Hstructure

Hstructure is a formal construct used in formal logic and computer science to model hierarchical data with structure-preserving mappings.

An Hstructure consists of a set of elements together with a hierarchy relation that orders elements into a tree-like or lattice-like arrangement, and a signature that includes relations and possibly functions to interpret attributes assigned to elements.

The defining feature is that homomorphisms between Hstructures preserve the hierarchy and the interpretations of relations, ensuring that the image of a related pair remains related in the same way.

Construction involves selecting a base hierarchy, specifying attribute interpretations on nodes, and enforcing constraints that substructures respect parent-child relations and other hierarchy-preserving conditions.

Properties commonly studied include closure under certain constructions, embeddability of substructures, and preservation of truth of formulas under homomorphisms. The exact properties depend on the chosen signature.

Applications include data modeling for XML and other hierarchical data formats, knowledge representation with layered semantics, and model-checking tasks for systems with hierarchical state spaces.

Variants include pure Hstructure (no node attributes), labeled Hstructure (attributes on nodes), and weighted or probabilistic Hstructure (weights on relations or transitions).

See also related concepts such as trees, graphs with hierarchy, homomorphism, and XML data models.