Round-Off Error Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Round-Off Error Ontology within the Kireth Research Semantic Library. Round-off error arises when exact mathematical values or arithmetic results are replaced by finite-precision representations. This ontology supports rounding models, machine epsilon, unit roundoff, ULP error, cumulative rounding effects, cancellation, error propagation, summation error, matrix computation error, mixed precision, numerical stability, precision floors and relationships with floating-point arithmetic, machine precision, numerical error and numerical convergence.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = RoundOffError
 ENTITY_CLASS = FinitePrecisionRoundingErrorFramework
 PRIMARY_CONTEXT = NumericalMethods
 DEFINITION = "Round-off error is the numerical discrepancy introduced when an exact value or arithmetic result is rounded to a finite-precision representable value."
 SYMBOLIC_FORMS = [
  "fl(x)=x(1+delta)",
  "|delta|<=u under standard conditions",
  "e_round=fl(x)-x",
  "gamma_n=(n*u)/(1-n*u)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for finite-precision rounding discrepancies, local and accumulated round-off, cancellation, error growth, precision floors and numerically stable computation."
}

SCOPE {
 INCLUDE = [
  RoundOffError, RoundingError, FloatingPointError, FinitePrecisionError,
  RepresentationError, QuantisationError, LocalRoundOffError,
  AccumulatedRoundOffError, RelativeRoundOffError, AbsoluteRoundOffError,
  UnitRoundoff, MachineEpsilon, ULP, UnitInLastPlace, MachinePrecision,
  FloatingPointArithmetic, FloatingPointNumber, Rounding, RoundingMode,
  RoundToNearest, DirectedRounding, CorrectRounding, FaithfulRounding,
  FinitePrecision, Significand, Exponent, Radix, NormalNumber,
  SubnormalNumber, Overflow, Underflow, GradualUnderflow, Cancellation,
  CatastrophicCancellation, LossOfSignificance, Absorption,
  ErrorPropagation, ErrorAmplification, ErrorAccumulation, ErrorBound,
  ForwardError, BackwardError, Residual, Conditioning, ConditionNumber,
  NumericalStability, BackwardStability, ForwardStability, StableAlgorithm,
  UnstableAlgorithm, Summation, NaiveSummation, PairwiseSummation,
  KahanSummation, CompensatedSummation, DotProduct, MatrixProduct,
  GaussianElimination, LUDecomposition, QRDecomposition,
  IterativeMethod, IterativeRefinement, MixedPrecision,
  AccumulationPrecision, StoragePrecision, ComputePrecision,
  PrecisionStudy, PrecisionSweep, ErrorFloor, RoundoffFloor,
  ConvergenceStagnation, Tolerance, AbsoluteTolerance, RelativeTolerance,
  Benchmark, ReferenceSolution, HighPrecisionReference, Verification,
  Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  TruncationErrorWithoutFinitePrecision,
  DiscretisationErrorWithoutRounding,
  MeasurementRoundingWithoutComputationalArithmetic,
  ModelErrorWithoutFinitePrecisionContext
 ]
 DISAMBIGUATION_RULE = "When discrepancy is caused specifically by rounding exact values or arithmetic results to finite machine-representable values, interpret the context as round-off error."
}

CORE_CONCEPTS {
 RoundOffError = "Error introduced by finite-precision rounding of exact values or arithmetic results."
 LocalRoundOffError = "Rounding error introduced by an individual representation or arithmetic operation."
 AccumulatedRoundOffError = "Combined effect of many local rounding errors through a computation."
 UnitRoundoff = "Standard bound used for relative rounding-error models under declared assumptions."
 MachineEpsilon = "Spacing-related quantity near one under a declared convention."
 ULP = "Local spacing unit between adjacent representable floating-point values."
 Cancellation = "Loss of significant information when nearly equal quantities are subtracted."
 ErrorPropagation = "Transmission and amplification of rounding errors through later computations."
 RoundoffFloor = "Accuracy limit at which further refinement is dominated by finite-precision effects."
}

