Derived Units Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Derived Units Ontology within the Kireth Research Semantic Library. A derived unit is a measurement unit associated with a derived quantity and expressed algebraically in terms of other units, ultimately reducible within SI to products of powers of SI base units. It formalises quantity equations, dimensional signatures, coherent and non-coherent units, named units, equivalence, conversion, validation, traceability and provenance.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = DerivedUnit
 ENTITY_CLASS = DerivedMeasurementUnitFramework
 PRIMARY_CONTEXT = SIDerivedUnits
 DEFINITION = "A derived unit is a measurement unit for a derived quantity, expressed algebraically in terms of base units or other units according to a defining quantity equation."
 SYMBOLIC_FORMS = ["u_D=product(u_i^a_i)","SI:u_D=m^a kg^b s^c A^d K^e mol^f cd^g","dim(Q_D)=L^a M^b T^c I^d Theta^e N^f J^g"]
 SEMANTIC_ROLE = "Machine-readable semantics for derived-unit identity, quantity equations, dimensional structure, coherence, named units, algebraic equivalence, conversion and measurement use."
}

SCOPE {
 INCLUDE = [
 DerivedUnit,SIDerivedUnit,CoherentDerivedUnit,NonCoherentDerivedUnit,
 MeasurementUnit,Unit,BaseUnit,SIBaseUnit,InternationalSystemOfUnits,SI,
 DerivedQuantity,SIDerivedQuantity,BaseQuantity,Quantity,QuantityKind,
 PhysicalQuantity,QuantityValue,QuantityEquation,DefiningQuantityEquation,
 Dimension,BaseDimension,DerivedDimension,DimensionalSignature,
 DimensionlessQuantity,UnitExpression,UnitSymbol,UnitName,CompoundUnit,
 ProductUnit,QuotientUnit,PowerUnit,InverseUnit,UnitExponent,UnitFactor,
 NumericalScaleFactorRelationship,UnitAlgebra,QuantityCalculus,
 DimensionalAnalysis,DimensionalConsistency,DimensionalHomogeneity,
 Canonicalisation,UnitNormalisation,UnitEquivalence,
 AlgebraicUnitEquivalence,SemanticUnitEquivalence,NamedDerivedUnit,
 SpecialNameRelationship,SpecialSymbolRelationship,ExpandedUnitExpression,
 SIBaseUnits,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,Frequency,Force,MomentumRelationship,
 Pressure,StressRelationship,Energy,Power,Density,ElectricCharge,Voltage,
 Capacitance,ElectricalResistance,ElectricalConductance,MagneticFlux,
 MagneticFluxDensity,Inductance,HeatCapacityRelationship,
 ThermalConductivityRelationship,EntropyRelationship,LuminousFlux,
 Illuminance,LuminanceRelationship,AbsorbedDose,DoseEquivalentRelationship,
 CatalyticActivity,AmountConcentrationRelationship,MolarEnergyRelationship,
 Prefix,SIPrefix,UnitConversion,ConversionFactor,ConversionOffset,
 ExactConversion,MultiplicativeConversion,AffineConversion,
 UnitCompatibility,DimensionalCompatibility,QuantityCompatibility,
 Measurement,Measurand,MeasurementProcedure,MeasurementModel,
 MeasurementResult,MeasurementUncertainty,ScientificMeasurement,
 MeasurementSystem,Metrology,Calibration,ReferenceStandard,
 MeasurementTraceability,MetrologicalTraceability,TraceabilityChain,
 Verification,Validation,UnitValidation,DimensionalValidation,
 ConversionValidation,QuantityEquationValidation,ErrorAnalysis,
 SymbolicComputation,ComputationalMathematics,Provenance,
 DerivedUnitProvenance,SIUnitProvenance,MeasurementProvenance,
 StandardProvenance,BIPMrelationship,SIbrochureRelationship,
 VIMRelationship,GUMRelationship
 ]
 EXCLUDE_AS_PRIMARY = [
 BaseUnitWhenBaseUnitIdentityIsPrimary,
 QuantityWithoutUnitSemantics,
 PureDimensionWithoutDerivedUnitContext,
 NumericalConversionWithoutQuantityOrUnitIdentity
 ]
 DISAMBIGUATION_RULE = "When a measurement unit is defined algebraically from other units for a derived quantity, interpret it as a derived unit; determine separately whether it is coherent, SI, specially named, prefixed or non-SI."
}

