Measurement & Units Ontologies

AI INTERPRETATION NOTE
This Flow Algebra defines the Measurement & Units Ontologies parent ontology within the Kireth Research Semantic Library. Measurement and units ontologies concern the formal representation of measurable quantities, dimensions, unit systems, scales, conversions, standards, uncertainty, calibration, dimensional analysis, derived units, prefixes, constants, tolerances and semantic interoperability between numerical values and their physical or abstract measurement meanings.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Measurement & Units Ontologies"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Measurement & Units"
 SPECIALISATION = "Measurement, Quantities, Dimensions, Units and Conversion / Parent Ontology"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/measurement-units-ontologies.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = MeasurementUnitsOntologies
 ENTITY_CLASS = MeasurementQuantityUnitSemanticFramework
 PRIMARY_CONTEXT = MathematicalOntologies
 DEFINITION = "Measurement and units ontologies formally represent measurable quantities, dimensions, units, scales, standards, conversions, uncertainty and relationships between numerical values and their measurement semantics."
 SYMBOLIC_FORMS = [
  "Q = {value, unit}",
  "[Q] = dimensional_signature",
  "value_target = conversion(value_source,unit_source,unit_target)",
  "measurement = estimate ± uncertainty"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for quantities, dimensions, units, measurement values, conversions, uncertainty, calibration and dimensional consistency."
}

