interpolant

An interpolant is a concept with several related meanings in mathematics and logic. In mathematics and numerical analysis, an interpolant is a function or rule that passes through or matches a set of known data points in order to estimate values at other points. Common interpolants include polynomial interpolation (such as Lagrange polynomials), spline interpolation, and piecewise linear interpolation. The aim is to produce values that are consistent with the given data within the domain of interest.

In logic and computer science, an interpolant is a formula that lies logically between two statements with shared vocabulary. Craig’s interpolation theorem states that if A implies B, there exists an interpolant I such that A implies I, I implies B, and the non-logical symbols of I are contained in the intersection of the symbols of A and B. An interpolant thus summarizes the portion of the information in A that is relevant to B, using only the common vocabulary.

Computationally, interpolants play a key role in automated reasoning and verification. In SAT/SMT solving and model checking, interpolants are computed from proofs of A ⇒ B and used to refine abstractions or abstractions-based analyses, guiding refinement loops and helping to generate invariants. In numerical contexts, interpolants are constructed from sample points, with considerations such as degree, smoothness, and error behavior shaping their selection.

The concept has broad applicability across domains such as data fitting, computer-aided design, and signal processing. An interpolant is not unique for a given data set or pair of formulas; different methods or schemes yield distinct interpolants with different properties.