Functional Relationship Ontology

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

flow "FunctionalRelationshipOntology" v1.0

ENTITY_TYPES:
  Set: {id, label, elements?, provenance?}
  Element: {id, member_of:Set}
  Function: {id, domain:Set, codomain:Set, mapping, partial:bool=false}
  FunctionalRelation: {id, source:Element, target:Element, function:Function}
  Evidence: {id, source, timestamp?, method?}

RELATIONSHIPS:
  DOMAIN_OF(Set, Function)
  CODOMAIN_OF(Set, Function)
  MAPS_TO(Element, Element, Function)
  COMPOSES_WITH(Function, Function)
  INVERSE_OF(Function, Function)
  EQUIVALENT_TO(Function, Function, context)

AXIOMS:
  A1: For a total function f:A -> B, every x in A maps to exactly one y in B.
  A2: For a partial function f:A ⇀ B, each x in A maps to at most one y in B.
  A3: A mapped output must belong to the declared codomain.
  A4: Equal inputs yield equal outputs for a deterministic mathematical function.
  A5: A relation R subset A×B is functional iff
      forall x in A, y1,y2 in B: (R(x,y1) and R(x,y2)) implies y1=y2.

INFERENCE_RULES:
  R1: MAPS_TO(x,y,f) and MAPS_TO(y,z,g) and compatible_codomain_domain(f,g)
      -> MAPS_TO(x,z,compose(g,f)).
  R2: bijective(f) and MAPS_TO(x,y,f) -> MAPS_TO(y,x,inverse(f)).
  R3: MAPS_TO(x,y1,f) and MAPS_TO(x,y2,f) and y1 != y2 -> FUNCTIONALITY_VIOLATION(f,x).
  R4: total(f) and exists x in domain(f) with no output -> TOTALITY_VIOLATION(f,x).
  R5: forall x in domain(f): f(x)=g(x), with compatible domains/codomains
      -> EXTENSIONALLY_EQUAL(f,g).

CONSTRAINTS:
  - Require domain/codomain type compatibility before composition.
  - Distinguish mathematical equality from approximate semantic similarity.
  - Do not infer invertibility without injectivity and surjectivity.
  - Do not infer causation from a functional association.
  - Record scope and assumptions for inferred equivalence.

SEMANTIC_EQUIVALENCE:
  exact: identical mapping on the same domain with compatible codomains
  contextual: equivalent outputs only under declared context and assumptions
  approximate: similarity score does not establish equality

CONFIDENCE:
  direct_axiom: 1.0
  formally_proved: 1.0
  evidence_based: calibrated_score_in_[0,1]
  rule: derived confidence must not exceed the weakest unverified premise

VALIDATION:
  - verify_unique_output_per_input
  - verify_domain_codomain_membership
  - verify_totality_if_declared
  - detect_undefined_compositions
  - verify_inverse_claims
  - audit_provenance_and_assumptions

FLOW_HASH:
  algorithm: SHA-256
  input: canonical_serialisation(ontology_version, entities, relations, rules, constraints)
  purpose: content integrity and reproducible version identification

UPDATE_MODEL:
  strategy: versioned_append_with_validation
  on_change: revalidate_affected_mappings_and_inferences
  retain: previous_version, provenance, change_log
  reject: conflicting_functional_outputs_without_resolution

AI_CONSUMPTION:
  representation: typed_entities_and_explicit_rules
  retrieval_keys: [function, domain, codomain, mapping, composition, inverse]
  reasoning: constraint_checked_deduction
  output: {result, supporting_rules, assumptions, confidence, provenance, flow_hash}
  safeguards: [no_unjustified_inverse, no_causation_from_association, no_equivalence_from_similarity]

end flow