Symbolic Differentiation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Symbolic Differentiation Ontology within the Kireth Research Semantic Library. Symbolic differentiation concerns the exact algorithmic derivation of derivatives from formal mathematical expressions. This ontology supports differentiation rules, expression trees, variables and constants, product and quotient rules, chain rule, higher and partial derivatives, gradients, Jacobians, Hessians, implicit differentiation, piecewise and branch-aware derivatives, simplification, exactness, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Symbolic Differentiation Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Symbolic Computation / Symbolic Differentiation"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/symbolic-computation/symbolic-differentiation-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SymbolicDifferentiation
 ENTITY_CLASS = ExactDerivativeComputationFramework
 PRIMARY_CONTEXT = SymbolicMathematics
 DEFINITION = "Symbolic differentiation is the exact derivation of derivative expressions from formal mathematical expressions using differentiation rules, algebraic structure and declared assumptions."
 SYMBOLIC_FORMS = [
  "D_x[f(x)]",
  "D_x[u+v]=D_x[u]+D_x[v]",
  "D_x[uv]=u'v+uv'",
  "D_x[f(g(x))]=f'(g(x))*g'(x)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for derivative operators, differentiation rules, higher derivatives, partial derivatives, gradient, Jacobian, Hessian, implicit differentiation, simplification and verification."
}

SCOPE {
 INCLUDE = [
  SymbolicDifferentiation,Differentiation,
  Derivative,SymbolicDerivative,
  DifferentialOperator,DOperator,
  Variable,IndependentVariable,
  DependentVariable,Parameter,
  Constant,ExactConstant,
  Expression,SymbolicExpression,
  ExpressionTree,AbstractSyntaxTree,AST,
  Function,SymbolicFunction,
  CompositeFunction,
  UnaryFunction,BinaryFunction,
  IdentityFunction,
  ConstantFunction,
  PolynomialFunction,
  RationalFunction,
  ExponentialFunction,
  LogarithmicFunction,
  TrigonometricFunction,
  InverseTrigonometricFunction,
  HyperbolicFunction,
  InverseHyperbolicFunction,
  AbsoluteValueFunction,
  PiecewiseFunction,
  PowerFunction,
  GeneralPower,
  SumRule,
  DifferenceRule,
  ConstantMultipleRule,
  ProductRule,
  QuotientRule,
  ChainRule,
  GeneralisedProductRule,
  LeibnizRule,
  PowerRule,
  GeneralPowerRule,
  ExponentialDerivativeRule,
  LogarithmicDerivativeRule,
  TrigonometricDerivativeRule,
  InverseTrigonometricDerivativeRule,
  HyperbolicDerivativeRule,
  InverseHyperbolicDerivativeRule,
  AbsoluteValueDerivativeRelationship,
  PiecewiseDerivative,
  HigherDerivative,
  SecondDerivative,
  ThirdDerivative,
  NthDerivative,
  PartialDerivative,
  MixedPartialDerivative,
  HigherPartialDerivative,
  DirectionalDerivative,
  Gradient,
  SymbolicGradient,
  Jacobian,
  SymbolicJacobian,
  Hessian,
  SymbolicHessian,
  LaplacianRelationship,
  DivergenceRelationship,
  CurlRelationship,
  TotalDerivative,
  ImplicitDifferentiation,
  LogarithmicDifferentiation,
  ParametricDifferentiation,
  VectorValuedFunction,
  ScalarValuedFunction,
  MultivariableFunction,
  MatrixValuedFunctionRelationship,
  TensorValuedFunctionRelationship,
  FréchetDerivativeRelationship,
  GateauxDerivativeRelationship,
  Differential,
  DifferentialFormRelationship,
  CriticalPointRelationship,
  StationaryPointRelationship,
  MonotonicityRelationship,
  ConvexityRelationship,
  ConcavityRelationship,
  TaylorSeriesRelationship,
  LinearApproximation,
  LocalLinearisation,
  SensitivityRelationship,
  ChainOfDependencies,
  DependencyGraphRelationship,
  SymbolOccurrence,
  FreeVariable,
  BoundVariable,
  Assumption,AssumptionContext,
  Domain,RealDomain,ComplexDomain,
  PositivityAssumption,
  NonzeroAssumption,
  BranchCondition,
  PrincipalBranchRelationship,
  RewriteRule,
  DifferentiationRule,
  RuleApplication,
  PatternMatching,
  StructuralRecursion,
  RecursiveDifferentiation,
  TreeTraversal,
  Memoisation,
  CommonSubexpression,
  CommonSubexpressionEliminationRelationship,
  Simplification,
  DerivativeSimplification,
  AlgebraicSimplification,
  TrigonometricSimplification,
  RationalSimplification,
  CanonicalForm,
  NormalForm,
  Expansion,
  Factorisation,
  Collection,
  Substitution,
  SymbolicAlgebra,
  SymbolicMathematics,
  SymbolicComputation,
  ComputerAlgebraSystem,CAS,
  ExactArithmetic,
  ExactDerivative,
  AutomaticDifferentiationRelationship,
  ForwardModeADRelationship,
  ReverseModeADRelationship,
  NumericalDifferentiationRelationship,
  FiniteDifferenceRelationship,
  SymbolicNumericDifferentiationRelationship,
  ExpressionSwell,
  IntermediateExpressionGrowth,
  ComputationalComplexity,
  TimeComplexity,
  SpaceComplexity,
  RewriteComplexity,
  Verification,
  DifferentiateAndSubstituteCheck,
  NumericSpotCheck,
  FiniteDifferenceCheckRelationship,
  KnownDerivative,
  ReferenceDerivative,
  IdentityCheck,
  ExactResidual,
  Reproducibility,
  Provenance,
  RuleSetVersion,
  CASVersion,
  AlgorithmVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  NumericalDifferentiationWhenApproximateFiniteDifferencesArePrimary,
  AutomaticDifferentiationWhenComputationalGraphPropagationIsPrimary,
  DifferentialGeometryWhenGeometricStructureIsPrimary,
  CalculusTheoryWithoutComputationalSymbolicDerivation
 ]
 DISAMBIGUATION_RULE = "When an exact derivative expression is derived from a formal symbolic expression using differentiation rules, interpret the context as symbolic differentiation."
}