SCOPE {
 INCLUDE = [
  MeasurementUnitsOntologies,
  MeasurementOntology,
  UnitsOntology,
  QuantityOntology,
  PhysicalQuantity,
  MeasurableQuantity,
  Measurement,
  MeasurementValue,
  NumericalValue,
  ScalarMeasurement,
  VectorMeasurementRelationship,
  TensorMeasurementRelationship,
  Dimension,
  PhysicalDimension,
  DimensionalSignature,
  BaseDimension,
  DerivedDimension,
  DimensionlessQuantity,
  Unit,
  MeasurementUnit,
  BaseUnit,
  DerivedUnit,
  CoherentUnit,
  NonCoherentUnitRelationship,
  StandardUnit,
  ReferenceUnit,
  CanonicalUnit,
  SIUnit,
  SIBaseUnit,
  SIDerivedUnit,
  ImperialUnitRelationship,
  USCUnitRelationship,
  MetricUnit,
  NaturalUnitRelationship,
  AtomicUnitRelationship,
  AstronomicalUnitRelationship,
  Prefix,
  SIPrefix,
  DecimalPrefix,
  BinaryPrefixRelationship,
  Conversion,
  UnitConversion,
  ConversionFactor,
  ConversionOffset,
  AffineConversion,
  MultiplicativeConversion,
  NonlinearConversionRelationship,
  Scale,
  MeasurementScale,
  RatioScale,
  IntervalScale,
  OrdinalScaleRelationship,
  NominalScaleRelationship,
  LogarithmicScale,
  LinearScale,
  TemperatureScale,
  TimeScaleRelationship,
  CoordinateScaleRelationship,
  QuantityKind,
  UnitKind,
  UnitSystem,
  SystemOfUnits,
  InternationalSystemOfUnits,
  SI,
  CGSrelationship,
  MKSrelationship,
  FPSrelationship,
  UnitSymbol,
  QuantitySymbol,
  DimensionSymbol,
  PrefixSymbol,
  UnitExpression,
  CompoundUnit,
  ProductUnit,
  QuotientUnit,
  PowerUnit,
  InverseUnit,
  UnitExponent,
  DimensionalAnalysis,
  DimensionalConsistency,
  DimensionalEquation,
  Homogeneity,
  UnitChecking,
  UnitInference,
  QuantityCalculus,
  UnitAlgebra,
  MeasurementEquation,
  Calibration,
  CalibrationStandard,
  ReferenceStandard,
  PrimaryStandardRelationship,
  SecondaryStandardRelationship,
  Traceability,
  MeasurementTraceability,
  Metrology,
  ScientificMeasurement,
  InstrumentMeasurementRelationship,
  SensorMeasurementRelationship,
  ObservationRelationship,
  ExperimentalMeasurementRelationship,
  Resolution,
  Precision,
  Accuracy,
  Repeatability,
  Reproducibility,
  Sensitivity,
  SpecificityRelationship,
  Range,
  DynamicRange,
  DetectionLimitRelationship,
  QuantisationRelationship,
  SignificantFigures,
  DecimalPlaces,
  Rounding,
  RoundOffError,
  MeasurementError,
  SystematicError,
  RandomError,
  Bias,
  Uncertainty,
  MeasurementUncertainty,
  StandardUncertainty,
  CombinedUncertainty,
  ExpandedUncertainty,
  ConfidenceIntervalRelationship,
  CoverageFactorRelationship,
  ErrorBound,
  Tolerance,
  AbsoluteTolerance,
  RelativeTolerance,
  PropagationOfUncertainty,
  UncertaintyBudget,
  ErrorBudget,
  Covariance,
  Correlation,
  ProbabilityDistributionRelationship,
  NormalDistributionRelationship,
  StandardDeviationRelationship,
  VarianceRelationship,
  ExactValue,
  TrueValueRelationship,
  ReferenceValue,
  ConventionalValueRelationship,
  MeasuredValue,
  EstimatedValue,
  NominalValue,
  RatedValueRelationship,
  Threshold,
  Limit,
  Minimum,
  Maximum,
  LowerBound,
  UpperBound,
  Interval,
  MeasurementInterval,
  QuantityInterval,
  Constant,
  PhysicalConstant,
  MathematicalConstantRelationship,
  DefinedConstant,
  ExactConstant,
  ExperimentalConstantRelationship,
  ConversionConstant,
  DimensionlessConstant,
  FundamentalConstantRelationship,
  SpeedOfLightRelationship,
  PlanckConstantRelationship,
  BoltzmannConstantRelationship,
  AvogadroConstantRelationship,
  ElementaryChargeRelationship,
  GravitationalConstantRelationship,
  Length,
  Mass,
  Time,
  ElectricCurrent,
  ThermodynamicTemperature,
  AmountOfSubstance,
  LuminousIntensity,
  Area,
  Volume,
  Velocity,
  Acceleration,
  Force,
  Pressure,
  Energy,
  Power,
  Frequency,
  Charge,
  Voltage,
  Resistance,
  Conductance,
  Capacitance,
  Inductance,
  MagneticFluxRelationship,
  MagneticFluxDensityRelationship,
  IlluminanceRelationship,
  RadioactivityRelationship,
  AbsorbedDoseRelationship,
  EquivalentDoseRelationship,
  CatalyticActivityRelationship,
  Angle,
  SolidAngle,
  Density,
  Concentration,
  MolarityRelationship,
  MolalityRelationship,
  FlowRateRelationship,
  ViscosityRelationship,
  EntropyRelationship,
  HeatCapacityRelationship,
  ThermalConductivityRelationship,
  DimensionlessNumber,
  ReynoldsNumberRelationship,
  MachNumberRelationship,
  PrandtlNumberRelationship,
  NusseltNumberRelationship,
  PecletNumberRelationship,
  UnitRegistry,
  UnitDictionary,
  QuantityRegistry,
  OntologyMapping,
  UnitMapping,
  SemanticMapping,
  UnitEquivalence,
  QuantityEquivalence,
  Synonym,
  Alias,
  StandardName,
  CanonicalName,
  UnitIdentifier,
  QuantityIdentifier,
  URIrelationship,
  CodeRelationship,
  Metadata,
  Provenance,
  Standard,
  InternationalStandardRelationship,
  ISOrelationship,
  BIPMrelationship,
  NISTrelationship,
  IECrelationship,
  OIMLrelationship,
  UCUMrelationship,
  QUDTRelationship,
  OMRelationship,
  Verification,
  Validation,
  UnitValidation,
  ConversionValidation,
  DimensionalValidation,
  ConsistencyCheck,
  RangeCheck,
  PlausibilityCheck,
  NumericalVerification,
  ErrorAnalysis,
  ReproducibilityOntologyRelationship,
  ComputationalScientificWorkflow,
  ScientificSimulation,
  ComputationalStatistics,
  AIComputationalMathematics
 ]
 EXCLUDE_AS_PRIMARY = [
  PureNumberTheoryWithoutMeasurementSemantics,
  GeometryWithoutQuantityOrUnitContext,
  SymbolicAlgebraWithoutDimensionalMeaning,
  DomainSpecificMeasurementsWhenMeasurementSemanticsAreSecondary
 ]
 DISAMBIGUATION_RULE = "When numerical values are interpreted through measurable quantities, dimensions, units, scales, conversions or uncertainty, interpret the context as measurement and units semantics."
}

