Partial Function Ontology

FLOW ALGEBRA
PARTIAL FUNCTION ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Function and Mapping Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/function-and-mapping-entities/partial-function-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Status: Formal mathematical specification; source page content not independently verified
Version: 1.0 | Date: 2026-10-13

01. PURPOSE
flow "PartialFunctionOntology" v1.0
  intent: model partial mappings with explicit definedness, typed inputs, and verifiable reasoning.
  mathematical_basis: classical set theory and function theory.
  convention: A is the declared source set; D subseteq A is the effective domain.

02. ENTITY TYPES
  PartialFunction: mapping f:A ⇀ B with effective_domain D subseteq A.
  SourceSet: A.
  Codomain: B.
  EffectiveDomain: D.
  DefinedInput: x in D.
  UndefinedInput: x in A\D.
  Graph: G_f subseteq A×B.
  FunctionValue: unique y in B for x in D.
  Restriction: f restricted to S.
  Extension: g extending f.
  TotalFunction: special case D=A.

03. FORMAL DEFINITION
  partial_function(f,A,B) iff:
    G_f subseteq A×B
    and forall x in A, forall y,z in B:
      ((x,y) in G_f and (x,z) in G_f) => y=z.
  D_f := {x in A | exists y in B: (x,y) in G_f}.
  f(x)=y iff (x,y) in G_f.
  defined(f,x) iff x in D_f.
  undefined(f,x) iff x in A and x notin D_f.
  total(f) iff D_f=A.
  NOTE: undefined is not the same as returning null, zero, or an error object.

04. CORE RELATIONSHIPS
  hasDeclaredSource(f,A)
  hasEffectiveDomain(f,D_f)
  hasCodomain(f,B)
  hasGraph(f,G_f)
  mapsTo(f,x,y) iff (x,y) in G_f
  isDefinedAt(f,x) iff x in D_f
  isUndefinedAt(f,x) iff x in A\D_f
  isRestrictionOf(g,f)
  isExtensionOf(f,g)
  isSpecialisationOf(TotalFunction,PartialFunction)

05. TYPE AND MEMBERSHIP RULES
  mapsTo(f,x,y) => x in A and x in D_f and y in B.
  x notin A => invalid_source_input, not merely undefined_in_source.
  x in A\D_f => undefined_in_source.
  x in D_f => exists unique y in B with mapsTo(f,x,y).
  graph_functionality: no two distinct outputs for one input.

06. EVALUATION MODEL
  evaluate(f,x):
    require x in A else TYPE_ERROR
    if x notin D_f then UNDEFINED
    else return unique y such that (x,y) in G_f.
  result_type: VALUE(y) | UNDEFINED | TYPE_ERROR.
  preserve distinction among these outcomes in AI consumption.

07. RESTRICTION
  for S subseteq A:
    (f|S) has declared source S, codomain B,
    graph G_f intersect (S×B),
    effective domain D_f intersect S.
  restriction preserves all previously defined values on D_f intersect S.

08. EXTENSION
  g:A ⇀ B extends f:A ⇀ B iff:
    D_f subseteq D_g and forall x in D_f: g(x)=f(x).
  total_extension(g,f) iff g extends f and D_g=A.
  total extension exists if B nonempty; when A\D_f empty, f already total.
  if B empty and A nonempty, no total function A -> B exists.

09. COMPOSITION
  f:A ⇀ B, g:B ⇀ C.
  (g∘f):A ⇀ C.
  D_(g∘f) = {x in D_f | f(x) in D_g}.
  (g∘f)(x)=g(f(x)) for x in D_(g∘f).
  composition is associative for compatible typed partial functions.
  identity_A:A -> A is total and is composition identity.

10. INVERSE RELATION
  inverse_graph(f) = {(y,x) | (x,y) in G_f}.
  inverse_graph is a partial function B ⇀ A iff f is injective on D_f.
  otherwise inverse_graph is a relation, not a single-valued function.

11. IMAGE AND PREIMAGE
  image(f,S) = {f(x) | x in S intersect D_f} for S subseteq A.
  preimage(f,T) = {x in D_f | f(x) in T} for T subseteq B.
  image(f,A) subseteq B.
  preimage(f,B) = D_f.
  preimage(f,empty_set) = empty_set.

12. EQUALITY AND EQUIVALENCE
  extensional_equal(f,g) for same declared source/codomain iff:
    D_f=D_g and forall x in D_f: f(x)=g(x).
  representation_equivalent(f,g) may hold despite different notation.
  do not equate undefined with a codomain value.
  distinguish same graph from same typed mapping when source/codomain differ.

13. DEFINEDNESS LOGIC
  defined(f,x) and f(x)=y => mapsTo(f,x,y).
  mapsTo(f,x,y) => defined(f,x).
  not defined(f,x) does not imply x outside A.
  defined(g∘f,x) iff defined(f,x) and defined(g,f(x)).
  last expression is evaluated with short-circuit definedness.