CORE_CONCEPTS {
 SymbolicDifferentiation = "Exact algorithmic derivation of derivative expressions."
 Derivative = "Rate-of-change expression associated with a function and variable."
 DifferentiationRule = "Formal transformation rule for deriving a derivative."
 ChainRule = "Rule differentiating function composition."
 PartialDerivative = "Derivative with respect to one variable while others are held fixed."
 Gradient = "Vector of first partial derivatives of a scalar-valued function."
 Jacobian = "Matrix of first partial derivatives of a vector-valued function."
 Hessian = "Matrix of second partial derivatives of a scalar-valued function."
}

RELATIONS {
 SymbolicDifferentiation --is_child_of--> SymbolicMathematics
 SymbolicDifferentiation --uses--> DifferentiationRules
 ExpressionTree --represents--> SymbolicExpression
 SumRule --applies_to--> SumExpression
 ProductRule --applies_to--> ProductExpression
 QuotientRule --applies_to--> QuotientExpression
 ChainRule --applies_to--> CompositeFunction
 HigherDerivative --derived_from--> RepeatedDifferentiation
 Gradient --contains--> PartialDerivatives
 Jacobian --contains--> PartialDerivatives
 Hessian --contains--> SecondPartialDerivatives
 ImplicitDifferentiation --uses--> EquationRelationship
 Simplification --reduces--> DerivativeExpression
 Verification --checks--> DerivedDerivative
}

OPERATORS {
 D(expr,x) := symbolic_derivative(expr,x)
 D_N(expr,x,n) := repeat(D(expr,x),n)
 PARTIAL(expr,x_i) := symbolic_partial_derivative(expr,x_i)
 GRADIENT(f,vars) := [PARTIAL(f,v) for v in vars]
 JACOBIAN(F,vars) := matrix(PARTIAL(F_i,v_j))
 HESSIAN(f,vars) := matrix(PARTIAL(PARTIAL(f,v_i),v_j))
 IMPLICIT_DERIVATIVE(F(x,y)=0,x,y) := -F_x/F_y
 SIMPLIFY_DERIVATIVE(expr) := simplify(symbolic_derivative_expression)
}

AXIOMS {
 A1 = "Symbolic differentiation transforms formal expressions into exact derivative expressions."
 A2 = "Differentiation rules apply according to expression structure."
 A3 = "The chain rule is required for nontrivial function composition."
 A4 = "Higher derivatives arise through repeated derivative operations."
 A5 = "Partial derivatives depend on the declared independent variables."
 A6 = "Branch-sensitive functions can require domain conditions for valid derivatives."
 A7 = "Equivalent derivative expressions can differ syntactically."
 A8 = "Symbolic-differentiation results require explicit variable, domain, rule, simplification and verification semantics."
}