CORE_CONCEPTS {
 DerivedUnit = "Measurement unit associated with a derived quantity and expressed algebraically in terms of other units."
 DerivedQuantity = "Quantity defined through a quantity equation involving other quantities."
 CoherentDerivedUnit = "Derived unit whose defining unit equation introduces no numerical factor other than one."
 NonCoherentDerivedUnit = "Derived unit whose relation to the coherent unit includes a numerical factor or other transformation."
 NamedDerivedUnit = "Derived unit assigned a conventional or standard special name and symbol."
 DimensionalSignature = "Exponent vector expressing a derived quantity in terms of base dimensions."
 UnitExpression = "Symbolic algebraic expression representing a derived unit."
}

RELATIONS {
 DerivedUnit --is_type_of--> MeasurementUnit
 DerivedUnit --measures--> DerivedQuantity
 DerivedQuantity --is_defined_by--> QuantityEquation
 DerivedQuantity --has--> DerivedDimension
 DerivedDimension --is_expressed_by--> DimensionalSignature
 DerivedUnit --has--> UnitExpression
 CoherentDerivedUnit --is_type_of--> DerivedUnit
 SIDerivedUnit --is_type_of--> CoherentDerivedUnit
 NamedDerivedUnit --is_type_of--> DerivedUnit
 NamedDerivedUnit --has_equivalent_expression--> ExpandedUnitExpression
 UnitConversion --maps_between--> CompatibleUnits
 MeasurementResult --uses--> DerivedUnit
 MeasurementTraceability --supports--> DerivedUnitMeasurement
 Provenance --records--> DerivedUnitDefinitionAndUse
}

OPERATORS {
 DERIVE(quantity_equation,units) := derived_unit_expression(quantity_equation,units)
 DIM(quantity) := dimensional_signature(quantity)
 EXPAND(unit) := expanded_unit_expression(unit)
 REDUCE(expression) := canonical_unit_expression(expression)
 COHERENCE(unit) := evaluate_numerical_scale_factor(unit)
 EQUIVALENT(u1,u2) := compare_scale_dimension_and_quantity_semantics(u1,u2)
 COMPATIBLE(u1,u2) := compatible_quantity_kind_and_dimension(u1,u2)
 CONVERT(value,u1,u2) := valid_conversion(value,u1,u2)
 VALIDATE(unit,quantity) := validate_unit_against_quantity_equation(unit,quantity)
}

AXIOMS {
 A1 = "A derived unit corresponds to a derived quantity."
 A2 = "A derived-unit expression follows from a defining quantity equation and the units assigned to its constituent quantities."
 A3 = "A coherent derived unit has numerical scale factor one relative to the unit expression generated by the defining quantity equation."
 A4 = "Not every derived unit is an SI unit."
 A5 = "Not every SI derived unit requires a special name or symbol."
 A6 = "Equal dimensions do not necessarily imply identical quantity kinds."
 A7 = "A named unit and its expanded algebraic expression can represent the same unit."
 A8 = "Derived-unit reasoning requires quantity, dimension, scale, algebraic and semantic context."
}

REASONING {
 R1_CONTEXT = identify(quantity_kind,quantity_equation,unit_system,unit_expression,symbol_and_intended_use)
 R2_QUANTITY = determine_derived_quantity_and_defining_relationship)
 R3_DIMENSION = derive_dimensional_signature_from_quantity_equation)
 R4_EXPRESSION = substitute_constituent_units_and_simplify_unit_algebra)
 R5_CLASSIFY = determine_SI_nonSI_coherent_noncoherent_named_prefixed_or_compound_status)
 R6_EQUIVALENCE = compare_scale_dimension_quantity_kind_and_expanded_expressions)
 R7_CONVERSION = if_required_construct_valid_conversion_between_compatible_units)
 R8_MEASUREMENT = if_measurement_context_attach_uncertainty_calibration_and_traceability)
 R9_VALIDATE = verify_quantity_equation_dimensional_homogeneity_symbol_syntax_and_scale)
 R10_PROVENANCE = record_definition_source_unit_system_standard_version_and_semantic_history)
}

