representationwhere

Representationwhere is a term used in discussions of representation in logic, mathematics, and computer science. It denotes a construction that pairs a representation of an object or structure with the condition or region of validity under which that representation holds. The idea is to capture not only how something is represented but also where that representation is applicable.

Definition and formalization: Given a representation function rep that maps objects to representations, and a predicate C(x) that specifies a domain or condition, a representationwhere is the pair (rep(x), C(x)). In formal notation one may write repwhere = {(x, rep(x)) | C(x) holds}. In practice, representationwhere can express that a particular encoding is valid only for inputs satisfying C.

Examples: In a graph database, a subgraph encoding might be valid only for graphs with at least k nodes. In machine learning, a feature representation learned on a dataset from distribution D is claimed as a representationwhere for data drawn from D. In logical frameworks, a model might be represented only for structures that satisfy a theory T.

Applications and relation: The concept helps formalize context-dependence of representations, distinguishing representation from its domain of validity. It relates to constraint-based representations, conditional embeddings, and knowledge representation. It may assist in reasoning about when different representations are interchangeable or incompatible.

History and usage: Representationwhere is a neologism not widely adopted in standard literature, but it appears in some theoretical discussions, proposals for formalizing context-aware representations, and as a pedagogical device. Users should specify the domain predicates clearly when employing the term.