SI Derived Units

AI INTERPRETATION NOTE
This Flow Algebra defines the SI Derived Units ontology within the Kireth Research Semantic Library. SI derived units are units formed coherently from products of powers of the seven SI base units according to defining quantity equations. It formalises derived quantities, dimensional algebra, specially named units, unit expressions, equivalence, conversion, validation, traceability and provenance.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "SI Derived Units"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Measurement & Units"
 SPECIALISATION = "SI Derived Units / Parent Ontology"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/si-derived-units.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SIDerivedUnits
 ENTITY_CLASS = CoherentDerivedUnitFramework
 PRIMARY_CONTEXT = MeasurementUnitsOntologies
 DEFINITION = "SI derived units are measurement units derived coherently from SI base units through algebraic products of powers corresponding to defining quantity equations."
 SYMBOLIC_FORMS = ["u_D=m^a kg^b s^c A^d K^e mol^f cd^g","dim(Q)=L^a M^b T^c I^d Theta^e N^f J^g","coherent_factor=1"]
 SEMANTIC_ROLE = "Machine-readable semantics for coherent SI derived units, named units, dimensional signatures, unit algebra, quantity equations, conversions and validation."
}

SCOPE {
 INCLUDE = [
 SIDerivedUnits,SIDerivedUnit,DerivedUnit,CoherentDerivedUnit,CoherentUnit,
 MeasurementUnit,Unit,InternationalSystemOfUnits,SI,InternationalSystemOfQuantities,ISQ,
 SIBaseUnits,SIBaseUnit,BaseUnit,DerivedQuantity,SIDerivedQuantity,BaseQuantity,
 Quantity,QuantityKind,PhysicalQuantity,QuantityValue,QuantityEquation,
 DefiningQuantityEquation,Dimension,BaseDimension,DerivedDimension,DimensionalSignature,
 DimensionlessQuantity,UnitExpression,UnitSymbol,UnitName,CompoundUnit,ProductUnit,
 QuotientUnit,PowerUnit,InverseUnit,UnitExponent,UnitAlgebra,QuantityCalculus,
 DimensionalAnalysis,DimensionalConsistency,DimensionalHomogeneity,Canonicalisation,
 UnitEquivalence,Second,Metre,Kilogram,Ampere,Kelvin,Mole,Candela,
 Radian,Steradian,Hertz,Newton,Pascal,Joule,Watt,Coulomb,Volt,Farad,Ohm,
 Siemens,Weber,Tesla,Henry,Lumen,Lux,Becquerel,Gray,Sievert,Katal,
 PlaneAngle,SolidAngle,Frequency,Force,Pressure,Energy,Power,ElectricCharge,
 Voltage,Capacitance,ElectricalResistance,ElectricalConductance,MagneticFlux,
 MagneticFluxDensity,Inductance,LuminousFlux,Illuminance,AbsorbedDose,
 CatalyticActivity,Area,Volume,Velocity,Acceleration,Density,CurrentDensityRelationship,
 DynamicViscosityRelationship,KinematicViscosityRelationship,SurfaceTensionRelationship,
 HeatCapacityRelationship,SpecificHeatCapacityRelationship,ThermalConductivityRelationship,
 EnergyDensityRelationship,ElectricFieldStrengthRelationship,ChargeDensityRelationship,
 AmountConcentrationRelationship,MolarEnergyRelationship,MolarEntropyRelationship,
 AngularVelocityRelationship,WaveNumberRelationship,Prefix,SIPrefix,
 UnitConversion,ConversionFactor,ExactConversion,MultiplicativeConversion,
 AffineConversion,UnitCompatibility,DimensionalCompatibility,QuantityCompatibility,
 NamedDerivedUnit,SpecialNameRelationship,ExpandedBaseUnitExpression,Measurement,
 Measurand,MeasurementResult,MeasurementUncertainty,ScientificMeasurement,
 MeasurementSystem,Metrology,Calibration,ReferenceStandard,MeasurementTraceability,
 MetrologicalTraceability,TraceabilityChain,Verification,Validation,UnitValidation,
 DimensionalValidation,ConversionValidation,QuantityEquationValidation,
 ErrorAnalysis,SymbolicComputation,ComputationalMathematics,Provenance,
 SIDerivedUnitProvenance,SIUnitProvenance,MeasurementProvenance,StandardProvenance,
 BIPMrelationship,SIbrochureRelationship,VIMRelationship,GUMRelationship
 ]
 EXCLUDE_AS_PRIMARY = [SIBaseUnitsWhenBaseUnitSemanticsArePrimary,NonSIUnitsWhenNonSISystemSemanticsArePrimary,PhysicalQuantityWithoutDerivedUnitSemantics]
 DISAMBIGUATION_RULE = "When a unit is algebraically constructed from SI base units or is a specially named coherent SI derived unit, interpret the context as an SI derived unit."
}

