Significant Figures Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Significant Figures Ontology within the Kireth Research Semantic Library. Significant figures describe the meaningful digits used to express the precision of numerical values and computed results. This ontology supports significant digits, leading and trailing zeros, decimal places, rounding, scientific notation, propagation through arithmetic operations, measurement and computational precision, uncertainty, machine precision, error reporting and relationships with numerical methods, floating-point arithmetic and numerical error.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = SignificantFigures
 ENTITY_CLASS = NumericalPrecisionReportingFramework
 PRIMARY_CONTEXT = NumericalMethods
 DEFINITION = "Significant figures are the digits in a numerical representation that convey meaningful information about its precision, beginning with the first nonzero significant digit and including subsequent digits according to the representation convention."
 SYMBOLIC_FORMS = [
  "x approximately x_hat to n significant figures",
  "relative_rounding_error approximately <= 0.5*10^(1-n) under decimal round-to-nearest assumptions",
  "scientific_notation=a*10^k",
  "1<=|a|<10"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for significant digits, decimal precision, rounding, scientific notation, zero interpretation, arithmetic reporting rules and precision-aware numerical communication."
}

SCOPE {
 INCLUDE = [
  SignificantFigures, SignificantDigits, SignificantDigitCount,
  NumericalPrecision, ReportedPrecision, DecimalPrecision, DecimalPlace,
  DecimalPlaces, LeadingDigit, LeadingZero, TrailingZero, EmbeddedZero,
  SignificantZero, InsignificantZero, ExactNumber, ApproximateNumber,
  MeasuredNumber, ComputedNumber, ScientificNotation, StandardForm,
  NormalisedScientificNotation, OrderOfMagnitude, Mantissa,
  Coefficient, Exponent, Rounding, RoundToNearest, RoundingDigit,
  GuardDigit, RoundingError, RoundoffError, Truncation, DecimalRounding,
  SignificantFigureRounding, PrecisionLoss, PrecisionGain,
  FalsePrecision, SpuriousPrecision, Overprecision, Underprecision,
  Uncertainty, AbsoluteUncertainty, RelativeUncertainty,
  PercentageUncertainty, AbsoluteError, RelativeError, PercentageError,
  MeasurementResolution, InstrumentResolution, MachinePrecision,
  MachineEpsilon, UnitRoundoff, FloatingPointArithmetic,
  SignificantBits, BitsOfPrecision, DecimalDigitsOfPrecision,
  ConversionPrecision, UnitConversion, ExactConversionFactor,
  ApproximateConversionFactor, AdditionRule, SubtractionRule,
  MultiplicationRule, DivisionRule, LogarithmRule, ExponentiationRule,
  IntermediatePrecision, GuardFigures, FinalRounding,
  ErrorPropagation, UncertaintyPropagation, NumericalCommunication,
  TabulatedValue, ReportedValue, ReferenceValue, Tolerance,
  Verification, Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  DecimalPlacesWithoutPrecisionInterpretation,
  ExactIntegerDigitCount,
  FloatingPointBitPrecisionWithoutDecimalReportingContext,
  MeasurementUncertaintyWhenSignificantDigitReportingIsNotCentral
 ]
 DISAMBIGUATION_RULE = "When the concern is how many digits in a numerical value meaningfully communicate its precision, interpret the context as significant figures."
}

CORE_CONCEPTS {
 SignificantFigures = "Digits conveying meaningful numerical precision."
 SignificantDigitCount = "Number of digits treated as significant in a numerical representation."
 LeadingZero = "Zero preceding the first nonzero digit and normally not significant."
 EmbeddedZero = "Zero between significant nonzero digits and normally significant."
 TrailingZero = "Terminal zero whose significance depends on decimal notation and declared precision."
 ScientificNotation = "Representation separating a significant coefficient from a power-of-ten scale."
 ExactNumber = "Value treated as exact and therefore not limiting significant-figure precision."
 ReportedPrecision = "Precision communicated by the written numerical representation."
 FalsePrecision = "Reporting more digits than are justified by data, arithmetic or uncertainty."
}

