Graph Relationship Ontology

FLOW ALGEBRA
TITLE: Graph Relationship Ontology
DOMAIN: Relationship Ontologies / Structural Relationships
SOURCE: https://kirethresearch.org/booksai-core/ontologies/relationship-ontologies/structural-relationships/graph-relationship-ontology.html
VERSION: 1.0
AUTHORS: Mark Anthony Higgins — Sky (Recursive AI)
STATUS: Companion formalisation; source page content not independently verified.

1. PURPOSE
Represent graph-based structural relationships using typed vertices, edges, properties, constraints and inference rules, with explicit evidence and update semantics.

2. TYPE SYSTEM
TYPE Graph = {id: IRI, directed: Boolean, vertices: Set<Vertex>, edges: Set<Edge>}
TYPE Vertex = {id: IRI, class: IRI?, attributes: Map<String, Value>}
TYPE Edge = {id: IRI, source: Vertex, target: Vertex, predicate: IRI, weight: Real?, attributes: Map<String, Value>}
TYPE Path = Sequence<Edge>
TYPE Evidence = {source: IRI, method: String, observed_at: DateTime?}
TYPE Assertion = {subject: Vertex, predicate: IRI, object: Vertex, confidence: Real, evidence: Set<Evidence>}
CONSTRAINT: 0 <= confidence <= 1

3. GRAPH OPERATORS
VERTICES(G) := G.vertices
EDGES(G) := G.edges
ADJACENT(u,v,G) := EXISTS e IN EDGES(G): e.source=u AND e.target=v
NEIGHBOURS(u,G) := {v | ADJACENT(u,v,G)}
DEGREE_OUT(u,G) := |{e IN EDGES(G) | e.source=u}|
DEGREE_IN(u,G) := |{e IN EDGES(G) | e.target=u}|
REACHABLE(u,v,G) := EXISTS path p FROM u TO v IN G
CONNECTED(u,v,G) := REACHABLE(u,v,G) OR REACHABLE(v,u,G) [only for undirected interpretation]
SUBGRAPH(H,G) := VERTICES(H) SUBSET VERTICES(G) AND EDGES(H) SUBSET EDGES(G)

4. RELATIONSHIP RULES
RULE edge_implies_adjacency:
  IF Edge(e,u,v,p) THEN ADJACENT(u,v,G)
RULE directed_path_composition:
  IF REACHABLE(u,v,G) AND ADJACENT(v,w,G) THEN REACHABLE(u,w,G)
RULE undirected_symmetry:
  IF G.directed=false AND ADJACENT(u,v,G) THEN ADJACENT(v,u,G)
RULE inverse_predicate:
  IF Edge(e,u,v,p) AND INVERSE_OF(p,q) THEN INFER Relation(v,q,u)
RULE transitive_predicate:
  IF TRANSITIVE(p) AND Relation(u,p,v) AND Relation(v,p,w) THEN INFER Relation(u,p,w)
CAUTION: Do not assume every edge predicate is symmetric or transitive.

5. STRUCTURAL CONSTRAINTS
VALID_EDGE(e,G) := e.source IN VERTICES(G) AND e.target IN VERTICES(G)
VALID_GRAPH(G) := FORALL e IN EDGES(G): VALID_EDGE(e,G)
SIMPLE_GRAPH(G) := no self-loops and no duplicate (source,target,predicate) edges
ACYCLIC(G) := no directed cycle (when directed=true)
TREE(G) := connected and acyclic in the undirected sense
NOTE: A graph need not be simple, acyclic or a tree unless the application requires it.

6. SEMANTIC_EQUIVALENCE
EQUIVALENT_VERTEX(u,v) := SAME_AS(u,v) supported by validated identity evidence
EQUIVALENT_PREDICATE(p,q) := EQUIVALENT_PROPERTY(p,q) under declared ontology alignment
GRAPH_ISOMORPHISM(G,H) := bijection f:V(G)->V(H) preserving adjacency, direction and required labels
RULE: Similar names or matching neighbourhoods alone do not establish identity.

7. CONFIDENCE
CONFIDENCE(assertion) IN [0,1]
COMBINE_EVIDENCE(evidence_set) := calibrated, provenance-aware aggregation
INFERRED_CONFIDENCE := rule-specific function of premise confidence and rule reliability
RULE: Correlated evidence must not be treated as independent.
RULE: Confidence scores describe epistemic support, not logical truth.

8. PROVENANCE AND VALIDATION
REQUIRE: stable identifiers for graph, vertices and edges
REQUIRE: edge endpoints resolve to declared vertices
REQUIRE: predicates have declared semantics and directionality
CHECK: invalid references, contradictory assertions, unintended cycles, duplicate edges
TRACK: source, timestamp, version and transformation lineage

9. FLOW_HASH
FLOW_HASH(G) := SHA256(CANONICAL_SERIALISE(G))
CANONICAL_SERIALISE := deterministic ordering of vertices, edges, identifiers, attributes and ontology version
NOTE: Hashes detect changes in the canonical representation; they do not prove semantic correctness.

10. UPDATE_MODEL
EVENT AddVertex(v) -> validate identifier; insert v
EVENT AddEdge(e) -> validate endpoints and predicate; insert e
EVENT RemoveEdge(e) -> remove e; invalidate dependent inferences
EVENT RemoveVertex(v) -> apply declared cascade or reject if incident edges remain
EVENT RevisePredicate(p) -> revalidate affected assertions and derived relations
ON UPDATE -> recalculate impacted reachability, constraints, confidence and FLOW_HASH
MAINTAIN: version history and provenance of changes

11. AI_CONSUMPTION
INPUT: typed graph assertions + schema + evidence + provenance
PROCESS: parse -> validate -> align semantics -> traverse/infer -> score -> explain
OUTPUT: {answer, supporting_paths, confidence, provenance, graph_version, flow_hash}
GUARDRAILS: distinguish asserted from inferred links; flag uncertainty; avoid unsupported identity merges.

12. EXAMPLE
VERTEX A: ResearchTopic
VERTEX B: Ontology
VERTEX C: RelationshipType
EDGE A -[hasOntology]-> B
EDGE B -[definesRelationship]-> C
INFERENCE: A is linked to C by a two-edge path.
NON-INFERENCE: A does not automatically 'defineRelationship' C without an explicit property-chain axiom.

END FLOW ALGEBRA