CORE_CONCEPTS {
 SIDerivedUnits = "Set of coherent units derived algebraically from SI base units."
 SIDerivedUnit = "SI unit derived from products of powers of SI base units according to a quantity equation."
 CoherentDerivedUnit = "Derived unit whose expression introduces no numerical factor other than one."
 DerivedQuantity = "Quantity defined in terms of base quantities or other quantities by a quantity equation."
 DimensionalSignature = "Vector of exponents describing dependence on the seven SI base dimensions."
 NamedDerivedUnit = "SI derived unit assigned a special name and symbol."
 UnitExpression = "Algebraic representation of a unit through products, quotients and powers."
}

RELATIONS {
 SIDerivedUnits --is_part_of--> InternationalSystemOfUnits
 SIDerivedUnit --is_derived_from--> SIBaseUnits
 SIDerivedUnit --measures--> DerivedQuantity
 DerivedQuantity --has--> DerivedDimension
 DerivedDimension --is_expressed_by--> DimensionalSignature
 NamedDerivedUnit --has_equivalent_expression--> ExpandedBaseUnitExpression
 Hertz --equals--> inverse_second
 Newton --equals--> kilogram_metre_per_second_squared
 Pascal --equals--> newton_per_square_metre
 Joule --equals--> newton_metre
 Watt --equals--> joule_per_second
 Coulomb --equals--> ampere_second
 Volt --equals--> watt_per_ampere
 Farad --equals--> coulomb_per_volt
 Ohm --equals--> volt_per_ampere
 Siemens --equals--> ampere_per_volt
 Weber --equals--> volt_second
 Tesla --equals--> weber_per_square_metre
 Henry --equals--> weber_per_ampere
 Lumen --equals--> candela_steradian
 Lux --equals--> lumen_per_square_metre
 Becquerel --equals--> inverse_second
 Gray --equals--> joule_per_kilogram
 Sievert --equals--> joule_per_kilogram
 Katal --equals--> mole_per_second
 MeasurementTraceability --links--> DerivedQuantityMeasurementToSI
}

OPERATORS {
 DERIVE_UNIT(exponents) := product_of_SI_base_units(exponents)
 DIMENSION(unit) := dimensional_signature(unit)
 EXPAND(unit) := canonical_base_unit_expression(unit)
 REDUCE(expression) := canonical_named_or_base_unit_expression(expression)
 COHERENT(expression) := numerical_scale_factor(expression)==1
 COMPATIBLE(u1,u2) := compatible_quantity_kind_and_dimension(u1,u2)
 CONVERT(value,u1,u2) := valid_compatible_unit_conversion(value,u1,u2)
 VALIDATE(expression) := dimensional_and_unit_algebra_validation(expression)
}

AXIOMS {
 A1 = "Every coherent SI derived unit can be expressed as a product of powers of the seven SI base units."
 A2 = "A coherent derived unit introduces no additional numerical factor in the defining quantity equation."
 A3 = "Special names and symbols do not alter the underlying base-unit expression."
 A4 = "Two quantities can share a dimensional signature without representing the same quantity kind."
 A5 = "Two specially named units can share an equivalent base-unit expression while retaining different semantic roles."
 A6 = "Unit symbols and their algebra are case-sensitive."
 A7 = "Dimensionless derived quantities can retain distinct semantic meanings."
 A8 = "Derived-unit reasoning requires explicit quantity equations, dimensions, unit expressions and provenance."
}

REASONING {
 R1_CONTEXT = identify(quantity_kind,quantity_equation,unit_expression,symbol_and_measurement_context)
 R2_QUANTITY = resolve_base_or_derived_quantity_and_semantic_role)
 R3_DIMENSION = derive_dimension_vector_from_quantity_equation)
 R4_UNIT = derive_coherent_SI_unit_from_base_unit_exponents)
 R5_NAMED = map_to_special_SI_name_and_symbol_when_defined)
 R6_EQUIVALENCE = compare_expanded_base_unit_expressions_and_quantity_semantics)
 R7_PREFIX = if_prefixed_apply_decimal_scaling_according_to_SI_rules)
 R8_CONVERSION = if_required_convert_between_compatible_units)
 R9_VALIDATE = check_dimensional_homogeneity_symbol_case_coherence_and_quantity_kind)
 R10_PROVENANCE = record_definition_source_standard_version_expression_and_traceability)
}

