Recursive Relationship Ontology

FLOW ALGEBRA
TITLE: Recursive Relationship Ontology
DOMAIN: Relationship Ontologies / Recursive Relationships
SOURCE: https://kirethresearch.org/booksai-core/ontologies/relationship-ontologies/recursive-relationships/recursive-relationship-ontology-2.html
VERSION: 1.0
AUTHORS: Mark Anthony Higgins — Sky (Recursive AI)
STATUS: Companion formalisation; source-page specification not independently verified.

flow "RecursiveRelationshipOntology" v1.0 {
  ontology:
    id: "recursive-relationship-ontology-2"
    parent: "recursive-relationships"
    purpose: "Represent self-referential and mutually recursive entity relationships with explicit base cases, evaluation semantics and traceable inference."

  types:
    Entity: {id: string, kind: string, domain: string?}
    RecursiveRule: {id: string, head: string, body: string[], base_case: bool, monotonic: bool, provenance: string}
    RelationAssertion: {id: string, subject: string, predicate: string, object: string, confidence: number, evidence: string[]}
    Evaluation: {id: string, strategy: enum[least_fixed_point, bounded_unfolding, well_founded], iteration_limit: int?, status: enum[pending, converged, bounded, failed]}
    DependencyEdge: {from_rule: string, to_rule: string, polarity: enum[positive, negative]}

  predicates:
    refers_to(a: Entity, b: Entity)
    depends_on(a: Entity, b: Entity)
    recursively_related_to(a: Entity, b: Entity)
    derives(rule: RecursiveRule, assertion: RelationAssertion)
    participates_in_cycle(rule: RecursiveRule, cycle_id: string)

  constraints:
    - "All rule references and assertion endpoints resolve to declared entities or predicates."
    - "Confidence lies within [0, 1]; evidence and provenance remain attached to derived assertions."
    - "A recursive rule requires explicit base-case facts or a documented fixed-point semantics."
    - "Do not assume every dependency is transitive or every cycle is erroneous."
    - "For least-fixed-point evaluation, recursion must be monotonic over a finite or otherwise convergent domain."
    - "Negation cycles require a defined non-classical semantics or must be rejected."
    - "A bounded iteration result is not labelled complete unless convergence is established."

  inference:
    rule recursive_closure:
      if recursively_related_to(a,b) and recursively_related_to(b,c) and transitive_relation_context(a,b,c)
      then recursively_related_to(a,c)
      confidence: "retain weakest supported evidence; avoid artificial certainty"
    rule base_case_seed:
      if valid_base_case(rule, assertion)
      then derives(rule, assertion)
      confidence: "source-evidence dependent"
    rule fixed_point:
      if monotonic_ruleset(rules) and iteration_stabilises(rules, facts)
      then converged_least_fixed_point(rules, facts)
      confidence: "proof and evaluation trace required"

  SEMANTIC_EQUIVALENCE:
    - "Distinguish recursive dependency, ordinary association, and identity."
    - "Normalise relation aliases without conflating direct and transitive relationships."
    - "Preserve scope, rule semantics, and version during alignment."

  CONFIDENCE:
    range: [0,1]
    factors: [evidence_quality, rule_validity, inference_depth, completeness, source_reliability]
    policy: "Repeated cycles do not independently corroborate a claim or increase confidence."

  VALIDATION:
    - "Check typing, reference integrity, rule safety, stratification and termination assumptions."
    - "Detect strongly connected components and classify intentional versus unsupported recursion."
    - "Verify base cases, convergence, provenance and derivation traces."
    - "Flag non-monotonic recursion and unresolved contradictory assertions."

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

  UPDATE_MODEL:
    strategy: "versioned incremental fixed-point recomputation"
    steps: [ingest, normalise, validate, identify_affected_cycles, retract_stale_derivations, recompute, verify, publish]
    rollback: true

  AI_CONSUMPTION:
    representation: "typed recursive knowledge graph with explicit rule provenance"
    outputs: [recursive_edges, dependency_graph, fixed_point_status, derivation_traces, uncertainty_annotations]
    safeguards: [bounded_execution, no_unsupported_transitivity, no_cycle_based_confidence_inflation, preserve_provenance]
}

END FLOW ALGEBRA