RELATIONS {
 RoundOffError --is_child_of--> NumericalError
 RoundOffError --is_caused_by--> FinitePrecision
 FloatingPointArithmetic --introduces--> RoundOffError
 RoundingMode --determines--> RoundedResult
 UnitRoundoff --bounds_under_conditions--> RelativeRoundOffError
 ULP --measures--> LocalRepresentableSpacing
 MachinePrecision --constrains--> MinimumAttainableRoundoff
 Cancellation --amplifies--> RelativeRoundOffError
 CatastrophicCancellation --causes--> LossOfSignificance
 Conditioning --amplifies--> PerturbationEffects
 NumericalStability --controls--> ErrorAmplification
 PairwiseSummation --reduces--> SummationRoundoffGrowth
 KahanSummation --compensates--> SummationRoundOffError
 HigherAccumulationPrecision --reduces--> AccumulatedRoundOffError
 MixedPrecision --changes--> RoundOffErrorDistribution
 RoundoffFloor --can_cause--> ConvergenceStagnation
 PrecisionStudy --diagnoses--> RoundOffSensitivity
 HighPrecisionReference --supports--> RoundOffVerification
}

OPERATORS {
 ROUND_ERROR(x,flx) := e_round=flx-x
 ABS_ROUND_ERROR(x,flx) := |flx-x|
 REL_ROUND_ERROR(x,flx) := |flx-x|/|x|
 STANDARD_MODEL(x,delta,u) := fl(x)=x*(1+delta), |delta|<=u
 OPERATION_MODEL(x,op,y,delta) := fl(x op y)=(x op y)*(1+delta)
 GAMMA_BOUND(n,u) := gamma_n=(n*u)/(1-n*u) when n*u<1
 ULP_ERROR(x,flx) := distance_in_ULPs(x,flx)
 SUM_ERROR_BOUND(n,u,values) := bound_depending_on_gamma_n_and_value_magnitudes
}

AXIOMS {
 A1 = "Round-off error arises from finite representability and finite arithmetic precision."
 A2 = "The standard relative-error model applies only under its stated normal-range and rounding assumptions."
 A3 = "Local round-off errors can accumulate or cancel across a computation."
 A4 = "Round-off error and truncation error are distinct."
 A5 = "Problem conditioning and algorithmic stability determine how round-off perturbations affect final outputs."
 A6 = "Cancellation can transform small absolute rounding errors into large relative errors."
 A7 = "Higher precision reduces round-off effects but does not remove discretisation, iteration or model error."
 A8 = "Round-off analysis requires explicit arithmetic format, rounding mode and precision assumptions."
}

REASONING {
 R1_CONTEXT = identify(format,precision,rounding_mode,algorithm,operation_count)
 R2_LOCAL = estimate(local_representation_and_operation_roundoff)
 R3_PROPAGATION = trace(error_propagation_through_algorithm)
 R4_CONDITIONING = assess(problem_sensitivity_to_roundoff_perturbations)
 R5_STABILITY = assess_algorithmic_error_amplification)
 R6_CANCELLATION = detect(subtractive_cancellation_and_loss_of_significance)
 R7_ACCUMULATION = estimate(accumulated_roundoff_using_operation_structure)
 R8_REFORMULATION = if_required use(scaling,pairwise_summation,compensation,FMA_or_higher_precision)
 R9_FLOOR = compare(roundoff_floor_with_truncation_discretisation_and_iteration_error)
 R10_VALIDATION = perform(precision_sweep,high_precision_reference,error_bound_and_reproducibility_checks)
}

CONSTRAINTS {
 C1 = "Relative round-off error can be misleading when the exact value is zero or near zero."
 C2 = "The simple fl(x)=x(1+delta) model does not directly cover overflow, underflow or all subnormal cases."
 C3 = "Operation count alone does not determine total round-off error; algorithmic structure matters."
 C4 = "Independent-error assumptions may be invalid for deterministic floating-point computations."
 C5 = "Cancellation can dominate otherwise small local rounding effects."
 C6 = "Increasing mesh or iteration resolution can eventually expose a round-off-dominated error floor."
 C7 = "Compensated algorithms reduce but do not eliminate all finite-precision error."
 C8 = "Parallel operation reordering can change accumulated round-off error."
 C9 = "Higher precision can improve round-off behaviour without fixing poor conditioning."
 C10 = "Empirical precision agreement does not replace a theoretical error analysis when guarantees are required."
}

