Truncation Error Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Truncation Error Ontology within the Kireth Research Semantic Library. Truncation error arises when an infinite, continuous or exact mathematical process is replaced by a finite approximation. This ontology supports local and global truncation error, Taylor-series truncation, finite-difference error, quadrature error, iterative and series truncation, discretisation relationships, order of accuracy, asymptotic error, consistency, convergence, Richardson extrapolation, adaptive refinement and relationships with numerical error, numerical approximation and numerical analysis.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Truncation Error Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Numerical Methods / Truncation Error"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/numerical-methods/truncation-error-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = TruncationError
 ENTITY_CLASS = FiniteApproximationOmissionErrorFramework
 PRIMARY_CONTEXT = NumericalMethods
 DEFINITION = "Truncation error is the error introduced when an infinite, continuous or exact mathematical process is replaced by a finite approximation."
 SYMBOLIC_FORMS = [
  "f(x+h)=sum_(k=0)^p f^(k)(x)h^k/k! + R_(p+1)",
  "tau_h=L_h(u)-L(u)",
  "error(h)=C*h^p+O(h^(p+1))",
  "E_N=sum_(k=N+1)^infinity a_k"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for errors caused by finite series, finite differences, finite quadrature, finite discretisation, omitted higher-order terms and asymptotic approximation."
}

SCOPE {
 INCLUDE = [
  TruncationError, LocalTruncationError, GlobalTruncationError,
  SeriesTruncationError, TaylorSeriesTruncation, TaylorRemainder,
  RemainderTerm, OmittedTerm, HigherOrderTerm, AsymptoticError,
  LeadingOrderError, ErrorConstant, OrderOfAccuracy, ConsistencyError,
  DiscretisationError, FiniteDifferenceError, ForwardDifferenceError,
  BackwardDifferenceError, CentralDifferenceError, QuadratureError,
  RectangleRuleError, MidpointRuleError, TrapezoidalRuleError,
  SimpsonRuleError, GaussianQuadratureError, InterpolationError,
  PolynomialApproximationError, SpectralTruncationError,
  FourierTruncationError, ModalTruncationError, SeriesApproximation,
  PartialSum, InfiniteSeries, PowerSeries, TaylorSeries, FourierSeries,
  NumericalDifferentiation, NumericalIntegration, NumericalODE,
  NumericalPDE, TimeDiscretisation, SpatialDiscretisation, StepSize,
  TimeStep, SpatialStep, MeshSize, GridSpacing, FiniteDifferenceMethod,
  FiniteVolumeMethod, FiniteElementMethod, SpectralMethod,
  EulerMethod, RungeKuttaMethod, MultistepMethod,
  LocalError, GlobalError, ErrorPropagation, ErrorAccumulation,
  Consistency, Stability, Convergence, ConvergenceRate,
  Refinement, MeshRefinement, TimestepRefinement, HRefinement,
  PRefinement, HPRefinement, RichardsonExtrapolation,
  RombergIntegration, GridConvergenceIndex, GCI, ErrorEstimator,
  APrioriErrorEstimate, APosterioriErrorEstimate, AdaptiveMethod,
  AdaptiveStepSize, AdaptiveMesh, ReferenceSolution, Verification,
  Benchmark, Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  RoundOffErrorWithoutApproximationTruncation,
  MeasurementErrorWithoutNumericalApproximation,
  ModelFormErrorWithoutFiniteApproximation,
  IterationErrorWhenNoTruncatedMathematicalProcessIsPresent
 ]
 DISAMBIGUATION_RULE = "When error is caused by replacing an infinite, continuous or exact mathematical process with a finite approximation, interpret the context as truncation error."
}

CORE_CONCEPTS {
 Truncation-Error = "Error introduced by terminating or discretising an exact mathematical process."
 LocalTruncation-Error = "Error introduced in one numerical step when exact prior information is assumed."
 GlobalTruncation-Error = "Accumulated effect of local truncation errors over the full numerical computation."
 TaylorRemainder = "Difference between a function and a finite Taylor polynomial."
 LeadingOrderError = "Dominant asymptotic truncation-error term as the discretisation parameter tends to zero."
 OrderOfAccuracy = "Power of the discretisation parameter governing leading truncation-error decay."
 Consistency = "Property that local discretisation error tends to zero as resolution is refined."
 RichardsonExtrapolation = "Technique using asymptotic error models at multiple resolutions to improve an approximation."
}

