Specificslength

Specificslength is a theoretical measure used in formal specification and search problems to denote the length of the shortest object that satisfies a given specification. For a specification S defined over a domain of objects, specificslength(S) is defined as the minimum length of any object o such that o satisfies S. If no object satisfies S, the specificslength is undefined.

In the common case where objects are strings over a finite alphabet and length is the usual string length, specificslength(S) = min{ length(o) : S(o) is true }. This concept emphasizes minimality with respect to a fixed constraint, rather than the intrinsic descriptive complexity of the object itself.

Examples help illustrate the idea. For a specification requiring a binary string that represents a prime number, the shortest such string has length 2, since 2 is represented as 10 in binary. In a programming context, the specificslength of a specification that asks for a program printing a given message is the length of the shortest program in the chosen language that outputs that message.

Computationally, determining specificslength can be challenging and is often undecidable in general. It is typically approached via search, constraint solving, or approximation within restricted classes of specifications.

Relation to other concepts: specificslength is related to, but distinct from, Kolmogorov complexity. Both involve minimal representations, but specificslength is anchored to a fixed specification, whereas Kolmogorov complexity uses a universal description language. Specificslength is also connected to the idea of a shortest witness in decision problems and to minimum description length in model selection.

See also: Kolmogorov complexity, shortest witness problem, minimum description length, constraint solving.