SEMANTIC_EQUIVALENCE {
 "RoundOffError" <=> "RoundingError"
 "AccumulatedRoundOffError" <=> "CumulativeRoundoffError"
 "UnitInLastPlace" <=> "ULP"
 "RoundoffFloor" <=> "FinitePrecisionErrorFloor"
 "CatastrophicCancellation" <=> "SevereCancellation"
}

VALIDATION {
 V1 = "Arithmetic format and precision are explicit."
 V2 = "Rounding mode is documented."
 V3 = "Machine epsilon and unit roundoff conventions are distinguished."
 V4 = "Local versus accumulated round-off is distinguished."
 V5 = "Cancellation-sensitive operations are identified."
 V6 = "Conditioning and numerical stability are assessed separately."
 V7 = "Precision studies are used where round-off dominance is plausible."
 V8 = "High-precision references are used where feasible."
 V9 = "Parallel or reordered reductions are documented."
 V10 = "Round-off conclusions are reproducible under documented arithmetic settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ROUND_OFF_ERROR_DEFINITION = 0.99
 STANDARD_MODEL_REASONING = 0.99
 ACCUMULATION_REASONING = 0.98
 CANCELLATION_REASONING = 0.99
 CONDITIONING_REASONING = 0.99
 STABILITY_REASONING = 0.99
 PRECISION_FLOOR_REASONING = 0.98
 MIXED_PRECISION_REASONING = 0.98
 POLICY = "Reduce confidence when arithmetic format, precision, rounding mode, operation order, conditioning, cancellation or high-precision reference behaviour are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, RoundOffErrorReasoning,
  RoundingErrorReasoning, FloatingPointErrorReasoning,
  UnitRoundoffReasoning, ULPReasoning, LocalRoundoffReasoning,
  AccumulatedRoundoffReasoning, CancellationReasoning,
  ErrorPropagationReasoning, ConditioningReasoning,
  NumericalStabilityReasoning, SummationErrorReasoning,
  MixedPrecisionReasoning, ErrorFloorReasoning,
  PrecisionStudyReasoning, VerificationReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "round-off error" -> RoundOffError
  "roundoff error" -> RoundOffError
  "rounding error" -> RoundOffError
  "ULP" -> UnitInLastPlace
  "roundoff floor" -> RoundoffFloor
  "catastrophic cancellation" -> CatastrophicCancellation
 }
}

EXECUTION_RULES {
 E1 = "If finite-precision rounding is the source of numerical discrepancy, activate RoundOffErrorOntology."
 E2 = "If using standard floating-point error models, verify normal-range and rounding assumptions."
 E3 = "If many operations accumulate, estimate structured error growth rather than only local error."
 E4 = "If near-equal subtraction occurs, test for cancellation."
 E5 = "If summation dominates, consider pairwise or compensated summation."
 E6 = "If round-off limits convergence, compare against discretisation and iteration error."
 E7 = "If precision sensitivity is uncertain, perform a precision sweep."
 E8 = "If mixed precision is used, track where each precision enters the error path."
 E9 = "Any round-off claim should be validated with format, precision, stability and reproducibility information."
}