REASONING {
 R1_CONTEXT = identify(expression,differentiation_variable,other_variables,domain,assumptions,target_derivative_order)
 R2_PARSE = construct_expression_tree)
 R3_DEPENDENCIES = identify_variable_occurrence_and_function_dependencies)
 R4_RULE = select_constant_sum_product_quotient_power_chain_or_function_specific_rule)
 R5_RECURSE = differentiate_subexpressions_recursively)
 R6_COMBINE = assemble_derivative_expression)
 R7_SIMPLIFY = simplify_under_declared_assumptions)
 R8_HIGHER = if_order_greater_than_one repeat_derivative_process)
 R9_VERIFY = compare_with_known_derivative_substitution_or_numeric_spot_checks)
 R10_REPRODUCE = document_rules_assumptions_CAS_and_versions)
}

CONSTRAINTS {
 C1 = "Differentiation with respect to an absent variable returns zero only when independence is declared."
 C2 = "Absolute-value and piecewise derivatives can require domain partitioning."
 C3 = "Complex logarithms and powers can require branch-aware differentiation."
 C4 = "Repeated symbolic differentiation can cause severe expression swell."
 C5 = "Unsimplified derivative expressions can be mathematically correct but computationally inefficient."
 C6 = "Implicit differentiation requires nonzero denominator conditions where division is introduced."
 C7 = "Mixed partial derivatives need not commute without sufficient regularity assumptions."
 C8 = "Symbolic differentiation does not by itself establish differentiability at every point."
 C9 = "A compact original expression can yield a large derivative expression."
 C10 = "Automatic and symbolic differentiation are related but distinct computational models."
}

SEMANTIC_EQUIVALENCE {
 "SymbolicDerivative" <=> "ExactDerivative"
 "DifferentialOperator" <=> "DOperator"
 "ComputerAlgebraSystem" <=> "CAS"
 "AbstractSyntaxTree" <=> "AST"
}

VALIDATION {
 V1 = "Differentiation variable is explicit."
 V2 = "Expression domain and assumptions are documented."
 V3 = "Each rule application matches the expression structure."
 V4 = "Function-specific derivatives use valid domain conditions."
 V5 = "Higher derivative order is documented."
 V6 = "Gradient, Jacobian and Hessian dimensions are checked."
 V7 = "Implicit derivatives preserve denominator conditions."
 V8 = "Results are compared with known derivatives or numerical spot checks where feasible."
 V9 = "Simplification preserves derivative equivalence."
 V10 = "Results are reproducible under documented rule set, CAS and algorithm versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SYMBOLIC_DIFFERENTIATION_DEFINITION = 0.99
 RULE_REASONING = 0.99
 CHAIN_RULE_REASONING = 0.99
 HIGHER_DERIVATIVE_REASONING = 0.99
 PARTIAL_DERIVATIVE_REASONING = 0.99
 IMPLICIT_DIFFERENTIATION_REASONING = 0.98
 BRANCH_REASONING = 0.98
 POLICY = "Reduce confidence when variable dependencies, domain, differentiability assumptions, branch conditions or derivative order are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,SymbolicDifferentiationReasoning,
  SumRuleReasoning,ProductRuleReasoning,QuotientRuleReasoning,
  ChainRuleReasoning,PowerRuleReasoning,
  HigherDerivativeReasoning,PartialDerivativeReasoning,
  GradientReasoning,JacobianReasoning,HessianReasoning,
  ImplicitDifferentiationReasoning,
  DerivativeSimplificationReasoning,
  BranchConditionReasoning,VerificationReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "symbolic differentiation" -> SymbolicDifferentiation
  "symbolic derivative" -> SymbolicDerivative
  "CAS" -> ComputerAlgebraSystem
  "AST" -> AbstractSyntaxTree
 }
}

EXECUTION_RULES {
 E1 = "If an exact derivative is derived from a symbolic expression, activate SymbolicDifferentiationOntology."
 E2 = "Identify differentiation variable and dependency structure before applying rules."
 E3 = "Apply differentiation rules according to expression-tree structure."
 E4 = "Use the chain rule for composed functions."
 E5 = "Preserve domain and branch conditions for logarithmic, radical, absolute-value and piecewise expressions."
 E6 = "Simplify derivative expressions without changing their mathematical meaning."
 E7 = "Monitor expression swell for higher-order derivatives."
 E8 = "Verify results using known identities, substitution or numerical spot checks where feasible."
 E9 = "Document assumptions, rule sets and CAS versions for reproducibility."
}