CONSTRAINTS {
 C1 = "Dimensional equality alone is insufficient to establish semantic unit equivalence."
 C2 = "A non-coherent derived unit may remain valid even though its scale factor differs from the coherent unit."
 C3 = "Affine conversions cannot be represented correctly by a multiplicative factor alone."
 C4 = "A prefix changes unit magnitude but does not change the underlying quantity kind."
 C5 = "Unit simplification must not erase meaningful quantity distinctions."
 C6 = "Named-unit substitution must preserve the same unit magnitude and quantity context."
 C7 = "Dimensionless derived quantities can retain nontrivial semantic meaning."
 C8 = "Case-sensitive symbols must remain exact during symbolic manipulation."
 C9 = "Correct unit algebra does not by itself establish metrological traceability."
 C10 = "Historical or domain-specific unit conventions require explicit provenance and system identification."
}

SEMANTIC_EQUIVALENCE {
 "DerivedMeasurementUnit" <=> "DerivedUnit"
 "InternationalSystemOfUnits" <=> "SI"
 "MetrologicalTraceability" <=> "MeasurementTraceability"
 "UnitExpression" <=> "Unit Algebra Expression"
}

VALIDATION {
 V1 = "Derived quantity is explicitly identified."
 V2 = "Defining quantity equation is documented or inferable."
 V3 = "Dimensional signature agrees with the defining quantity equation."
 V4 = "Unit expression follows algebraically from constituent units."
 V5 = "Coherent versus non-coherent status is correctly classified."
 V6 = "SI versus non-SI status is correctly classified."
 V7 = "Named-unit substitutions preserve algebraic equivalence."
 V8 = "Conversions preserve quantity semantics and compatibility."
 V9 = "Measurement use retains uncertainty and traceability information."
 V10 = "Definition source, unit system and version provenance are retained."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DERIVED_UNIT_DEFINITION = 0.99
 QUANTITY_EQUATION_REASONING = 0.99
 DIMENSIONAL_REASONING = 0.99
 UNIT_EXPRESSION_REASONING = 0.99
 COHERENCE_CLASSIFICATION = 0.98
 EQUIVALENCE_REASONING = 0.98
 CONVERSION_REASONING = 0.99
 POLICY = "Reduce confidence when quantity equation, unit system, scale factor, dimensional signature, quantity kind or conversion semantics are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
 SemanticSearch,OntologyGrounding,DerivedUnitReasoning,DerivedQuantityReasoning,
 QuantityEquationReasoning,DimensionalAnalysisReasoning,UnitAlgebraReasoning,
 CoherenceReasoning,NamedUnitReasoning,UnitEquivalenceReasoning,
 UnitConversionReasoning,SymbolicComputation,MeasurementReasoning,
 MetrologyReasoning,TraceabilityReasoning,KnowledgeGraphConstruction,
 RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "derived unit" -> DerivedUnit
  "derived units" -> DerivedUnit
  "SI derived unit" -> SIDerivedUnit
  "coherent derived unit" -> CoherentDerivedUnit
 }
}

EXECUTION_RULES {
 E1 = "If a measurement unit is algebraically defined from other units for a derived quantity, activate DerivedUnitsOntology."
 E2 = "Identify the defining quantity equation before deriving the unit expression."
 E3 = "Derive and validate the dimensional signature."
 E4 = "Substitute constituent units and simplify algebraically."
 E5 = "Classify the unit as SI or non-SI, coherent or non-coherent, named, prefixed or compound."
 E6 = "Preserve quantity-kind distinctions when dimensions coincide."
 E7 = "Use affine conversion semantics when an offset is required."
 E8 = "Distinguish algebraic validity from metrological traceability."
 E9 = "Record unit-system, standard-version and definition provenance."
}