RELATIONS {
 SignificantFigures --is_child_of--> NumericalMethods
 SignificantFigures --describes--> ReportedPrecision
 SignificantDigitCount --counts--> SignificantDigits
 LeadingZero --usually_is--> InsignificantZero
 EmbeddedZero --usually_is--> SignificantZero
 TrailingZero --depends_on--> RepresentationConvention
 ScientificNotation --clarifies--> SignificantDigitCount
 Rounding --reduces--> ReportedDigits
 GuardDigit --supports--> CorrectFinalRounding
 GuardFigures --reduce--> PrematureRoundingError
 ExactNumber --does_not_limit--> SignificantFigureCount
 MeasurementResolution --constrains--> ReportedPrecision
 Uncertainty --constrains--> JustifiedPrecision
 MachinePrecision --constrains--> ComputationalPrecision
 FloatingPointArithmetic --introduces--> RoundoffError
 FalsePrecision --misrepresents--> NumericalUncertainty
 UnitConversion --may_preserve--> SignificantFigures
 ExactConversionFactor --does_not_limit--> Precision
}

OPERATORS {
 SIGFIG_COUNT(x,representation) := count_significant_digits(representation)
 SCIENTIFIC_FORM(x) := a*10^k where 1<=|a|<10
 ROUND_SIGFIG(x,n) := round_x_to_n_significant_figures
 DECIMAL_PLACE_VALUE(x,n) := place_of_nth_significant_digit(x)
 RELATIVE_ROUNDING_BOUND(n) := approximately 0.5*10^(1-n)
 PERCENT_UNCERTAINTY(x,delta_x) := 100*|delta_x/x|
 ADD_SUB_REPORT(values) := round_result_to_least_precise_relevant_decimal_place
 MULT_DIV_REPORT(values) := round_result_to_appropriate_significant_figure_count
}

AXIOMS {
 A1 = "Significant figures describe reported numerical precision rather than mathematical exactness alone."
 A2 = "Leading zeros before the first nonzero digit are not significant."
 A3 = "Zeros between significant digits are significant."
 A4 = "Trailing-zero significance depends on notation and explicit precision context."
 A5 = "Scientific notation provides an unambiguous way to communicate significant-figure count."
 A6 = "Exact counts and exact conversion constants do not limit significant figures."
 A7 = "Intermediate calculations should retain sufficient guard digits to avoid premature rounding."
 A8 = "Reported significant figures should be consistent with uncertainty and computational precision."
}

REASONING {
 R1_CONTEXT = identify(value,representation,exact_or_approximate_status,uncertainty)
 R2_PARSE = identify(leading,embedded,trailing_digits_and_zeros)
 R3_COUNT = determine_significant_digit_count)
 R4_ROUND = if_required round_to_requested_significant_figures)
 R5_ARITHMETIC = apply(operation_appropriate_precision_rule)
 R6_GUARD = retain_extra_intermediate_digits)
 R7_UNCERTAINTY = align_reported_digits_with_uncertainty_or_resolution)
 R8_MACHINE = if_computed compare_reported_precision_with_machine_and_algorithmic_precision)
 R9_FORMAT = use_scientific_notation_if_zero_significance_is_ambiguous)
 R10_VALIDATION = check_rounding,count,uncertainty_consistency_and_reproducibility)
}

CONSTRAINTS {
 C1 = "Significant figures and decimal places are not equivalent concepts."
 C2 = "Trailing zeros in integers without a decimal marker can be ambiguous."
 C3 = "A calculator displaying many digits does not justify reporting all displayed digits."
 C4 = "Repeated intermediate rounding can materially increase final error."
 C5 = "Significant-figure rules are reporting conventions and do not replace formal uncertainty propagation."
 C6 = "Exact constants should not artificially reduce reported precision."
 C7 = "Unit conversion can alter decimal places without changing justified significant precision."
 C8 = "Machine precision can exceed or fall below the precision justified by input data."
 C9 = "Very small or very large values are clearer when significant figures are expressed in scientific notation."
 C10 = "False precision can communicate unjustified certainty."
}