RELATIONS {
 TruncationError --is_child_of--> NumericalError
 TruncationError --is_analysed_by--> NumericalAnalysis
 TaylorSeriesTruncation --produces--> TaylorRemainder
 FiniteDifferenceMethod --introduces--> FiniteDifferenceError
 NumericalIntegration --introduces--> QuadratureError
 TimeDiscretisation --introduces--> TemporalTruncationError
 SpatialDiscretisation --introduces--> SpatialTruncationError
 LocalTruncationError --propagates_to--> GlobalTruncationError
 LeadingOrderError --determines--> OrderOfAccuracy
 Consistency --requires_under_context--> VanishingTruncationError
 Refinement --reduces_under_conditions--> TruncationError
 RichardsonExtrapolation --uses--> LeadingOrderErrorModel
 RombergIntegration --uses--> RichardsonExtrapolation
 GridConvergenceIndex --uses--> RefinementData
 ErrorEstimator --estimates--> TruncationError
 AdaptiveMethod --uses--> TruncationErrorEstimate
}

OPERATORS {
 TAYLOR_REMAINDER(f,x,h,p) := f(x+h)-sum_(k=0)^p f^(k)(x)h^k/k!
 LEADING_ERROR(h,C,p) := E(h)=C*h^p+O(h^(p+1))
 LOCAL_TRUNCATION(exact,scheme,h) := tau_h=(exact_one_step-scheme_one_step)/appropriate_scale
 FINITE_DIFFERENCE_ERROR(f,x,h) := exact_derivative-approximate_derivative
 SERIES_TAIL(a,N) := E_N=sum_(k=N+1)^infinity a_k
 OBSERVED_ORDER(E_h,E_h2,r) := p_obs=log(E_h/E_h2)/log(r)
 RICHARDSON(A_h,A_h2,p) := A_star=A_h2+(A_h2-A_h)/(2^p-1)
}

AXIOMS {
 A1 = "Truncation error is distinct from round-off error."
 A2 = "Truncation error requires a finite approximation to an infinite, continuous or exact mathematical process."
 A3 = "Local and global truncation error are distinct."
 A4 = "Order of accuracy is determined by the leading nonzero truncation-error term."
 A5 = "Consistency requires truncation error to vanish under the relevant refinement limit."
 A6 = "Reducing step size typically reduces truncation error within the method's asymptotic regime."
 A7 = "Truncation-error reduction can eventually expose round-off or other error sources."
 A8 = "Truncation-error claims require explicit approximation, step size, order and regularity assumptions."
}

REASONING {
 R1_CONTEXT = identify(exact_process,finite_approximation,discretisation_parameter,target_quantity)
 R2_EXPANSION = derive(Taylor_series_asymptotic_series_or_exact_remainder_relation)
 R3_CANCEL = identify_terms_reproduced_by_the_numerical_scheme)
 R4_LEADING = identify(first_nonzero_omitted_or_uncancelled_term)
 R5_ORDER = determine(order_of_accuracy)
 R6_LOCAL_GLOBAL = distinguish(local_and_global_truncation_behaviour)
 R7_REFINEMENT = predict_error_decay_under_step_mesh_or_order_refinement)
 R8_EXTRAPOLATION = if_applicable apply_Richardson_or_related_method)
 R9_BALANCE = compare_truncation_with_roundoff_iteration_and_discretisation_components)
 R10_VALIDATION = perform_refinement_study_observed_order_reference_solution_and_reproducibility_checks)
}

CONSTRAINTS {
 C1 = "Taylor-based truncation analysis requires sufficient smoothness."
 C2 = "The leading asymptotic term may not dominate on coarse grids or large timesteps."
 C3 = "Observed error reduction can differ from theoretical order before the asymptotic regime."
 C4 = "Discontinuities, singularities and low regularity can reduce the effective convergence order."
 C5 = "Global error may accumulate local truncation errors differently depending on stability."
 C6 = "Richardson extrapolation requires a valid asymptotic error model."
 C7 = "Reducing step size cannot indefinitely improve total accuracy because other errors can dominate."
 C8 = "Adaptive refinement requires suitable local error indicators."
 C9 = "Series-tail bounds depend on convergence properties of the underlying series."
 C10 = "Truncation error must not be conflated with model-form or measurement error."
}

SEMANTIC_EQUIVALENCE {
 "TruncationError" <=> "FiniteApproximationTruncationError"
 "TaylorRemainder" <=> "TaylorSeriesRemainder"
 "LeadingOrderError" <=> "DominantTruncationErrorTerm"
 "OrderOfAccuracy" <=> "TruncationOrder"
 "GridConvergenceIndex" <=> "GCI"
}

