Dimension Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Dimension Ontology 2 within the Kireth Research Semantic Library. Dimension represents how a quantity depends on selected base quantities independently of the units used to express it. It formalises base and derived dimensions, dimensional signatures, dimensionless quantities, dimensional algebra, homogeneity, compatibility, inference, validation and provenance.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Dimension Ontology 2"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Measurement & Units"
 SPECIALISATION = "Foundations of Measurement / Dimension"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/foundations-of-measurement/dimension-ontology-2.html"
 LANGUAGE = "en-GB"
 VERSION = "2.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Dimension
 ENTITY_CLASS = QuantityDimensionalStructureFramework
 PRIMARY_CONTEXT = QuantityOntology
 DEFINITION = "A dimension is the expression of the dependence of a quantity on the base quantities of a chosen system of quantities, represented as products of powers of base dimensions."
 SYMBOLIC_FORMS = ["[Q]=L^a M^b T^c I^d Theta^e N^f J^g","[Q1*Q2]=[Q1][Q2]","[Q1/Q2]=[Q1][Q2]^-1","[Q^n]=[Q]^n"]
 SEMANTIC_ROLE = "Machine-readable semantics for dimensional signatures, base dimensions, derived dimensions, dimensional algebra, homogeneity, compatibility and inference."
}

SCOPE {
 INCLUDE = [
 Dimension,PhysicalDimension,QuantityDimension,DimensionalSignature,
 DimensionalFormula,DimensionVector,DimensionTuple,DimensionExpression,
 BaseDimension,DerivedDimension,DimensionlessQuantity,DimensionOne,
 DimensionlessDimension,DimensionalExponent,DimensionExponentVector,
 BaseQuantity,DerivedQuantity,PhysicalQuantity,Quantity,QuantityKind,
 PhysicalQuantityKind,QuantityValue,Unit,MeasurementUnit,
 UnitCompatibility,QuantityCompatibility,DimensionalCompatibility,
 DimensionalEquivalence,DimensionEquality,DimensionComparison,
 DimensionMapping,DimensionInference,DimensionalAnalysis,
 DimensionalAlgebra,DimensionAlgebra,DimensionalEquation,
 DimensionalConstraint,DimensionalConsistency,DimensionalHomogeneity,
 PrincipleOfDimensionalHomogeneity,DimensionalValidation,DimensionCheck,
 UnitCheck,ConsistencyCheck,DimensionalError,DimensionMismatch,
 ProductDimension,QuotientDimension,PowerDimension,InverseDimension,
 CompositeDimension,Multiplication,Division,Exponentiation,Addition,
 Subtraction,SystemOfQuantities,QuantitySystem,BaseQuantitySet,
 InternationalSystemOfQuantities,ISQ,InternationalSystemOfUnits,SI,
 SIBaseQuantity,SIBaseUnit,LengthDimension,MassDimension,TimeDimension,
 ElectricCurrentDimension,ThermodynamicTemperatureDimension,
 AmountOfSubstanceDimension,LuminousIntensityDimension,AreaDimension,
 VolumeDimension,VelocityDimension,AccelerationDimension,
 FrequencyDimension,ForceDimension,PressureDimension,EnergyDimension,
 PowerDimension,DensityDimension,ChargeDimension,VoltageDimension,
 ResistanceDimension,ConductanceDimension,CapacitanceDimension,
 InductanceDimension,MagneticFluxDimension,MagneticFluxDensityDimension,
 DimensionlessNumber,ReynoldsNumber,MachNumber,PrandtlNumber,
 NusseltNumber,PecletNumber,BuckinghamPiTheorem,BuckinghamPiGroup,
 PiGroup,DimensionalReduction,NonDimensionalisation,
 DimensionlessVariable,CharacteristicScale,ReferenceScale,
 ScalingVariable,NormalisedQuantity,ScaledQuantity,SimilarityRelationship,
 ScalingLaw,DimensionMatrix,RankRelationship,NullSpaceRelationship,
 DimensionalBasisRelationship,DimensionalInferenceRule,
 SymbolicDimensionInference,AutomaticUnitCheckingRelationship,
 DimensionalTypeSafetyRelationship,Measurement,Measurand,
 MeasurementResult,MeasurementUncertainty,ScientificMeasurement,
 Metrology,Calibration,MeasurementTraceability,QuantityEquation,
 ConservationLaw,PhysicalConstant,DimensionedConstant,
 DimensionlessConstant,ConversionFactor,UnitConversion,CoherentUnit,
 DerivedUnit,CompoundUnit,ProductUnit,QuotientUnit,PowerUnit,
 UnitExpression,Provenance,DimensionProvenance,QuantityProvenance,
 DefinitionSource,BIPMrelationship,VIMRelationship,SIbrochureRelationship,
 QUDTRelationship,OMRelationship,Verification,Validation,ErrorAnalysis,
 ScientificSimulation,ComputationalScientificWorkflow,SymbolicComputation
 ]
 EXCLUDE_AS_PRIMARY = [
 SpatialDimensionWhenGeometricCoordinateAxesArePrimary,
 MatrixDimensionWhenRowsAndColumnsArePrimary,
 FractalDimensionWhenScalingGeometryIsPrimary,
 DatabaseDimensionWhenDataWarehousingIsPrimary,
 UnitDefinitionWhenDimensionalStructureIsSecondary
 ]
 DISAMBIGUATION_RULE = "When dimension means the dependence of a measurable quantity on base quantities, rather than spatial axes, matrix size or fractal dimension, interpret the context as quantity dimension."
}