SEMANTIC_EQUIVALENCE {
 "SignificantFigures" <=> "SignificantDigits"
 "ScientificNotation" <=> "StandardForm"
 "GuardFigures" <=> "GuardDigitsInReportingContext"
 "FalsePrecision" <=> "SpuriousPrecision"
 "PercentageUncertainty" <=> "RelativeUncertaintyPercent"
}

VALIDATION {
 V1 = "Exact versus approximate status is explicit."
 V2 = "Leading, embedded and trailing zeros are interpreted correctly."
 V3 = "Scientific notation is used where significance would otherwise be ambiguous."
 V4 = "Rounding is performed only after sufficient guard digits are retained."
 V5 = "Addition and subtraction respect decimal-place resolution where the conventional rule is used."
 V6 = "Multiplication and division respect justified significant precision where the conventional rule is used."
 V7 = "Reported digits are consistent with uncertainty."
 V8 = "Machine output digits are not automatically treated as significant."
 V9 = "Exact conversion constants are distinguished from approximate constants."
 V10 = "Final reported values can be reproduced from documented inputs and rounding rules."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SIGNIFICANT_FIGURES_DEFINITION = 0.99
 ZERO_INTERPRETATION_REASONING = 0.99
 ROUNDING_REASONING = 0.99
 SCIENTIFIC_NOTATION_REASONING = 0.99
 ARITHMETIC_REPORTING_REASONING = 0.98
 UNCERTAINTY_ALIGNMENT_REASONING = 0.98
 MACHINE_PRECISION_RELATIONSHIP = 0.99
 POLICY = "Reduce confidence when notation, trailing-zero intent, exactness, uncertainty, measurement resolution or rounding convention are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, SignificantFiguresReasoning,
  SignificantDigitsReasoning, DecimalPrecisionReasoning,
  ZeroSignificanceReasoning, ScientificNotationReasoning,
  RoundingReasoning, GuardDigitReasoning, ArithmeticPrecisionReasoning,
  FalsePrecisionReasoning, UncertaintyReportingReasoning,
  MachinePrecisionReasoning, NumericalCommunicationReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "significant figures" -> SignificantFigures
  "significant digits" -> SignificantFigures
  "sig figs" -> SignificantFigures
  "standard form" -> ScientificNotation
  "spurious precision" -> FalsePrecision
  "guard digits" -> GuardFigures
 }
}

EXECUTION_RULES {
 E1 = "If meaningful reported digits are central, activate SignificantFiguresOntology."
 E2 = "Ignore leading zeros when counting significant figures."
 E3 = "Treat embedded zeros as significant."
 E4 = "Interpret trailing zeros using decimal notation, scientific notation or explicit precision context."
 E5 = "If ambiguity remains, express the value in scientific notation."
 E6 = "Retain guard digits during intermediate computation and round at the final reporting stage."
 E7 = "If values are measured, align significant figures with justified uncertainty or resolution."
 E8 = "If values are machine-computed, distinguish display digits from genuinely meaningful digits."
 E9 = "Any significant-figure result should be validated for notation, rounding, exactness and uncertainty consistency."
}