CORE_CONCEPTS {
 Measurement = "Assignment or estimation of a value to a quantity according to a measurement procedure."
 Quantity = "Property that can be expressed by a number and a reference such as a unit."
 Dimension = "Abstract representation of the physical nature of a quantity independent of unit choice."
 Unit = "Reference quantity used to express values of quantities of the same kind."
 UnitSystem = "Coherent or conventional set of units and defining relationships."
 Conversion = "Transformation of a numerical value between compatible units."
 Uncertainty = "Quantified doubt associated with a measurement result."
 DimensionalAnalysis = "Reasoning about mathematical expressions through quantity dimensions."
}

RELATIONS {
 MeasurementUnitsOntologies --is_child_of--> MathematicalOntologies
 Measurement --measures--> Quantity
 Measurement --produces--> MeasurementValue
 MeasurementValue --uses--> Unit
 Quantity --has--> Dimension
 Unit --realises--> QuantityKind
 BaseUnit --belongs_to--> UnitSystem
 DerivedUnit --derives_from--> BaseUnits
 UnitConversion --maps--> CompatibleUnits
 ConversionFactor --scales--> MeasurementValue
 Calibration --links--> InstrumentToReferenceStandard
 MeasurementTraceability --links_to--> ReferenceStandard
 Uncertainty --qualifies--> MeasurementValue
 DimensionalAnalysis --checks--> MathematicalExpression
 Provenance --records--> MeasurementAndConversionContext
}

OPERATORS {
 QUANTITY(value,unit) := {value,unit}
 DIMENSION(quantity) := dimensional_signature(quantity)
 CONVERT(value,u1,u2) := conversion_rule(u1,u2)(value)
 MULTIPLY(q1,q2) := combine_values_and_units(q1,q2)
 DIVIDE(q1,q2) := divide_values_and_units(q1,q2)
 POWER(q,n) := raise_value_and_unit(q,n)
 CHECK_DIM(expr) := dimensional_consistency(expr)
 PROPAGATE_UNCERTAINTY(f,x,Ux) := uncertainty_of_function(f,x,Ux)
}

AXIOMS {
 A1 = "A measurement value without a quantity or unit context is semantically incomplete."
 A2 = "Only dimensionally compatible quantities can be directly converted between units."
 A3 = "Unit conversion must preserve represented quantity."
 A4 = "Derived units inherit dimensions from their defining quantity equations."
 A5 = "Measurement uncertainty is distinct from numerical rounding error."
 A6 = "Dimensional consistency is necessary but not sufficient for physical correctness."
 A7 = "Calibration and traceability determine how measurements relate to reference standards."
 A8 = "Measurement-and-unit reasoning requires explicit quantity, dimension, unit, scale, uncertainty and provenance semantics."
}

REASONING {
 R1_CONTEXT = identify(quantity,value,unit,dimension,scale,measurement_method,uncertainty_and_target_use)
 R2_QUANTITY = determine_quantity_kind_and_dimensional_signature)
 R3_UNIT = identify_unit_system_symbol_prefix_and_conversion_semantics)
 R4_CONVERT = if_required convert_to_target_compatible_unit)
 R5_CHECK = verify_dimensional_consistency_and_unit_compatibility)
 R6_UNCERTAINTY = propagate_measurement_and_conversion_uncertainty)
 R7_CALIBRATION = if_instrumental identify_standard_traceability_and_calibration_status)
 R8_NORMALISE = map_aliases_symbols_and_units_to_canonical_identifiers)
 R9_VALIDATE = apply_range_plausibility_conversion_and_consistency_checks)
 R10_PROVENANCE = record_source_standard_versions_conversion_rules_and_measurement_context)
}

CONSTRAINTS {
 C1 = "Unit symbols can be ambiguous without context or standards."
 C2 = "Affine units such as temperature scales require offsets as well as scale factors."
 C3 = "Dimensionless quantities can still represent semantically different quantity kinds."
 C4 = "Equivalent dimensions do not imply equivalent physical quantities."
 C5 = "Rounding during repeated conversions can accumulate numerical error."
 C6 = "Uncertainty propagation depends on correlations between inputs."
 C7 = "Prefix interpretation must distinguish decimal and binary conventions."
 C8 = "Unit aliases can create semantic duplication if canonical mapping is absent."
 C9 = "Historical or domain-specific units can have context-dependent definitions."
 C10 = "Conversion correctness depends on both unit compatibility and exact conversion semantics."
}

SEMANTIC_EQUIVALENCE {
 "InternationalSystemOfUnits" <=> "SI"
 "MeasurementUnit" <=> "Unit"
 "UnitConversion" <=> "Conversion"
 "MeasurementUncertainty" <=> "Uncertainty"
}