VALIDATION {
 V1 = "Exact or reference mathematical process is explicit."
 V2 = "Finite approximation is explicit."
 V3 = "Step size, mesh size or truncation index is documented."
 V4 = "Regularity assumptions are stated."
 V5 = "Leading truncation term is derived or estimated."
 V6 = "Theoretical order is compared with observed order."
 V7 = "Local and global error are distinguished."
 V8 = "Refinement studies verify asymptotic behaviour where feasible."
 V9 = "Round-off dominance is checked at fine resolutions."
 V10 = "Truncation-error results are reproducible under documented settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TRUNCATION_ERROR_DEFINITION = 0.99
 TAYLOR_REMAINDER_REASONING = 0.99
 LOCAL_TRUNCATION_REASONING = 0.99
 GLOBAL_TRUNCATION_REASONING = 0.99
 ORDER_REASONING = 0.99
 REFINEMENT_REASONING = 0.99
 RICHARDSON_REASONING = 0.98
 SERIES_TAIL_REASONING = 0.98
 POLICY = "Reduce confidence when smoothness, discretisation parameter, approximation formula, asymptotic regime, error norm or reference solution are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, TruncationErrorReasoning,
  LocalTruncationErrorReasoning, GlobalTruncationErrorReasoning,
  TaylorRemainderReasoning, FiniteDifferenceErrorReasoning,
  QuadratureErrorReasoning, SeriesTruncationReasoning,
  OrderOfAccuracyReasoning, ConsistencyReasoning,
  RefinementReasoning, RichardsonExtrapolationReasoning,
  GridConvergenceReasoning, AdaptiveErrorReasoning,
  NumericalErrorReasoning, NumericalConvergenceReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "truncation error" -> TruncationError
  "local truncation error" -> LocalTruncationError
  "global truncation error" -> GlobalTruncationError
  "Taylor remainder" -> TaylorRemainder
  "order of accuracy" -> OrderOfAccuracy
  "GCI" -> GridConvergenceIndex
 }
}

EXECUTION_RULES {
 E1 = "If finite approximation of an infinite or continuous process causes the error, activate TruncationErrorOntology."
 E2 = "If a numerical formula is derived from Taylor expansion, identify the first omitted nonzero term."
 E3 = "If a one-step method is analysed, distinguish local from global truncation error."
 E4 = "If resolution is refined, estimate observed order and compare with theory."
 E5 = "If an asymptotic error model is valid, consider Richardson extrapolation."
 E6 = "If error reduction stalls at fine resolution, test for round-off domination."
 E7 = "If regularity is limited, reduce expected convergence order appropriately."
 E8 = "If adaptive methods are used, connect local error estimates to refinement."
 E9 = "Any truncation-error claim should be validated using refinement, observed order and reference solutions."
}

