Recursive Graph Ontology

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

flow "RecursiveGraphOntology" v1.0 {
  ontology:
    id: "recursive-graph-ontology"
    parent: "recursive-relationships"
    purpose: "Represent recursive graph structure, traversal, dependencies, cycles and evidence-backed reachability without assuming all edges are transitive."

  types:
    Graph: {id: string, directed: boolean, version: string, provenance: string}
    Node: {id: string, graph_id: string, kind: string, attributes: map?}
    Edge: {id: string, graph_id: string, source: string, target: string, predicate: string, confidence: number, provenance: string}
    Path: {id: string, graph_id: string, nodes: string[], edges: string[], length: integer}
    Traversal: {id: string, graph_id: string, start_node: string, strategy: enum[depth_first, breadth_first, fixed_point], visited: string[], max_depth: integer?}
    RecursiveRule: {id: string, base_case: string, recursive_step: string, termination_condition: string}

  predicates:
    has_node(graph: Graph, node: Node)
    has_edge(graph: Graph, edge: Edge)
    connects(edge: Edge, source: Node, target: Node)
    directly_reaches(a: Node, b: Node)
    reachable(a: Node, b: Node, graph: Graph)
    participates_in_cycle(node: Node, graph: Graph)
    depends_on(a: Node, b: Node)

  constraints:
    - "Every edge endpoint must resolve to a node in the same graph."
    - "Graph directionality must be respected in traversal."
    - "Reachability does not imply causal, semantic or identity equivalence."
    - "Cycles are permitted unless the declared graph subtype requires acyclicity."
    - "Traversal must track visited states or otherwise guarantee termination."
    - "Confidence values must lie within [0,1]."
    - "Recursive inferences must retain source edges and provenance."

  inference:
    rule base_reachability:
      if connects(e,a,b) and has_edge(g,e)
      then reachable(a,b,g)
      confidence: "derived from source edge"
    rule recursive_reachability:
      if reachable(a,b,g) and reachable(b,c,g)
      then reachable(a,c,g)
      confidence: "preserve path evidence; no automatic confidence increase"
    rule cycle_detection:
      if reachable(a,b,g) and reachable(b,a,g) and a != b
      then participates_in_cycle(a,g) and participates_in_cycle(b,g)
      confidence: "requires verified directed paths"

  SEMANTIC_EQUIVALENCE:
    - "Normalise node and predicate identifiers only against declared, versioned mappings."
    - "Distinguish graph isomorphism, structural similarity and entity identity."
    - "Do not merge nodes solely because they share labels or neighbourhoods."

  CONFIDENCE:
    range: [0,1]
    factors: [edge_evidence, provenance_quality, mapping_reliability, inference_depth, graph_version]
    policy: "Track path-specific confidence; do not manufacture certainty by chaining."

  VALIDATION:
    - "Check endpoint integrity, direction, edge type and graph membership."
    - "Detect duplicate edges, dangling references and unexpected cycles."
    - "Test base cases, termination conditions and fixed-point convergence."
    - "Record unresolved contradictions and alternative graph interpretations."

  FLOW_HASH:
    algorithm: "SHA-256"
    input: "canonicalised ontology version, graph assertions, recursive rules and provenance"
    note: "Compute at execution; no hash value is claimed here."

  UPDATE_MODEL:
    strategy: "versioned incremental graph maintenance"
    steps: [ingest, normalise, validate, detect_changes, invalidate_affected_paths, recompute_fixed_point, publish]
    rollback: true

  AI_CONSUMPTION:
    representation: "typed, provenance-preserving recursive knowledge graph"
    outputs: [nodes, edges, reachability_paths, cycle_reports, inferred_relations, confidence_annotations, provenance_records]
    safeguards: [bounded_traversal, no_unsupported_transitivity, no_unqualified_identity_merges, preserve_graph_versions]
}

END FLOW ALGEBRA