VALIDATION {
 V1 = "Quantity kind is documented."
 V2 = "Unit symbol and canonical unit identity are documented."
 V3 = "Dimensional signature is documented or derivable."
 V4 = "Conversion factors and offsets are explicit."
 V5 = "Measurement scale semantics are documented where relevant."
 V6 = "Uncertainty and tolerance are distinguished."
 V7 = "Calibration and traceability are documented for instrument-derived measurements where applicable."
 V8 = "Canonical identifiers and aliases are mapped."
 V9 = "Range and dimensional consistency checks are applied."
 V10 = "Results are reproducible under documented standards, conversion rules and ontology versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MEASUREMENT_UNITS_DEFINITION = 0.99
 DIMENSION_REASONING = 0.99
 UNIT_CONVERSION_REASONING = 0.99
 UNCERTAINTY_REASONING = 0.98
 CALIBRATION_REASONING = 0.98
 SEMANTIC_MAPPING_REASONING = 0.98
 POLICY = "Reduce confidence when quantity kind, unit identity, scale type, conversion rule, uncertainty or standard provenance are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,MeasurementReasoning,
  QuantityReasoning,DimensionReasoning,UnitReasoning,
  UnitConversionReasoning,DimensionalAnalysisReasoning,
  MeasurementScaleReasoning,UncertaintyReasoning,
  CalibrationReasoning,TraceabilityReasoning,
  UnitNormalisationReasoning,SemanticMappingReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "measurement units ontologies" -> MeasurementUnitsOntologies
  "measurement unit" -> Unit
  "SI" -> InternationalSystemOfUnits
  "unit conversion" -> UnitConversion
  "measurement uncertainty" -> MeasurementUncertainty
 }
}

EXECUTION_RULES {
 E1 = "If a numerical value is interpreted as a measured or quantified property, activate MeasurementUnitsOntologies."
 E2 = "Identify the quantity kind and dimension before interpreting or converting units."
 E3 = "Convert only between dimensionally compatible units."
 E4 = "Use affine conversion semantics where offsets are required."
 E5 = "Preserve uncertainty and significant-figure semantics through conversions."
 E6 = "Use canonical unit identifiers while retaining source aliases and symbols."
 E7 = "Apply dimensional checks to equations involving measured quantities."
 E8 = "Record calibration and traceability for measurement-system outputs where relevant."
 E9 = "Document standards, unit definitions, conversion rules and provenance for reproducibility."
}