RECURSION {
 ENTRY = TruncationError
 EXPAND(TruncationError) -> [
  LocalTruncationError, GlobalTruncationError, SeriesTruncationError,
  TaylorSeriesTruncation, TaylorRemainder, RemainderTerm, OmittedTerm,
  HigherOrderTerm, AsymptoticError, LeadingOrderError, ErrorConstant,
  OrderOfAccuracy, ConsistencyError, DiscretisationError,
  FiniteDifferenceError, ForwardDifferenceError, BackwardDifferenceError,
  CentralDifferenceError, QuadratureError, RectangleRuleError,
  MidpointRuleError, TrapezoidalRuleError, SimpsonRuleError,
  GaussianQuadratureError, InterpolationError,
  PolynomialApproximationError, SpectralTruncationError,
  FourierTruncationError, SeriesApproximation, PartialSum,
  InfiniteSeries, PowerSeries, TaylorSeries, FourierSeries,
  NumericalDifferentiation, NumericalIntegration, NumericalODE,
  NumericalPDE, TimeDiscretisation, SpatialDiscretisation,
  StepSize, TimeStep, SpatialStep, MeshSize, GridSpacing,
  FiniteDifferenceMethod, FiniteVolumeMethod, FiniteElementMethod,
  SpectralMethod, EulerMethod, RungeKuttaMethod, MultistepMethod,
  LocalError, GlobalError, ErrorPropagation, Consistency, Stability,
  Convergence, ConvergenceRate, Refinement, MeshRefinement,
  TimestepRefinement, HRefinement, PRefinement, HPRefinement,
  RichardsonExtrapolation, RombergIntegration, GridConvergenceIndex,
  ErrorEstimator, APrioriErrorEstimate, APosterioriErrorEstimate,
  AdaptiveMethod, AdaptiveStepSize, AdaptiveMesh, ReferenceSolution,
  Verification, Benchmark, Reproducibility
 ]
 STOP_CONDITION = "Stop when exact process, finite approximation, leading omitted term, error order, refinement behaviour and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Numerical Error Ontology"
 RELATED_ONTOLOGIES = [
  "Numerical Methods",
  "Numerical Error Ontology",
  "Round-Off Error Ontology",
  "Numerical Analysis Ontology 2",
  "Numerical Approximation Ontology",
  "Numerical Convergence Ontology",
  "Numerical Stability Ontology 2",
  "Numerical Computation Ontology",
  "Differential Equation Ontologies"
 ]
 ROUTE_RULES = [
  "Truncation Error -> Numerical Error for parent error semantics",
  "Truncation Error -> Round-Off Error for total-error tradeoffs",
  "Truncation Error -> Numerical Analysis for formal asymptotic analysis",
  "Truncation Error -> Numerical Approximation for approximation-space error",
  "Truncation Error -> Numerical Convergence for observed order and refinement",
  "Truncation Error -> Numerical Stability for local-to-global error propagation",
  "Truncation Error -> Differential Equations for ODE and PDE discretisation error"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible truncation classes, asymptotic models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve truncation-versus-round-off distinction.",
  "Preserve local-versus-global truncation distinction.",
  "Preserve leading-term-versus-total-error distinction.",
  "Preserve theoretical-versus-observed-order distinction.",
  "Preserve consistency-versus-convergence distinction.",
  "Preserve series-versus-discretisation truncation distinction.",
  "Preserve machine-readable truncation-error semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Truncation error is the discrepancy introduced by replacing an infinite, continuous or exact mathematical process with a finite approximation."
 LOCAL_SUMMARY = "Local truncation error measures one-step approximation discrepancy under exact prior information."
 GLOBAL_SUMMARY = "Global truncation error describes accumulated approximation discrepancy over the complete numerical solution."
 ORDER_SUMMARY = "The first nonzero omitted or uncancelled term determines the leading asymptotic error and formal order of accuracy."
 REFINEMENT_SUMMARY = "Mesh, timestep and approximation-order refinement should reduce truncation error within the asymptotic regime."
 EXTRAPOLATION_SUMMARY = "Richardson and Romberg techniques exploit leading-order error models to improve approximations."
 BALANCE_SUMMARY = "At sufficiently fine resolution, round-off or other errors can dominate decreasing truncation error."
 VALIDATION_SUMMARY = "Truncation-error claims should be checked through derivation, refinement studies, observed order, reference solutions and reproducibility."
}

FLOW {
 INPUT
 -> detect_truncation_error_context
 -> identify_exact_process_finite_approximation_discretisation_parameter_and_target_quantity
 -> derive_Taylor_series_asymptotic_series_or_remainder_relation
 -> identify_cancelled_and_reproduced_terms
 -> identify_first_nonzero_omitted_term
 -> determine_local_or_global_truncation_error
 -> derive_order_of_accuracy_and_error_constant
 -> perform_mesh_timestep_series_or_order_refinement
 -> if_valid_apply_Richardson_Romberg_or_related_extrapolation
 -> compare_truncation_with_roundoff_iteration_and_other_error_sources
 -> validate_with_observed_order_reference_solution_and_reproducibility
 -> route_to_numerical_error_roundoff_analysis_approximation_convergence_stability_computation_or_differential_equation_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Truncation Error Ontology defines error caused by finite approximation of infinite, continuous or exact mathematical processes. It supports local and global truncation error, Taylor-series truncation and remainder terms, omitted higher-order terms, asymptotic and leading-order error, order of accuracy, finite-difference and quadrature error, interpolation and spectral truncation, series tails and partial sums, numerical differentiation and integration, ODE and PDE discretisation, temporal and spatial truncation, consistency, stability and convergence relationships, mesh, timestep, h, p and hp refinement, Richardson extrapolation, Romberg integration, grid convergence indices, a-priori and a-posteriori error estimates, adaptive refinement, reference solutions, verification, benchmarking and reproducibility."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-MATHEMATICS-TRUNCATION-ERROR-ONTOLOGY-V1"
}

}

END_FLOW