RECURSION {
 ENTRY = RoundOffError
 EXPAND(RoundOffError) -> [
  RoundingError, FloatingPointError, FinitePrecisionError,
  RepresentationError, QuantisationError, LocalRoundOffError,
  AccumulatedRoundOffError, RelativeRoundOffError, AbsoluteRoundOffError,
  UnitRoundoff, MachineEpsilon, ULP, MachinePrecision,
  FloatingPointArithmetic, RoundingMode, RoundToNearest,
  DirectedRounding, CorrectRounding, FinitePrecision, Significand,
  Exponent, Radix, NormalNumber, SubnormalNumber, Overflow, Underflow,
  Cancellation, CatastrophicCancellation, LossOfSignificance,
  ErrorPropagation, ErrorAmplification, ErrorAccumulation, ErrorBound,
  ForwardError, BackwardError, Residual, Conditioning, ConditionNumber,
  NumericalStability, BackwardStability, ForwardStability,
  StableAlgorithm, Summation, NaiveSummation, PairwiseSummation,
  KahanSummation, CompensatedSummation, DotProduct, MatrixProduct,
  GaussianElimination, LUDecomposition, QRDecomposition,
  IterativeMethod, IterativeRefinement, MixedPrecision,
  AccumulationPrecision, StoragePrecision, ComputePrecision,
  PrecisionStudy, PrecisionSweep, ErrorFloor, RoundoffFloor,
  ConvergenceStagnation, Tolerance, HighPrecisionReference,
  Verification, Reproducibility
 ]
 STOP_CONDITION = "Stop when arithmetic format, local error, accumulation, cancellation, conditioning, stability, precision floor and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Numerical Error Ontology"
 RELATED_ONTOLOGIES = [
  "Numerical Methods",
  "Floating Point Arithmetic Ontology",
  "Machine Precision Ontology",
  "Numerical Error Ontology",
  "Numerical Stability Ontology 2",
  "Numerical Convergence Ontology",
  "Numerical Analysis Ontology 2",
  "Numerical Computation Ontology",
  "Numerical Linear Algebra Ontologies"
 ]
 ROUTE_RULES = [
  "Round-Off Error -> Numerical Error for parent error semantics",
  "Round-Off Error -> Floating Point Arithmetic for rounding semantics",
  "Round-Off Error -> Machine Precision for finite resolution",
  "Round-Off Error -> Numerical Stability for amplification behaviour",
  "Round-Off Error -> Numerical Convergence for error floors and stagnation",
  "Round-Off Error -> Numerical Analysis for formal finite-precision bounds",
  "Round-Off Error -> Numerical Linear Algebra for matrix error accumulation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible round-off models, compensation methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve round-off-versus-truncation distinction.",
  "Preserve local-versus-accumulated round-off distinction.",
  "Preserve absolute-versus-relative round-off distinction.",
  "Preserve conditioning-versus-stability distinction.",
  "Preserve machine-epsilon-versus-unit-roundoff distinction.",
  "Preserve serial-versus-reordered accumulation distinction.",
  "Preserve machine-readable round-off-error semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Round-off error is the discrepancy introduced when exact values or arithmetic results are rounded to finite-precision representable values."
 LOCAL_SUMMARY = "Each finite-precision representation or arithmetic operation can introduce local round-off."
 ACCUMULATION_SUMMARY = "Local errors can accumulate, cancel or amplify according to algorithm structure."
 CANCELLATION_SUMMARY = "Subtractive cancellation can produce severe relative loss of significant information."
 CONDITIONING_SUMMARY = "Problem conditioning determines how perturbations caused by rounding affect the exact solution."
 STABILITY_SUMMARY = "Algorithmic stability determines how strongly round-off perturbations are amplified by computation."
 PRECISION_FLOOR_SUMMARY = "Round-off can establish a practical error floor that limits further useful refinement."
 VALIDATION_SUMMARY = "Round-off analysis should be checked through explicit arithmetic assumptions, precision studies, high-precision references and reproducible computation."
}

FLOW {
 INPUT
 -> detect_round_off_error_context
 -> identify_arithmetic_format_precision_rounding_mode_algorithm_and_operation_structure
 -> estimate_local_rounding_error
 -> propagate_and_accumulate_error_through_computation
 -> analyse_conditioning_and_numerical_stability
 -> detect_cancellation_loss_of_significance_overflow_and_underflow
 -> estimate_roundoff_bounds_or_error_floor
 -> apply_scaling_compensation_pairwise_summation_FMA_or_higher_precision_if_required
 -> compare_roundoff_with_truncation_discretisation_and_iteration_error
 -> perform_precision_sweep_or_high_precision_reference_if_required
 -> route_to_numerical_error_floating_point_machine_precision_stability_convergence_analysis_or_linear_algebra_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Round-Off Error Ontology defines finite-precision rounding discrepancies in numerical computation. It supports local and accumulated round-off, absolute and relative rounding error, machine epsilon, unit roundoff and ULPs, floating-point representation, rounding modes, normal and subnormal values, overflow and underflow, cancellation and loss of significance, error propagation and accumulation, forward and backward error, residuals, conditioning and numerical stability, summation and compensated summation, dot products and matrix computation, iterative refinement, mixed precision, storage, compute and accumulation precision, precision studies, round-off floors, convergence stagnation, tolerances, high-precision references, verification and reproducibility."
}

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

}

END_FLOW