RECURSION {
 ENTRY = MeasurementUnitsOntologies
 EXPAND(MeasurementUnitsOntologies) -> [
  MeasurementOntology,UnitsOntology,QuantityOntology,
  Measurement,MeasurementValue,PhysicalQuantity,MeasurableQuantity,
  Dimension,PhysicalDimension,BaseDimension,DerivedDimension,
  DimensionlessQuantity,Unit,MeasurementUnit,BaseUnit,DerivedUnit,
  StandardUnit,ReferenceUnit,SIUnit,SIBaseUnit,SIDerivedUnit,
  MetricUnit,UnitSystem,SystemOfUnits,InternationalSystemOfUnits,
  Prefix,SIPrefix,UnitSymbol,QuantitySymbol,
  UnitExpression,CompoundUnit,ProductUnit,QuotientUnit,PowerUnit,
  Conversion,UnitConversion,ConversionFactor,ConversionOffset,
  MeasurementScale,RatioScale,IntervalScale,LogarithmicScale,
  DimensionalAnalysis,DimensionalConsistency,UnitChecking,
  QuantityCalculus,Calibration,ReferenceStandard,
  MeasurementTraceability,Metrology,ScientificMeasurement,
  Precision,Accuracy,Resolution,SignificantFigures,Rounding,
  MeasurementError,SystematicError,RandomError,Uncertainty,
  MeasurementUncertainty,StandardUncertainty,CombinedUncertainty,
  ExpandedUncertainty,Tolerance,PropagationOfUncertainty,
  ErrorBudget,ReferenceValue,MeasuredValue,NominalValue,
  PhysicalConstant,DefinedConstant,Length,Mass,Time,
  ElectricCurrent,ThermodynamicTemperature,AmountOfSubstance,
  LuminousIntensity,Area,Volume,Velocity,Acceleration,Force,
  Pressure,Energy,Power,Frequency,Density,Concentration,
  UnitRegistry,UnitDictionary,OntologyMapping,UnitEquivalence,
  Metadata,Provenance,Standard,BIPMrelationship,NISTrelationship,
  UCUMrelationship,QUDTRelationship,Verification,Validation
 ]
 EXPAND(MeasurementUnitsFamilies) -> [
  QuantitiesAndMeasurementValues,
  DimensionsAndDimensionalAnalysis,
  UnitsPrefixesAndUnitSystems,
  UnitExpressionsAndQuantityCalculus,
  ConversionScalesAndNormalisation,
  UncertaintyAccuracyPrecisionAndError,
  CalibrationTraceabilityAndStandards,
  SemanticMappingValidationAndProvenance
 ]
 STOP_CONDITION = "Stop when quantity, dimension, unit, scale, conversion, uncertainty, standard and provenance are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Ontologies"
 RELATED_ONTOLOGIES = [
  "Algebra Ontologies",
  "Geometry Ontologies",
  "Applied Mathematics Ontologies",
  "Computational Mathematics Ontologies",
  "Scientific Measurement Ontology",
  "Probability & Statistics Ontologies",
  "Calculus & Analysis Ontologies",
  "Physical Ontologies",
  "Scientific Ontologies",
  "Verification & Validation",
  "Reproducibility Ontology",
  "Computational Scientific Workflows"
 ]
 ROUTE_RULES = [
  "Measurement & Units Ontologies -> Mathematical Ontologies for parent semantics",
  "Measurement & Units Ontologies -> Algebra for unit algebra and symbolic quantity expressions",
  "Measurement & Units Ontologies -> Geometry for spatial quantities and coordinate measurement",
  "Measurement & Units Ontologies -> Applied Mathematics for domain-specific quantity use",
  "Measurement & Units Ontologies -> Computational Mathematics for numerical unit handling and conversion",
  "Measurement & Units Ontologies -> Scientific Measurement for instrumentation, calibration and metrology",
  "Measurement & Units Ontologies -> Probability & Statistics for uncertainty and error propagation",
  "Measurement & Units Ontologies -> Calculus & Analysis for rates, derivatives and dimensional quantities",
  "Measurement & Units Ontologies -> Physical and Scientific Ontologies for domain quantity semantics",
  "Measurement & Units Ontologies -> Verification & Validation for measurement and conversion checks",
  "Measurement & Units Ontologies -> Reproducibility and Computational Scientific Workflows for traceable computational measurement semantics"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible quantity kinds, unit systems, conversion rules or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve quantity-versus-unit distinction.",
  "Preserve dimension-versus-unit distinction.",
  "Preserve value-versus-measurement-result distinction.",
  "Preserve dimensionless-versus-unitless-semantic distinction.",
  "Preserve multiplicative-versus-affine conversion distinction.",
  "Preserve accuracy-versus-precision distinction.",
  "Preserve uncertainty-versus-error distinction.",
  "Preserve machine-readable measurement-and-unit semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Measurement and units ontologies formally represent quantities, dimensions, units, scales, conversions, uncertainty, calibration and standards."
 QUANTITY_SUMMARY = "A quantity combines a measurable property with a numerical value and a reference unit or scale."
 DIMENSION_SUMMARY = "Dimensions describe the physical nature of quantities independently of specific unit choices."
 UNIT_SUMMARY = "Units provide agreed references for expressing values of compatible quantity kinds."
 CONVERSION_SUMMARY = "Conversions preserve quantity meaning while transforming numerical representation between compatible units."
 UNCERTAINTY_SUMMARY = "Measurement results should include uncertainty, precision, calibration and traceability semantics where applicable."
 VALIDATION_SUMMARY = "Measurement-and-unit data should be dimensionally consistent, conversion-validated, provenance-linked and reproducible under declared standards."
}

FLOW {
 INPUT
 -> detect_measurement_units_context
 -> identify_quantity_value_unit_dimension_scale_measurement_method_uncertainty_and_target_use
 -> determine_quantity_kind_and_dimensional_signature
 -> identify_unit_system_symbol_prefix_and_canonical_unit
 -> if_conversion_required_validate_compatibility_and_apply_scale_factor_offset_or_declared_rule
 -> verify_dimensional_consistency_and_quantity_semantics
 -> propagate_measurement_conversion_and_input_uncertainty
 -> if_instrumental_record_calibration_standard_and_traceability
 -> normalise_aliases_symbols_and_identifiers_to_canonical_semantics
 -> apply_range_plausibility_conversion_and_consistency_checks
 -> record_source_standard_versions_conversion_rules_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-MEASUREMENT-UNITS-ONTOLOGIES-V1"
}

}

END_FLOW