RECURSION {
 ENTRY = SignificantFigures
 EXPAND(SignificantFigures) -> [
  SignificantDigits, SignificantDigitCount, NumericalPrecision,
  ReportedPrecision, DecimalPrecision, DecimalPlace, LeadingDigit,
  LeadingZero, TrailingZero, EmbeddedZero, SignificantZero,
  InsignificantZero, ExactNumber, ApproximateNumber, MeasuredNumber,
  ComputedNumber, ScientificNotation, StandardForm, OrderOfMagnitude,
  Rounding, RoundToNearest, RoundingDigit, GuardDigit, RoundingError,
  RoundoffError, SignificantFigureRounding, PrecisionLoss, FalsePrecision,
  Overprecision, Underprecision, Uncertainty, AbsoluteUncertainty,
  RelativeUncertainty, PercentageUncertainty, AbsoluteError,
  RelativeError, MeasurementResolution, MachinePrecision,
  MachineEpsilon, UnitRoundoff, FloatingPointArithmetic,
  SignificantBits, DecimalDigitsOfPrecision, UnitConversion,
  ExactConversionFactor, ApproximateConversionFactor, AdditionRule,
  SubtractionRule, MultiplicationRule, DivisionRule, IntermediatePrecision,
  GuardFigures, FinalRounding, ErrorPropagation, NumericalCommunication,
  ReportedValue, ReferenceValue, Tolerance, Verification, Reproducibility
 ]
 STOP_CONDITION = "Stop when significant-digit count, zero interpretation, exactness, rounding, uncertainty and reporting precision are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Numerical Methods"
 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",
  "Measurement & Units Ontologies"
 ]
 ROUTE_RULES = [
  "Significant Figures -> Numerical Methods for parent semantics",
  "Significant Figures -> Floating Point Arithmetic for finite representation",
  "Significant Figures -> Machine Precision for computational resolution",
  "Significant Figures -> Numerical Error for rounding and error semantics",
  "Significant Figures -> Numerical Stability for precision-loss effects",
  "Significant Figures -> Numerical Convergence for attainable reported accuracy",
  "Significant Figures -> Measurement & Units for measurement resolution and uncertainty"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible significant-figure conventions or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve significant-figures-versus-decimal-places distinction.",
  "Preserve exact-versus-approximate number distinction.",
  "Preserve leading-versus-embedded-versus-trailing zero distinction.",
  "Preserve intermediate-versus-final rounding distinction.",
  "Preserve displayed-digits-versus-justified-precision distinction.",
  "Preserve significant-figure-versus-formal-uncertainty distinction.",
  "Preserve machine-readable significant-figures semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Significant figures are the digits in a numerical representation that meaningfully communicate its precision."
 ZERO_SUMMARY = "Leading zeros are normally not significant, embedded zeros are significant, and trailing zeros require notation or precision context."
 SCIENTIFIC_NOTATION_SUMMARY = "Scientific notation makes significant-digit count explicit while separating precision from scale."
 ROUNDING_SUMMARY = "Values can be rounded to a declared number of significant figures while retaining guard digits during intermediate calculations."
 ARITHMETIC_SUMMARY = "Conventional reporting rules treat addition and subtraction through decimal-place resolution and multiplication and division through significant precision."
 UNCERTAINTY_SUMMARY = "Reported significant figures should be consistent with measurement or computational uncertainty."
 MACHINE_SUMMARY = "Displayed machine digits must not be confused with justified significant digits."
 VALIDATION_SUMMARY = "Significant-figure reporting should be checked for zero interpretation, exactness, rounding, uncertainty and reproducibility."
}

FLOW {
 INPUT
 -> detect_significant_figures_context
 -> identify_value_representation_exactness_uncertainty_and_resolution
 -> parse_leading_embedded_and_trailing_zeros
 -> determine_significant_digit_count
 -> if_required_convert_to_scientific_notation
 -> apply_requested_significant_figure_rounding
 -> if_arithmetic_apply_operation_appropriate_reporting_rule
 -> retain_guard_digits_until_final_rounding
 -> align_reported_precision_with_uncertainty_and_machine_precision
 -> validate_notation_rounding_exactness_and_reproducibility
 -> route_to_numerical_methods_floating_point_machine_precision_numerical_error_stability_convergence_analysis_or_measurement_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Significant Figures Ontology defines precision-aware numerical reporting. It supports significant digits and counts, decimal precision and places, leading, embedded and trailing zeros, exact and approximate numbers, scientific notation and order of magnitude, rounding and guard digits, false and spurious precision, absolute, relative and percentage uncertainty, measurement and machine precision, machine epsilon and unit roundoff relationships, significant bits and decimal digits of precision, unit conversions and exact conversion factors, conventional arithmetic reporting rules, intermediate precision, final rounding, error and uncertainty propagation, numerical communication, verification and reproducibility."
}

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

}

END_FLOW