CONSTRAINTS {
 C1 = "Dimensional equivalence does not by itself prove quantity-kind equivalence."
 C2 = "Becquerel and hertz both reduce to s^-1 but represent different quantity kinds."
 C3 = "Gray and sievert both reduce to J kg^-1 but retain different measurement semantics."
 C4 = "Radian and steradian are dimensionless derived units with distinct geometric meanings."
 C5 = "Special unit names must not obscure the underlying coherent base-unit expression."
 C6 = "Prefix scaling must follow SI syntax."
 C7 = "Affine units require scale-origin semantics and cannot be handled as purely multiplicative units."
 C8 = "Unit cancellation must preserve quantity meaning and not only symbolic algebra."
 C9 = "Traceability to SI requires measurement evidence, not merely a correct unit expression."
 C10 = "Non-SI conventions must not be silently treated as coherent SI derived units."
}

SEMANTIC_EQUIVALENCE {
 "InternationalSystemOfUnits" <=> "SI"
 "InternationalSystemOfQuantities" <=> "ISQ"
 "MetrologicalTraceability" <=> "MeasurementTraceability"
 "SIDerivedUnit" <=> "SI Derived Unit"
}

VALIDATION {
 V1 = "Derived quantity and defining quantity equation are identified."
 V2 = "Dimensional signature is algebraically correct."
 V3 = "Derived-unit expression follows from the corresponding quantity equation."
 V4 = "Coherent expressions have numerical scale factor one."
 V5 = "Special names and symbols map to correct expanded SI expressions."
 V6 = "Quantity-kind distinctions are preserved when dimensions coincide."
 V7 = "Prefix scaling and symbol case follow SI rules."
 V8 = "Conversions occur only between semantically and dimensionally compatible quantities."
 V9 = "Measurement applications retain uncertainty and traceability evidence."
 V10 = "Definitions and expressions are traceable to documented standards and versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DERIVED_UNIT_DEFINITION = 0.99
 QUANTITY_EQUATION_REASONING = 0.99
 DIMENSIONAL_REASONING = 0.99
 UNIT_ALGEBRA_REASONING = 0.99
 NAMED_UNIT_REASONING = 0.99
 EQUIVALENCE_REASONING = 0.98
 CONVERSION_REASONING = 0.99
 POLICY = "Reduce confidence when quantity kind, defining equation, dimension, scale factor, prefix semantics or measurement context are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
 SemanticSearch,OntologyGrounding,SIDerivedUnitReasoning,DerivedQuantityReasoning,
 QuantityEquationReasoning,DimensionalAnalysisReasoning,UnitAlgebraReasoning,
 NamedUnitReasoning,UnitEquivalenceReasoning,UnitConversionReasoning,
 SymbolicComputation,MeasurementReasoning,MetrologyReasoning,TraceabilityReasoning,
 KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "SI derived units" -> SIDerivedUnits
  "SI derived unit" -> SIDerivedUnit
  "SI" -> InternationalSystemOfUnits
  "ISQ" -> InternationalSystemOfQuantities
 }
}

EXECUTION_RULES {
 E1 = "If a unit is coherently derived from SI base units, activate SIDerivedUnitsOntology."
 E2 = "Resolve the defining quantity equation before deriving the unit expression."
 E3 = "Derive the dimensional signature from quantity semantics."
 E4 = "Expand named derived units to canonical base-unit expressions when needed."
 E5 = "Preserve quantity-kind distinctions when dimensions or unit expressions coincide."
 E6 = "Apply prefixes and conversions according to SI scaling rules."
 E7 = "Check dimensional homogeneity and symbol case."
 E8 = "Distinguish unit-expression correctness from metrological traceability."
 E9 = "Record definition sources, standard versions and provenance."
}

