Symbolic Integration Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Symbolic Integration Ontology within the Kireth Research Semantic Library. Symbolic integration concerns the exact algorithmic derivation of antiderivatives, definite integrals and transformed integral expressions from formal mathematical expressions. This ontology supports elementary integration rules, substitution, integration by parts, rational-function integration, partial fractions, trigonometric and exponential integrals, definite integration, parameterised integrals, improper integrals, special functions, the Risch-algorithm relationship, assumptions, branch conditions, simplification, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = SymbolicIntegration
 ENTITY_CLASS = ExactAntiderivativeComputationFramework
 PRIMARY_CONTEXT = SymbolicMathematics
 DEFINITION = "Symbolic integration is the exact algorithmic derivation or transformation of integral expressions using formal integration rules, algebraic structure, assumptions and symbolic algorithms."
 SYMBOLIC_FORMS = [
  "∫ f(x) dx",
  "D_x[F(x)] = f(x)",
  "∫_a^b f(x) dx = F(b)-F(a)",
  "∫ u dv = uv - ∫ v du"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for indefinite and definite integration, antiderivatives, substitution, integration by parts, rational integration, special functions, branch conditions, simplification and verification."
}

SCOPE {
 INCLUDE = [
  SymbolicIntegration,Integration,
  Integral,SymbolicIntegral,
  IndefiniteIntegral,DefiniteIntegral,
  Antiderivative,PrimitiveFunction,
  Integrand,IntegrationVariable,
  LowerLimit,UpperLimit,
  Bounds,BoundedIntegral,
  ImproperIntegral,
  ParameterisedIntegral,
  MultipleIntegralRelationship,
  RepeatedIntegralRelationship,
  LineIntegralRelationship,
  SurfaceIntegralRelationship,
  VolumeIntegralRelationship,
  ExactIntegration,
  ElementaryAntiderivative,
  ClosedFormIntegral,
  NonElementaryIntegral,
  UnevaluatedIntegral,
  IntegralTransformRelationship,
  Expression,SymbolicExpression,
  ExpressionTree,AbstractSyntaxTree,AST,
  Function,SymbolicFunction,
  PolynomialFunction,
  RationalFunction,
  ExponentialFunction,
  LogarithmicFunction,
  TrigonometricFunction,
  InverseTrigonometricFunction,
  HyperbolicFunction,
  AlgebraicFunction,
  RadicalFunction,
  PiecewiseFunction,
  AbsoluteValueFunction,
  ProductFunction,
  CompositeFunction,
  SumRuleForIntegration,
  ConstantMultipleRuleForIntegration,
  PowerRuleForIntegration,
  SubstitutionRule,
  USubstitution,
  ChangeOfVariable,
  IntegrationByParts,
  TabularIntegrationRelationship,
  ReductionFormulaRelationship,
  PartialFractionDecomposition,
  RationalFunctionIntegration,
  PolynomialIntegration,
  TrigonometricIntegration,
  TrigonometricSubstitution,
  ExponentialIntegration,
  LogarithmicIntegration,
  HyperbolicIntegration,
  AlgebraicFunctionIntegration,
  RadicalIntegration,
  CompletingTheSquareRelationship,
  WeierstrassSubstitutionRelationship,
  EulerSubstitutionRelationship,
  HermiteReductionRelationship,
  RothsteinTragerRelationship,
  RischAlgorithm,
  LiouvilleTheoremRelationship,
  ElementaryFunctionIntegration,
  DifferentialFieldRelationship,
  SpecialFunction,
  ErrorFunction,Erf,
  ExponentialIntegral,Ei,
  LogarithmicIntegralRelationship,
  SineIntegral,Si,
  CosineIntegral,Ci,