14. PARTIALITY AND TOTALITY
  partial_function includes total functions as a special case.
  strictly_partial(f) iff D_f proper_subset A.
  empty_graph:A ⇀ B has D_f=empty_set.
  empty_graph is total iff A=empty_set.

15. ALGEBRAIC EXAMPLES
  reciprocal:R ⇀ R, D=R\{0}, reciprocal(x)=1/x.
  sqrt_real:R ⇀ R, D={x in R | x>=0}, sqrt_real(x)=sqrt(x).
  division:R×R ⇀ R, D={(a,b) | b!=0}, division(a,b)=a/b.
  logarithm:R ⇀ R, D={x in R | x>0}, logarithm(x)=ln(x).
  note: these domains depend on specified real-valued codomains.

16. PARTIAL FUNCTIONS AS RELATIONS
  every partial function is a functional binary relation.
  not every binary relation is functional.
  total function requires existence and uniqueness for every x in A.
  partial function requires uniqueness wherever a value exists.

17. KNOWLEDGE GRAPH SCHEMA
  node_types: PartialFunction, Set, Input, Output, Graph, Constraint.
  edge_types: declaredSource, effectiveDomain, codomain, definedAt,
              undefinedAt, mapsTo, restricts, extends, composesWith.
  identity: use stable identifiers for function, source, codomain, graph.
  provenance: retain mathematical assumptions and source version.

18. SEMANTIC_EQUIVALENCE
  exact: partial map; partially defined function; partial mapping.
  contextual: partial transformation; partial operator.
  non_equivalent: multivalued relation; arbitrary relation; total-only mapping.
  warning: "partial" can mean incomplete information in other domains.

19. INFERENCE RULES
  R1: mapsTo(f,x,y) => defined(f,x).
  R2: defined(f,x) and mapsTo(f,x,y) and mapsTo(f,x,z) => y=z.
  R3: total(f) and x in A => defined(f,x).
  R4: isRestrictionOf(g,f) and x in D_g => g(x)=f(x).
  R5: isExtensionOf(f,g) and x in D_f => g(x)=f(x).
  R6: defined(g∘f,x) => defined(f,x) and defined(g,f(x)).
  R7: x in A and x notin D_f => undefined(f,x).

20. CONSTRAINT VALIDATION
  V1: D_f subseteq A.
  V2: G_f subseteq A×B.
  V3: forall x, at most one output y.
  V4: projection_first(G_f)=D_f.
  V5: all recorded outputs belong to B.
  V6: composition types match on intermediate set B.
  V7: no value assertion for undefined input.

21. COUNTEREXAMPLES
  {(1,2),(1,3)} is not a partial function from {1} to {2,3}.
  reciprocal(0) is undefined, not zero.
  f:A ⇀ B can be non-total even if every observed input has a value.
  inverse of non-injective f is generally not a partial function.

22. CONFIDENCE
  axiomatic_definition: 1.00 when assumptions explicitly satisfied.
  derived_rule: 1.00 for sound symbolic deduction.
  empirical_inference: evidence-dependent; never assume full definedness.
  confidence_is_not_truth: store provenance separately from proof status.

23. FLOW_HASH
  canonicalise: normalise identifiers, bound variables, ordering of set literals,
                typed signature, effective domain, graph and rule version.
  hash_input: canonical ontology specification plus version.
  hash_algorithm: SHA-256 or declared equivalent.
  do_not_fabricate_hash: compute hash only over actual serialised content.

24. UPDATE_MODEL
  track changes to A, B, D_f, G_f, type constraints and rules.
  on_domain_change: revalidate definedness and composition.
  on_graph_change: revalidate uniqueness and all derived images/preimages.
  versioning: preserve historical snapshots and change provenance.

25. AI_CONSUMPTION
  parse typed partial-function declarations.
  require explicit definedness checks before evaluation.
  distinguish TYPE_ERROR, UNDEFINED, and VALUE.
  propagate domain restrictions through composition and substitution.
  explain proof obligations and report counterexamples.
  expose provenance, confidence, and ontology version.

26. TEST CASES
  T1: reciprocal(2)=0.5 -> VALUE.
  T2: reciprocal(0) -> UNDEFINED.
  T3: reciprocal("text") -> TYPE_ERROR.
  T4: D_sqrt_real=[0,infinity) -> valid.
  T5: preimage(reciprocal,{1})={1} -> valid.
  T6: image(reciprocal,R)=R\{0} -> valid.
  T7: f with (1,2) and (1,3) -> REJECT.
  T8: empty source and empty graph -> total.
  T9: composition defined only when intermediate value in next effective domain.

27. INTEROPERABILITY
  related_ontologies: Function, Mapping, Domain, Codomain, Input,
    Output, Image, Preimage, InjectiveFunction, SurjectiveFunction,
    BijectiveFunction, TotalFunction, InverseFunction, CompositeFunction.
  linked_foundations: Set, BinaryRelation, MathematicalConstraint.

28. TERMINATION
  status: COMPLETE
  validation: definition, types, composition, restrictions and examples included.
  end_flow: PartialFunctionOntology v1.0