RECURSION {
 ENTRY = SIDerivedUnits
 EXPAND(SIDerivedUnits) -> [
 SIUnits,SIBaseUnits,SIBaseUnit,DerivedQuantity,SIDerivedQuantity,Quantity,
 QuantityKind,QuantityEquation,Dimension,DerivedDimension,DimensionalSignature,
 Unit,DerivedUnit,CoherentUnit,UnitExpression,UnitAlgebra,QuantityCalculus,
 DimensionalAnalysis,Second,Metre,Kilogram,Ampere,Kelvin,Mole,Candela,
 Radian,Steradian,Hertz,Newton,Pascal,Joule,Watt,Coulomb,Volt,Farad,Ohm,
 Siemens,Weber,Tesla,Henry,Lumen,Lux,Becquerel,Gray,Sievert,Katal,
 Area,Volume,Velocity,Acceleration,Density,Pressure,Energy,Power,
 ElectricCharge,Voltage,Capacitance,ElectricalResistance,MagneticFlux,
 MagneticFluxDensity,Inductance,LuminousFlux,Illuminance,AbsorbedDose,
 CatalyticActivity,SIPrefix,UnitConversion,Measurement,MeasurementUncertainty,
 Metrology,ReferenceStandard,MeasurementTraceability,Verification,Validation,
 SIDerivedUnitProvenance
 ]
 EXPAND(SIDerivedUnitFamilies) -> [
 GeometryAngleAreaAndVolumeUnits,
 MechanicsFrequencyForcePressureEnergyAndPowerUnits,
 ElectricalChargeVoltageResistanceCapacitanceAndConductanceUnits,
 MagnetismFluxFluxDensityAndInductanceUnits,
 ThermalAndMaterialPropertyDerivedUnits,
 PhotometricLumenLuxAndRelatedUnits,
 RadiationActivityDoseAndEquivalentDoseUnits,
 ChemistryConcentrationMolarAndCatalyticUnits,
 DimensionalAlgebraConversionValidationAndProvenance
 ]
 STOP_CONDITION = "Stop when quantity equation, dimensional signature, coherent unit expression, named-unit mapping, compatibility, validation and provenance are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Measurement & Units Ontologies"
 RELATED_ONTOLOGIES = [
 "Measurement & Units Ontologies","SI Base Units","SI Units Ontology",
 "Foundations of Measurement","Quantity Ontology","Physical Quantity Ontology",
 "Dimension Ontology 2","Unit Ontology","Measurement Ontology",
 "Scientific Measurement Ontology 2","Metrology Ontology",
 "Reference Standard Ontology","Measurement System Ontology",
 "Measurement Uncertainty Ontology","Dimensional Analysis Ontology",
 "Symbolic Computation","Verification & Validation"
 ]
 ROUTE_RULES = [
 "SI Derived Units -> SI Base Units for base-unit dependencies",
 "SI Derived Units -> SI Units Ontology for system-wide SI semantics",
 "SI Derived Units -> Quantity Ontology for derived-quantity equations",
 "SI Derived Units -> Dimension Ontology for dimensional signatures",
 "SI Derived Units -> Unit Ontology for unit identity, algebra and conversion",
 "SI Derived Units -> Dimensional Analysis for exponent and homogeneity reasoning",
 "SI Derived Units -> Symbolic Computation for algebraic expansion and reduction",
 "SI Derived Units -> Metrology and Reference Standard for measurement traceability",
 "SI Derived Units -> Measurement Uncertainty for uncertainty-bearing results",
 "SI Derived Units -> Verification & Validation for expression and conversion checks"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible derived units, quantity equations or reasoning relationships"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
 "Preserve base-unit-versus-derived-unit distinction.",
 "Preserve coherent-versus-noncoherent distinction.",
 "Preserve unit-versus-quantity distinction.",
 "Preserve dimension-versus-quantity-kind distinction.",
 "Preserve named-unit-versus-expanded-expression equivalence.",
 "Preserve dimensionless-versus-semantically-empty distinction.",
 "Preserve unit-expression-versus-metrological-traceability distinction.",
 "Preserve machine-readable SI-derived-unit semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "SI derived units are coherent units formed from algebraic products of powers of the seven SI base units according to defining quantity equations."
 DIMENSION_SUMMARY = "Each derived quantity has a dimensional signature expressed through exponents of the seven SI base dimensions."
 COHERENCE_SUMMARY = "A coherent SI derived unit follows its defining quantity equation without an additional numerical factor."
 NAMED_UNIT_SUMMARY = "Selected derived units have special SI names and symbols such as hertz, newton, pascal, joule, watt, volt and tesla."
 EQUIVALENCE_SUMMARY = "Named units can be expanded into base-unit expressions, but equal dimensions do not necessarily imply equal quantity semantics."
 VALIDATION_SUMMARY = "Derived-unit reasoning should preserve quantity equations, dimensional homogeneity, SI symbol rules, compatibility, uncertainty and provenance."
}

FLOW {
 INPUT
 -> detect_SI_derived_unit_context
 -> identify_quantity_kind_quantity_equation_unit_expression_symbol_and_measurement_context
 -> resolve_base_or_derived_quantity_and_semantic_role
 -> derive_dimension_vector_from_quantity_equation
 -> derive_coherent_SI_unit_from_base_unit_exponents
 -> map_to_special_SI_name_and_symbol_when_defined
 -> compare_expanded_base_unit_expressions_and_quantity_semantics
 -> if_prefixed_apply_decimal_scaling_according_to_SI_rules
 -> if_required_convert_between_compatible_units
 -> check_dimensional_homogeneity_symbol_case_coherence_and_quantity_kind
 -> record_definition_source_standard_version_expression_and_traceability
 -> 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-SI-DERIVED-UNITS-V1"
}

}

END_FLOW