removing any point satisfying its declared target constraints.
Core Idea¶
An interval contractor safely narrows a rectangular box \(B\) of possible real-valued inputs for a declared feasible set \(X\). Its output \(C(B)\) must remain inside the input box, and it must retain every point of \(X\) that the input contained: \(C(B)\subseteq B\) and \(C(B)\cap X=B\cap X\). It may leave impossible points behind; it may not discard a possible solution.[^ref-9ba5a42cb926]
The abstraction is the sound box-to-box operator, not any one implementation or a complete solver. Methods based on interval arithmetic can construct contractors for equations and inequalities. Larger solvers may compose them, repeat them, or split surviving boxes. Those steps can improve search, but they are not conditions that every single contractor must perform.[ref-9ba5a42cb926][ref-e3027855cf79]
Scope of Application¶
For the equation \(x_1=\exp(x_2)\), Chabert and Jaulin give a contractor that intersects the first coordinate interval with the exponential of the second and the second with the logarithm of the first, on an appropriate positive domain. Starting with \([2,3]\times[0,2]\) narrows the second interval to \([\log 2,\log 3]\) without losing any pair on the equation's curve. The remaining box is not claimed to consist entirely of solutions.[^ref-9ba5a42cb926]
For a planar polygon, IBEX documents one forward–backward contractor per half-space inequality and composes those contractors. The combined rule narrows a candidate rectangle while keeping every point satisfying the polygon constraints. Equation and inequality settings share the same target-set, box, contraction and non-loss roles, although their implementations differ.[^ref-e3027855cf79]
Contractors also appear inside branch-and-prune searches and interval-based set estimation. Bounded-error robot localization illustrates why several feasible regions may need to remain represented; the complete localization method should not be mistaken for one contractor.[ref-9ba5a42cb926][ref-ce6d40277ef6]
Clarity¶
“Smaller” is not enough. A heuristic could reduce a box and still delete the very root or feasible pose being sought. The preservation equation distinguishes a sound contractor from that heuristic. Conversely, returning the input box unchanged is sound but uninformative. Strength of narrowing and soundness are separate properties.[^ref-9ba5a42cb926]
Nor does a nonempty output prove that a feasible point exists. The output is an outer box for possible target points, not an inner set of confirmed solutions. An empty sound output, however, establishes that the input box had no target point under the stated model and arithmetic assumptions.[^ref-9ba5a42cb926]
Manages Complexity¶
The operator turns a potentially complex nonlinear feasible set into a tractable local question: which coordinate ranges can be removed with proof? A solver can carry boxes rather than enumerate infinitely many points. Different constraint-specific contractors can then be composed while preserving all feasible points.[ref-9ba5a42cb926][ref-e3027855cf79]
This compact representation is imperfect. A rectangle can cover empty space around a curved or disconnected set. More propagation or bisection may narrow the region further, but those steps cost computation. Forward–backward evaluation can also be weak when variables repeat in an expression.[^ref-e3027855cf79]
Abstract Reasoning¶
First declare the target set \(X\) and input box \(B\). Check that the rule's output is a box contained in \(B\). Then check that every target point initially in \(B\) remains in the output. This invariant supports safe elimination of a box region even when its exact feasible subset is hard to compute.[^ref-9ba5a42cb926]
Next ask what the result actually warrants. A contracted box may justify pruning, not root existence or uniqueness. If narrowing is weak, a different contractor, composition, or splitting may help. None can legitimately relax the non-loss guarantee while continuing to claim the same contractor identity.[ref-9ba5a42cb926][ref-e3027855cf79]
Knowledge Transfer¶
The exact invariant transfers among interval methods for equations, inequalities and bounded-error inverse problems: each declares feasible points in a box, then removes only points known not to be feasible. What does not transfer automatically is a particular formula, numerical precision, fixed-point guarantee, or full-solver conclusion.[ref-9ba5a42cb926][ref-e3027855cf79]
The live Mathematical Operator entry supplies the broader typed-map parent; Interval Contractor adds interval boxes and target-preserving contraction. Staged Interval Arithmetic is a common computational ingredient, not a synonym. Broad talk of “conservative filtering” is only an analogy outside box-based numerical reasoning.[^ref-9ba5a42cb926]
[^ref-9ba5a42cb926]: Gilles Chabert and Luc Jaulin, “Contractor Programming,” Artificial Intelligence 173 (2009), 1079–1100, §§1.1–1.2 and §3.2. Author-hosted paper. [^ref-e3027855cf79]: IBEX team, “Contractors,” IBEX 2.9 project documentation, Forward-Backward, Intersection/Union/Composition and Propagation sections. Official documentation. [^ref-ce6d40277ef6]: Michel Kieffer, Luc Jaulin, Éric Walter and Dominique Meizel, “Localisation et suivi robustes d’un robot mobile grâce à l’analyse par intervalles,” Traitement du Signal 17 (2000), 207–219, abstract and §§3–5. Publisher page and article.
Relationships to Other Abstractions¶
Current abstraction Interval Contractor Domain-specific
Parents (1) — more general patterns this builds on
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Interval Contractor is a kind of Mathematical Operator Domain-specific
An interval contractor is a mathematical operator specialized to solution-preserving box contraction.
Hierarchy path (1) — routes to 1 parentless root
- Interval Contractor → Mathematical Operator → Function (Mapping)
Neighborhood in Abstraction Space¶
Interval Contractor sits in a sparse region of the domain-specific corpus (75th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Feasible Region — 0.84
- Interior-Point Method — 0.84
- Space-Filling Curve — 0.83
- Means-End Analysis — 0.83
- Individual-Pieces Set — 0.82
Computed from structural-signature embeddings · 2026-10-08