diagramindependence

Diagramindependence is a property of a formal system in which the semantic content or computed results are invariant under diagrammatic representations that are equivalent under a specified set of diagrammatic rewrite rules. The concept is used in areas where diagrams serve as a compact syntax for complex structures, such as tensor networks, circuit diagrams, or categorical diagrams.

Formalization can be described by an equivalence relation on diagrams generated by the allowed local moves. A diagrammatic interpretation maps diagrams into a semantic space; diagramindependence holds if this interpretation factors through the quotient by the equivalence relation. In practice, diagramindependence is supported by confluence and termination of the rewriting system and by the functoriality of the interpretation, ensuring that all equivalent diagrams yield the same interpretation.

Examples include tensor networks, where different layouts can represent the same linear operator; if the interpretation is invariant to layout, diagramindependence holds. In knot theory, invariants that survive Reidemeister moves exhibit diagramindependence. In graphical models, two equivalent graph structures that yield the same factorization provide diagramindependence under the chosen semantics.

Implications of diagramindependence include greater modularity and interchangeability of representations, which can simplify optimization and reasoning about complex systems. Failures of diagramindependence often arise when the rewrite rules are incomplete or when extra diagrammatic structure (such as orientation or planarity) carries essential semantic information.

See also: diagrammatic reasoning, invariants, equivalence relations, category theory, tensor networks.