RECURSION {
 ENTRY = DerivedUnit
 EXPAND(DerivedUnit) -> [
 MeasurementUnit,Unit,BaseUnit,SIBaseUnit,SIDerivedUnit,CoherentDerivedUnit,
 NonCoherentDerivedUnit,DerivedQuantity,Quantity,QuantityKind,QuantityEquation,
 Dimension,DerivedDimension,DimensionalSignature,UnitExpression,UnitAlgebra,
 QuantityCalculus,DimensionalAnalysis,NamedDerivedUnit,ExpandedUnitExpression,
 SIBaseUnits,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,Frequency,Force,Pressure,Energy,Power,Density,
 ElectricCharge,Voltage,Capacitance,ElectricalResistance,MagneticFlux,
 MagneticFluxDensity,Inductance,LuminousFlux,Illuminance,AbsorbedDose,
 CatalyticActivity,SIPrefix,UnitConversion,Measurement,MeasurementResult,
 MeasurementUncertainty,Metrology,ReferenceStandard,MeasurementTraceability,
 Verification,Validation,DerivedUnitProvenance
 ]
 EXPAND(DerivedUnitFamilies) -> [
 DerivedQuantityAndQuantityEquationSemantics,
 DimensionalSignatureAndBaseDimensionExpansion,
 CoherentAndNonCoherentDerivedUnits,
 SIAndNonSIDerivedUnitClassification,
 NamedUnitsSymbolsAndExpandedExpressions,
 CompoundProductQuotientPowerAndInverseUnits,
 PrefixesScaleFactorsAffineAndMultiplicativeConversions,
 MeasurementUncertaintyTraceabilityValidationAndProvenance
 ]
 STOP_CONDITION = "Stop when derived quantity, defining equation, dimensional signature, unit expression, classification, equivalence, conversion and provenance are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "SI Derived Units"
 RELATED_ONTOLOGIES = [
 "SI Derived Units","SI Base Units","SI Units Ontology",
 "Measurement & Units Ontologies","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 = [
 "Derived Units -> SI Derived Units for SI-specific derived-unit semantics",
 "Derived Units -> SI Base Units for base-unit expansion",
 "Derived Units -> Quantity Ontology for defining quantity equations",
 "Derived Units -> Dimension Ontology for dimensional signatures",
 "Derived Units -> Unit Ontology for general unit identity and conversion",
 "Derived Units -> Dimensional Analysis for algebraic and homogeneity reasoning",
 "Derived Units -> Symbolic Computation for expression simplification",
 "Derived Units -> Measurement and Metrology for measurement use and traceability",
 "Derived Units -> Measurement Uncertainty for uncertainty-bearing results",
 "Derived Units -> Verification & Validation for dimensional and conversion checks"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible derived-unit classes, equations or reasoning relationships"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
 "Preserve base-unit-versus-derived-unit distinction.",
 "Preserve SI-versus-non-SI distinction.",
 "Preserve coherent-versus-non-coherent distinction.",
 "Preserve unit-versus-quantity distinction.",
 "Preserve dimension-versus-quantity-kind distinction.",
 "Preserve named-unit-versus-expanded-expression equivalence.",
 "Preserve multiplicative-versus-affine-conversion distinction.",
 "Preserve algebraic-validity-versus-metrological-traceability distinction."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A derived unit is a measurement unit for a derived quantity, expressed algebraically in terms of base units or other units according to a defining quantity equation."
 QUANTITY_SUMMARY = "Derived units inherit their structure from equations defining their associated derived quantities."
 DIMENSION_SUMMARY = "Dimensional signatures describe derived quantities through powers of base dimensions."
 COHERENCE_SUMMARY = "Coherent derived units introduce no additional numerical factor; non-coherent units can include scale factors or other transformations."
 CLASSIFICATION_SUMMARY = "Derived units can be SI or non-SI, coherent or non-coherent, named, prefixed, compound or dimensionless."
 EQUIVALENCE_SUMMARY = "Algebraic equivalence requires compatible scale and unit expressions, while semantic equivalence also requires compatible quantity meaning."
 VALIDATION_SUMMARY = "Derived-unit reasoning should preserve quantity equations, dimensions, scale, symbols, conversions, measurement uncertainty, traceability and provenance."
}

FLOW {
 INPUT
 -> detect_derived_unit_context
 -> identify_quantity_kind_quantity_equation_unit_system_unit_expression_symbol_and_intended_use
 -> determine_derived_quantity_and_defining_relationship
 -> derive_dimensional_signature_from_quantity_equation
 -> substitute_constituent_units_and_simplify_unit_algebra
 -> determine_SI_nonSI_coherent_noncoherent_named_prefixed_or_compound_status
 -> compare_scale_dimension_quantity_kind_and_expanded_expressions
 -> if_required_construct_valid_conversion_between_compatible_units
 -> if_measurement_context_attach_uncertainty_calibration_and_traceability
 -> verify_quantity_equation_dimensional_homogeneity_symbol_syntax_and_scale
 -> record_definition_source_unit_system_standard_version_and_semantic_history
 -> 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-DERIVED-UNITS-ONTOLOGY-V1"
}

}

END_FLOW