Variable Dependency Ontology

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

flow "VariableDependencyOntology" v1.0

DOMAIN:
  ontology: Mathematical Relationships / Variable Dependency
  purpose: Represent, validate and reason about dependencies between mathematical variables.

TYPES:
  Variable: {id: str, symbol: str, domain: Set, unit: str?}
  Expression: {id: str, formula: str, free_variables: Set[Variable]}
  Dependency: {dependent: Variable, independent: Variable, kind: DependencyKind, context: str?, confidence: float}
  DependencyKind: [functional, statistical, causal, conditional, temporal, implicit]
  Evidence: {source: str, method: str, timestamp: str?}

RELATIONSHIPS:
  DEPENDS_ON(y, x, context): y varies as a function of or is otherwise dependent on x in context.
  INDEPENDENT_OF(y, x, context): y is independent of x under an explicitly stated model.
  DETERMINED_BY(y, inputs, f): y = f(inputs).
  CONDITIONED_ON(y, x, condition): dependency applies only when condition holds.
  DERIVED_FROM(z, expression): z is defined by expression.

RULES:
  R1_FUNCTIONAL: y = f(x) and x is a free variable of f => candidate DEPENDS_ON(y,x).
  R2_CONSTANT: f(x) constant over stated domain => no varying functional dependence on x.
  R3_CHAIN: y=f(z), z=g(x) => y=(f∘g)(x); potential indirect dependence, subject to non-constant composition.
  R4_MULTIVARIATE: y=f(x1,...,xn) => record each effective input separately.
  R5_PARTIAL_DERIVATIVE: differentiable f and ∂f/∂x != 0 at a point => local sensitivity to x at that point.
  R6_CONTEXT: dependency claims must retain domain, assumptions, units and model context.
  R7_NONCAUSAL: functional or statistical dependence alone does not imply causation.
  R8_CYCLES: cyclic definitions require simultaneous-equation or fixed-point semantics; do not assume acyclic evaluation.
  R9_UNCERTAINTY: estimated dependencies carry confidence and evidence; logical deductions retain their assumptions.

INVARIANTS:
  I1: Every dependency references declared variables.
  I2: Confidence is within [0,1].
  I3: Function outputs respect declared codomains and units.
  I4: Independence and dependence cannot both be asserted for identical semantics and context without conflict annotation.
  I5: Statistical, functional and causal dependency kinds are not interchangeable.

SEMANTIC_EQUIVALENCE:
  exact: Same dependency kind, variables, direction, domain and assumptions.
  contextual: Equivalent only within a shared explicit context.
  non_equivalence: Correlation != causation; notation similarity != dependency identity.

CONFIDENCE:
  representation: float in [0,1]
  propagation: Rule-dependent; do not treat dependent evidence as independent.
  provenance_required: true

VALIDATION:
  - Check identifiers, types, domains, units and reference integrity.
  - Test model assumptions and distinguish local from global claims.
  - Flag contradictions, unsupported causal interpretations and unresolved cycles.
  - Preserve provenance for all asserted or inferred dependencies.

FLOW_HASH:
  algorithm: SHA-256
  input: Canonical UTF-8 serialisation of ontology version, entities, dependencies, rules and context.
  note: Compute at implementation time; no hash value is asserted here.

UPDATE_MODEL:
  strategy: Versioned, provenance-preserving updates.
  on_change: Revalidate affected dependencies and downstream inferences.
  conflict_policy: Retain competing claims with context and evidence until resolved.

AI_CONSUMPTION:
  format: Typed dependency graph with explicit contexts and provenance.
  tasks: [symbolic_reasoning, model_validation, sensitivity_analysis, dependency_tracing, knowledge_graph_alignment]
  safeguards: [no_unjustified_causality, no_context_loss, uncertainty_preservation]

END FLOW ALGEBRA