  GammaFunctionRelationship,
  IncompleteGammaRelationship,
  BetaFunctionRelationship,
  PolylogarithmRelationship,
  EllipticIntegralRelationship,
  FresnelIntegralRelationship,
  MeijerGRelationship,
  HypergeometricFunctionRelationship,
  PiecewiseIntegral,
  ConditionalIntegral,
  PrincipalValueIntegralRelationship,
  CauchyPrincipalValueRelationship,
  BranchCut,
  PrincipalBranch,
  BranchCondition,
  Assumption,AssumptionContext,
  RealAssumption,
  PositivityAssumption,
  NonzeroAssumption,
  ParameterAssumption,
  ConvergenceCondition,
  Domain,RealDomain,ComplexDomain,
  SingularPoint,
  Pole,
  EndpointSingularity,
  InfiniteInterval,
  ConvergentIntegral,
  DivergentIntegral,
  AbsoluteConvergenceRelationship,
  ConditionalConvergenceRelationship,
  FundamentalTheoremOfCalculus,
  NewtonLeibnizFormula,
  DifferentiationUnderIntegralSignRelationship,
  LeibnizIntegralRuleRelationship,
  IntegrationConstant,
  ConstantOfIntegration,
  SymbolicDifferentiation,
  DerivativeCheck,
  IntegrateThenDifferentiate,
  DifferentiateThenIntegrateRelationship,
  Simplification,
  IntegralSimplification,
  AlgebraicSimplification,
  TrigonometricSimplification,
  RationalSimplification,
  Expansion,
  Factorisation,
  Substitution,
  CanonicalForm,
  NormalForm,
  PatternMatching,
  RewriteRule,
  IntegrationRule,
  RuleBasedIntegration,
  HeuristicIntegration,
  TableLookupRelationship,
  RecursiveIntegration,
  Memoisation,
  CommonSubexpressionRelationship,
  ExactArithmetic,
  AlgebraicNumberRelationship,
  RationalArithmetic,
  ComputerAlgebraSystem,CAS,
  SymbolicComputation,
  SymbolicMathematics,
  SymbolicAlgebra,
  SymbolicNumericIntegrationRelationship,
  NumericalIntegrationRelationship,
  QuadratureRelationship,
  AdaptiveIntegrationRelationship,
  ExpressionSwell,
  IntermediateExpressionGrowth,
  ComputationalComplexity,
  TimeComplexity,
  SpaceComplexity,
  Verification,
  DifferentiateResult,
  BoundaryEvaluationCheck,
  NumericSpotCheck,
  ReferenceIntegral,
  KnownIntegral,
  ExactResidual,
  Reproducibility,
  Provenance,
  RuleSetVersion,
  CASVersion,
  AlgorithmVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  NumericalIntegrationWhenQuadratureApproximationIsPrimary,
  SymbolicDifferentiationWhenDerivativeDerivationIsPrimary,
  MeasureTheoryWithoutComputationalIntegrationContext,
  IntegralTransformsWhenTransformSemanticsArePrimaryRatherThanAntiderivativeComputation
 ]
 DISAMBIGUATION_RULE = "When an exact antiderivative or symbolic integral transformation is derived from a formal expression using integration rules and symbolic algorithms, interpret the context as symbolic integration."
}

CORE_CONCEPTS {
 SymbolicIntegration = "Exact algorithmic manipulation and evaluation of integrals."
 IndefiniteIntegral = "Family of antiderivatives differing by a constant."
 DefiniteIntegral = "Integral evaluated over specified bounds."
 Antiderivative = "Function whose derivative equals the integrand."
 SubstitutionRule = "Integration transformation based on a change of variable."
 IntegrationByParts = "Integration transformation derived from the product rule."
 RationalFunctionIntegration = "Exact integration of rational functions through algebraic decomposition."
 NonElementaryIntegral = "Integral not representable by elementary functions in the chosen domain."
}