RECURSION {
 ENTRY = SymbolicDifferentiation
 EXPAND(SymbolicDifferentiation) -> [
  Derivative,DifferentialOperator,Variable,Constant,ExpressionTree,
  Function,CompositeFunction,SumRule,DifferenceRule,ConstantMultipleRule,
  ProductRule,QuotientRule,ChainRule,PowerRule,
  ExponentialDerivativeRule,LogarithmicDerivativeRule,
  TrigonometricDerivativeRule,InverseTrigonometricDerivativeRule,
  HigherDerivative,SecondDerivative,NthDerivative,
  PartialDerivative,MixedPartialDerivative,DirectionalDerivative,
  Gradient,Jacobian,Hessian,TotalDerivative,
  ImplicitDifferentiation,LogarithmicDifferentiation,
  PiecewiseDerivative,Assumption,BranchCondition,
  RewriteRule,RecursiveDifferentiation,Simplification,
  CanonicalForm,SymbolicAlgebra,SymbolicMathematics,CAS,
  ExpressionSwell,Verification,KnownDerivative,Reproducibility
 ]
 EXPAND(SymbolicDifferentiationFamilies) -> [
  ElementaryDifferentiationRules,
  CompositionAndChainRule,
  HigherOrderDifferentiation,
  MultivariableDifferentiation,
  GradientJacobianAndHessianComputation,
  ImplicitAndParametricDifferentiation,
  PiecewiseAndBranchAwareDifferentiation,
  VerificationAndSimplification
 ]
 STOP_CONDITION = "Stop when variable dependencies, rules, derivative order, domain conditions, simplification and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Symbolic Mathematics Ontology"
 RELATED_ONTOLOGIES = [
  "Symbolic Mathematics Ontology",
  "Symbolic Computation",
  "Symbolic Algebra Ontology",
  "Calculus and Analysis Ontologies",
  "Derivative Ontology",
  "Function Ontology",
  "Automatic Differentiation Ontology",
  "Numerical Differentiation Ontology 2",
  "Symbolic Manipulation Ontology",
  "Computer Algebra Systems Ontology",
  "Computational Complexity Ontology"
 ]
 ROUTE_RULES = [
  "Symbolic Differentiation -> Symbolic Mathematics for parent semantics",
  "Symbolic Differentiation -> Symbolic Computation for execution semantics",
  "Symbolic Differentiation -> Symbolic Algebra for expression simplification",
  "Symbolic Differentiation -> Calculus and Derivative ontologies for mathematical derivative foundations",
  "Symbolic Differentiation -> Function Ontology for function composition and dependency semantics",
  "Symbolic Differentiation -> Automatic Differentiation for computational-graph derivative relationships",
  "Symbolic Differentiation -> Numerical Differentiation for approximate derivative comparison",
  "Symbolic Differentiation -> Symbolic Manipulation and CAS for rewrite implementation",
  "Symbolic Differentiation -> Computational Complexity for expression-growth costs"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible derivative rules, function classes or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve exact-symbolic-versus-numerical differentiation distinction.",
  "Preserve symbolic-versus-automatic differentiation distinction.",
  "Preserve first-versus-higher derivative distinction.",
  "Preserve ordinary-versus-partial derivative distinction.",
  "Preserve formal-derivative-versus-pointwise-differentiability distinction.",
  "Preserve syntax-versus-derivative-equivalence distinction.",
  "Preserve machine-readable symbolic-differentiation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Symbolic differentiation derives exact derivative expressions by recursively applying formal differentiation rules to symbolic expressions."
 RULE_SUMMARY = "Constant, sum, product, quotient, power and chain rules form the core transformation system."
 MULTIVARIABLE_SUMMARY = "Partial derivatives combine into gradients, Jacobians and Hessians for multivariable functions."
 HIGHER_ORDER_SUMMARY = "Repeated symbolic differentiation produces higher-order derivatives but can cause expression growth."
 CONDITION_SUMMARY = "Piecewise, absolute-value, logarithmic and complex-valued expressions can require explicit domain and branch conditions."
 VALIDATION_SUMMARY = "Symbolic derivatives should be checked through known identities, substitution, inverse reasoning or numerical spot checks under declared assumptions."
}

FLOW {
 INPUT
 -> detect_symbolic_differentiation_context
 -> identify_expression_variable_dependencies_domain_assumptions_and_derivative_order
 -> parse_expression_into_tree_or_DAG
 -> classify_expression_node
 -> select_constant_sum_product_quotient_power_chain_or_function_specific_rule
 -> recursively_differentiate_required_subexpressions
 -> combine_derivative_terms
 -> apply_domain_and_branch_conditions
 -> simplify_derivative_expression
 -> if_higher_order_repeat_differentiation
 -> if_multivariable_construct_gradient_Jacobian_or_Hessian_as_required
 -> verify_against_known_derivative_substitution_or_numeric_spot_check
 -> document_assumptions_rule_set_CAS_versions_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-SYMBOLIC-COMPUTATION-SYMBOLIC-DIFFERENTIATION-ONTOLOGY-V1"
}

}

END_FLOW