Structural Relationships

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
Structural Relationships — Kireth Semantic Library
Version: 1.0 | Locale: en-GB
Authors: Mark Anthony Higgins — Sky (Recursive AI)
Source: https://kirethresearch.org/booksai-core/ontologies/relationship-ontologies/structural-relationships.html

PURPOSE
Define a typed, verifiable structural relationship model spanning parent, child, hierarchy, tree, network, graph, node, edge, root and branch ontologies. This is a companion formal specification, not a transcription of page content.

ENTITY_TYPES
  Entity(id: URI, label: String, kind: String)
  Node(id: URI, entity: Entity)
  Edge(id: URI, source: Node, target: Node, relation: RelationType)
  Structure(id: URI, nodes: Set[Node], edges: Set[Edge], kind: StructureKind)
  Evidence(id: URI, source: URI, observed_at: DateTime)

ENUM RelationType = {PARENT_OF, CHILD_OF, ANCESTOR_OF, DESCENDANT_OF, CONNECTED_TO, BRANCH_OF, ROOT_OF, LINKED_TO}
ENUM StructureKind = {HIERARCHY, TREE, NETWORK, GRAPH}

TYPED_RELATIONSHIPS
  parent_of(parent: Node, child: Node) -> Boolean
  child_of(child: Node, parent: Node) -> Boolean
  ancestor_of(ancestor: Node, descendant: Node) -> Boolean
  descendant_of(descendant: Node, ancestor: Node) -> Boolean
  connected_to(a: Node, b: Node) -> Boolean
  root_of(root: Node, structure: Structure) -> Boolean
  branch_of(branch: Node, parent: Node) -> Boolean

AXIOMS
  A1: parent_of(p,c) <=> child_of(c,p).
  A2: parent_of(p,c) => p.id != c.id.
  A3: parent_of(p,c) => ancestor_of(p,c).
  A4: ancestor_of(a,b) AND ancestor_of(b,c) => ancestor_of(a,c).
  A5: ancestor_of(a,b) => descendant_of(b,a).
  A6: structure.kind = TREE => directed parent edges are acyclic.
  A7: structure.kind = TREE => exactly one root and each non-root node has exactly one parent.
  A8: structure.kind = TREE => all nodes are reachable from the root.
  A9: structure.kind = HIERARCHY => ancestor_of(a,b) implies NOT ancestor_of(b,a).
  A10: structure.kind = GRAPH => each edge endpoint belongs to structure.nodes.
  A11: structure.kind = NETWORK => connections may be cyclic and nodes may have multiple neighbours.
  A12: branch_of(b,p) => parent_of(p,b) within a rooted tree context.

INFERENCE
  infer_child: PARENT_OF(p,c) -> CHILD_OF(c,p).
  infer_ancestor: PARENT_OF(p,c) -> ANCESTOR_OF(p,c).
  infer_transitive_ancestor: ANCESTOR_OF(a,b) & ANCESTOR_OF(b,c) -> ANCESTOR_OF(a,c).
  infer_root: in a valid non-empty tree, the unique node with indegree 0 is ROOT_OF(tree).
  infer_connection: an edge between a and b establishes LINKED_TO(a,b); CONNECTED_TO is symmetric only when the edge is declared undirected.

SEMANTIC_EQUIVALENCE
  CHILD_OF(c,p) == inverse(PARENT_OF(p,c)).
  DESCENDANT_OF(d,a) == inverse(ANCESTOR_OF(a,d)).
  PARENT_OF != ANCESTOR_OF: direct versus transitive relation.
  EDGE != RELATIONSHIP: edge is a representation of a relationship.
  GRAPH != TREE: every tree is a graph, but not every graph is a tree.

CONFIDENCE
  score: Float in [0,1]
  asserted: require evidence source or explicitly labelled author assertion.
  derived: attach proof path and supporting premises.
  conflicting: retain competing assertions; flag for review rather than silently selecting one.
  threshold: 0.80 for automatic consumption of uncertain inferred claims (configurable).

VALIDATION
  V1: unique entity, node, edge and structure identifiers.
  V2: all edge endpoints resolve to declared nodes.
  V3: no self-parent relationship.
  V4: reject parent cycles in TREE or strict HIERARCHY.
  V5: enforce unique parent and unique root for TREE.
  V6: ensure root reaches all TREE nodes.
  V7: verify provenance and timestamps for externally asserted claims.
  V8: do not infer undirected symmetry from directed edges.

FLOW_HASH
  algorithm: SHA-256
  input: canonical UTF-8 serialisation of version, entity types, relations, axioms and validation rules; sort sets and keys deterministically.
  output: lowercase hexadecimal digest computed by implementing system.

UPDATE_MODEL
  policy: versioned, append-only change history.
  on_change: validate -> recompute inferences -> recompute FLOW_HASH -> publish version.
  invalid_change: reject and preserve previous valid snapshot.
  provenance: retain origin, timestamp, revision and evidence links.

AI_CONSUMPTION
  mode: typed semantic graph with deterministic inference and explicit validation.
  sequence: parse -> type-check -> validate -> infer -> score confidence -> export.
  outputs: entities, edges, hierarchy paths, roots, branches, proof traces, confidence and version hash.
  safeguards: distinguish asserted from inferred facts; do not invent absent edges or evidence.

END FLOW ALGEBRA