RELATIONS {
 SymbolicIntegration --is_child_of--> SymbolicMathematics
 SymbolicIntegration --uses--> IntegrationRules
 IndefiniteIntegral --produces--> AntiderivativeFamily
 DefiniteIntegral --uses--> Bounds
 FundamentalTheoremOfCalculus --relates--> DifferentiationAndIntegration
 SubstitutionRule --transforms--> IntegrationVariable
 IntegrationByParts --derives_from--> ProductRule
 PartialFractionDecomposition --supports--> RationalFunctionIntegration
 RischAlgorithm --decides_or_constructs--> ElementaryAntiderivativeRelationship
 SpecialFunction --can_represent--> NonElementaryIntegral
 Assumption --constrains--> IntegralTransformation
 Simplification --reduces--> IntegralResultExpression
 Verification --checks--> SymbolicIntegralResult
}

OPERATORS {
 INTEGRATE(expr,x) := symbolic_antiderivative(expr,x)
 DEFINITE_INTEGRAL(expr,x,a,b) := evaluate_symbolic_integral(expr,x,a,b)
 SUBSTITUTE_INTEGRAL(expr,x,g(t)) := transform_integrand_and_differential(expr,x,g,t)
 INTEGRATE_BY_PARTS(u,dv) := u*v - INTEGRATE(v*du)
 PARTIAL_FRACTION_INTEGRATE(rational_expr,x) := integrate(partial_fraction_decompose(rational_expr),x)
 VERIFY_ANTIDERIVATIVE(F,f,x) := simplify(D(F,x)-f) == 0
 EVALUATE_BOUNDS(F,a,b) := F(b)-F(a)
}

AXIOMS {
 A1 = "Symbolic integration seeks exact antiderivatives or exact transformed integral representations."
 A2 = "An indefinite integral is defined only up to an additive constant."
 A3 = "Integration rules must preserve equality under declared domain and convergence conditions."
 A4 = "Not every elementary integrand has an elementary antiderivative."
 A5 = "Definite integration can be possible even when no elementary antiderivative exists."
 A6 = "Branch cuts and singularities can affect complex and improper integrals."
 A7 = "Differentiating a valid antiderivative recovers the integrand on its valid domain."
 A8 = "Symbolic-integration results require explicit variable, domain, assumptions, convergence, transformation and verification semantics."
}

REASONING {
 R1_CONTEXT = identify(integrand,integration_variable,bounds,domain,assumptions,target_form,exactness_goal)
 R2_PARSE = construct_expression_tree)
 R3_ANALYSE = classify(polynomial,rational,exponential,logarithmic,trigonometric,algebraic,composite_or_special_form)
 R4_RULE = choose(direct_rule,substitution,parts,partial_fractions,trigonometric_substitution,Hermite,Risch_or_special_function_path)
 R5_CONDITIONS = derive_domain_branch_singularity_and_convergence_conditions)
 R6_EXECUTE = apply_symbolic_integration_algorithm)
 R7_SIMPLIFY = simplify_result_under_declared_assumptions)
 R8_BOUNDS = if_definite evaluate_bounds_or_preserve_exact_integral_representation)
 R9_VERIFY = differentiate_result_or_compare_with_reference_and_numeric_spot_checks)
 R10_REPRODUCE = document_rules_assumptions_CAS_and_versions)
}

CONSTRAINTS {
 C1 = "An antiderivative may require special functions rather than elementary functions."
 C2 = "Substitution requires correct transformation of the differential and bounds."
 C3 = "Integration by parts can increase complexity if the decomposition is poorly chosen."
 C4 = "Improper integrals require explicit convergence analysis."
 C5 = "Complex logarithms, powers and radicals require branch-aware integration."
 C6 = "Piecewise and absolute-value integrands can require interval decomposition."
 C7 = "Parameterised integrals can change convergence regime across parameter values."
 C8 = "Expression swell can make exact symbolic integration expensive."
 C9 = "Differentiation verification establishes local antiderivative correctness but does not alone settle endpoint or branch issues."
 C10 = "Numerical agreement at sample points does not prove symbolic equality."
}

SEMANTIC_EQUIVALENCE {
 "SymbolicIntegral" <=> "ExactSymbolicIntegral"
 "Antiderivative" <=> "PrimitiveFunction"
 "ComputerAlgebraSystem" <=> "CAS"
 "AbstractSyntaxTree" <=> "AST"
 "ConstantOfIntegration" <=> "IntegrationConstant"
}

VALIDATION {
 V1 = "Integration variable is explicit."
 V2 = "Bounds are documented for definite integrals."
 V3 = "Domain and assumptions are documented."
 V4 = "Substitution transformations include the differential and transformed bounds."
 V5 = "Improper integrals include convergence conditions."
 V6 = "Antiderivatives are checked by differentiation where feasible."
 V7 = "Definite results are checked by boundary evaluation or independent methods where feasible."
 V8 = "Branch-sensitive results document conditions or piecewise cases."
 V9 = "Exact output remains exact unless numerical approximation is explicitly requested."
 V10 = "Results are reproducible under documented rule set, CAS and algorithm versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SYMBOLIC_INTEGRATION_DEFINITION = 0.99
 SUBSTITUTION_REASONING = 0.99
 INTEGRATION_BY_PARTS_REASONING = 0.99
 RATIONAL_INTEGRATION_REASONING = 0.99
 RISCH_RELATIONSHIP_REASONING = 0.98
 IMPROPER_INTEGRAL_REASONING = 0.98
 BRANCH_REASONING = 0.98
 POLICY = "Reduce confidence when domain, assumptions, convergence, branch conditions, parameter regimes or transformation details are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,SymbolicIntegrationReasoning,
  AntiderivativeReasoning,DefiniteIntegralReasoning,
  SubstitutionReasoning,IntegrationByPartsReasoning,
  PartialFractionReasoning,RationalIntegrationReasoning,
  TrigonometricIntegrationReasoning,SpecialFunctionReasoning,
  ImproperIntegralReasoning,ConvergenceReasoning,
  BranchConditionReasoning,IntegralSimplificationReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "symbolic integration" -> SymbolicIntegration
  "symbolic integral" -> SymbolicIntegral
  "antiderivative" -> Antiderivative
  "primitive function" -> PrimitiveFunction
  "CAS" -> ComputerAlgebraSystem
  "AST" -> AbstractSyntaxTree
 }
}

EXECUTION_RULES {
 E1 = "If an exact antiderivative or symbolic integral transformation is derived, activate SymbolicIntegrationOntology."
 E2 = "Identify integration variable, bounds, domain and assumptions before choosing an algorithm."
 E3 = "Classify the integrand before applying direct, substitution, parts, rational or special-function methods."
 E4 = "Transform both differential and bounds during variable substitution."
 E5 = "Preserve branch and convergence conditions for singular, complex and improper integrals."
 E6 = "Return unevaluated or special-function forms rather than inventing an elementary antiderivative."
 E7 = "Simplify results without losing domain conditions."
 E8 = "Verify antiderivatives by differentiation where feasible."
 E9 = "Document assumptions, rule sets and CAS versions for reproducibility."
}

RECURSION {
 ENTRY = SymbolicIntegration
 EXPAND(SymbolicIntegration) -> [
  Integral,IndefiniteIntegral,DefiniteIntegral,Antiderivative,Integrand,
  IntegrationVariable,Bounds,ImproperIntegral,ParameterisedIntegral,
  ElementaryAntiderivative,NonElementaryIntegral,UnevaluatedIntegral,
  SumRuleForIntegration,ConstantMultipleRuleForIntegration,PowerRuleForIntegration,
  SubstitutionRule,USubstitution,ChangeOfVariable,IntegrationByParts,
  PartialFractionDecomposition,RationalFunctionIntegration,PolynomialIntegration,
  TrigonometricIntegration,TrigonometricSubstitution,ExponentialIntegration,
  LogarithmicIntegration,HermiteReduction,RischAlgorithm,SpecialFunction,
  ErrorFunction,ExponentialIntegral,SineIntegral,CosineIntegral,
  PiecewiseIntegral,BranchCondition,ConvergenceCondition,
  FundamentalTheoremOfCalculus,ConstantOfIntegration,
  SymbolicDifferentiation,DerivativeCheck,Simplification,
  RewriteRule,RuleBasedIntegration,ComputerAlgebraSystem,CAS,
  ExpressionSwell,Verification,ReferenceIntegral,Reproducibility
 ]
 EXPAND(SymbolicIntegrationFamilies) -> [
  ElementaryIntegrationRules,
  SubstitutionMethods,
  IntegrationByPartsMethods,
  RationalAndPartialFractionIntegration,
  TrigonometricAndAlgebraicIntegration,
  SpecialFunctionAndNonElementaryIntegration,
  DefiniteImproperAndParameterisedIntegration,
  VerificationAndSimplification
 ]
 STOP_CONDITION = "Stop when variable, bounds, domain, convergence, algorithm, result form, simplification and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Symbolic Mathematics Ontology"
 RELATED_ONTOLOGIES = [
  "Symbolic Mathematics Ontology",
  "Symbolic Computation",
  "Symbolic Algebra Ontology",
  "Symbolic Differentiation Ontology",
  "Calculus and Analysis Ontologies",
  "Integral Ontology",
  "Function Ontology",
  "Numerical Integration Ontology 2",
  "Quadrature Ontology",
  "Adaptive Integration Ontology",
  "Symbolic Manipulation Ontology",
  "Computer Algebra Systems Ontology",
  "Computational Complexity Ontology"
 ]
 ROUTE_RULES = [
  "Symbolic Integration -> Symbolic Mathematics for parent semantics",
  "Symbolic Integration -> Symbolic Computation for execution semantics",
  "Symbolic Integration -> Symbolic Algebra for simplification and rational decomposition",
  "Symbolic Integration -> Symbolic Differentiation for antiderivative verification",
  "Symbolic Integration -> Calculus and Integral ontologies for mathematical foundations",
  "Symbolic Integration -> Function Ontology for integrand structure",
  "Symbolic Integration -> Numerical Integration, Quadrature and Adaptive Integration for approximate comparison or fallback relationships",
  "Symbolic Integration -> Symbolic Manipulation and CAS for rewrite implementation",
  "Symbolic Integration -> Computational Complexity for expression-growth costs"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible integration algorithms, function classes or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve indefinite-versus-definite integral distinction.",
  "Preserve elementary-versus-special-function antiderivative distinction.",
  "Preserve symbolic-versus-numerical integration distinction.",
  "Preserve local-antiderivative-versus-global-domain validity distinction.",
  "Preserve proper-versus-improper integral distinction.",
  "Preserve exact-result-versus-approximation distinction.",
  "Preserve machine-readable symbolic-integration semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Symbolic integration derives exact antiderivatives or exact transformed integral representations from symbolic expressions."
 RULE_SUMMARY = "Direct rules, substitution, integration by parts and partial fractions form major elementary integration pathways."
 ALGORITHM_SUMMARY = "Hermite reduction, Risch-style reasoning and special-function representations address more difficult integrands."
 DEFINITE_SUMMARY = "Definite and improper integrals require bound, singularity and convergence semantics in addition to antiderivative computation."
 CONDITION_SUMMARY = "Complex, logarithmic, radical, absolute-value and piecewise integrands can require explicit branch and domain conditions."
 VALIDATION_SUMMARY = "Symbolic integrals should be checked through differentiation, boundary evaluation, independent identities and reproducible CAS settings."
}

FLOW {
 INPUT
 -> detect_symbolic_integration_context
 -> identify_integrand_variable_bounds_domain_assumptions_target_form_and_exactness_goal
 -> parse_expression_into_tree_or_DAG
 -> classify_integrand_structure
 -> analyse_singularities_branches_and_convergence_conditions
 -> choose_direct_rule_substitution_parts_partial_fractions_trigonometric_Hermite_Risch_or_special_function_path
 -> apply_symbolic_integration_algorithm
 -> simplify_result_under_declared_assumptions
 -> if_definite_evaluate_bounds_or_preserve_exact_integral_representation
 -> verify_by_differentiation_boundary_check_reference_identity_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-INTEGRATION-ONTOLOGY-V1"
}

}

END_FLOW