CORE_CONCEPTS {
 Dimension = "Expression of a quantity's dependence on selected base quantities."
 BaseDimension = "Dimension associated with a base quantity."
 DerivedDimension = "Dimension obtained algebraically from powers of base dimensions."
 DimensionalSignature = "Canonical exponent representation of a quantity dimension."
 DimensionalExponent = "Exponent of a base dimension in a dimensional signature."
 DimensionlessDimension = "Dimension whose base-dimensional exponents are all zero."
 DimensionalHomogeneity = "Property that terms in a valid quantity equation have compatible dimensions."
 DimensionalAnalysis = "Use of dimensions to analyse equations, derive relationships and construct dimensionless groups."
}

RELATIONS {
 Dimension --is_child_of--> QuantityOntology
 PhysicalQuantity --has_dimension--> Dimension
 BaseQuantity --has_dimension--> BaseDimension
 DerivedQuantity --has_dimension--> DerivedDimension
 DimensionalSignature --contains--> DimensionalExponents
 DerivedDimension --is_composed_of--> BaseDimensions
 Unit --has_dimension--> Dimension
 DimensionlessQuantity --has_dimension--> DimensionlessDimension
 QuantityEquation --must_satisfy--> DimensionalHomogeneity
 DimensionalAnalysis --uses--> DimensionalSignatures
 BuckinghamPiTheorem --produces--> DimensionlessGroups
 NonDimensionalisation --produces--> DimensionlessVariables
 DimensionalValidation --detects--> DimensionMismatch
 Provenance --records--> DimensionalDefinitionAndSource
}

OPERATORS {
 DIM(q) := dimensional_signature(q)
 PRODUCT_DIM(d1,d2) := add_exponent_vectors(d1,d2)
 QUOTIENT_DIM(d1,d2) := subtract_exponent_vectors(d1,d2)
 POWER_DIM(d,n) := scalar_multiply_exponents(d,n)
 INVERSE_DIM(d) := scalar_multiply_exponents(d,-1)
 DIMENSIONLESS(d) := all_exponents_zero(d)
 COMPATIBLE(d1,d2) := exponent_vectors_equal(d1,d2)
 CHECK_HOMOGENEITY(equation) := all_additive_terms_dimensionally_compatible(equation)
}

AXIOMS {
 A1 = "Dimension depends on the chosen system of base quantities but not on particular units."
 A2 = "Multiplication of quantities adds dimensional exponents."
 A3 = "Division of quantities subtracts dimensional exponents."
 A4 = "Exponentiation multiplies dimensional exponents by the power."
 A5 = "Quantities with identical dimensions need not be quantities of the same kind."
 A6 = "Addition and subtraction require dimensionally compatible terms."
 A7 = "Dimensionless quantities can remain semantically distinct."
 A8 = "Dimensional consistency is necessary but not sufficient for physical correctness."
}

REASONING {
 R1_CONTEXT = identify(quantity_or_expression,quantity_system,base_dimensions,units_and_intended_analysis)
 R2_BASES = identify_selected_base_quantities_and_corresponding_dimensions)
 R3_SIGNATURE = derive_exponent_vector_for_each_quantity_or_expression)
 R4_ALGEBRA = propagate_dimensions_through_products_quotients_powers_and_functions)
 R5_HOMOGENEITY = check_additive_terms_equation_sides_and_constraints)
 R6_COMPATIBILITY = distinguish_dimensional_equivalence_from_quantity_kind_equivalence)
 R7_NONDIMENSIONALISE = if_required_construct_characteristic_scales_and_dimensionless_groups)
 R8_INFER = infer_unknown_dimensions_from_valid_quantity_equations)
 R9_VALIDATE = detect_dimension_mismatch_unit_mismatch_and_invalid_operations)
 R10_PROVENANCE = record_quantity_system_base_set_definitions_equations_and_standard_source)
}

CONSTRAINTS {
 C1 = "Identical dimensions do not imply identical physical meaning."
 C2 = "A dimensionless result can still require quantity-kind semantics."
 C3 = "Transcendental functions generally require dimensionless arguments unless a specialised convention is defined."
 C4 = "Addition of unequal dimensions is invalid."
 C5 = "Affine scale values require care because scale origins are not represented by dimensional signatures."
 C6 = "Changing the base-quantity system can change representation without changing physical relationships."
 C7 = "Dimensional analysis cannot determine arbitrary dimensionless numerical constants."
 C8 = "Dimensional homogeneity cannot establish correctness of signs, coefficients or functional forms."
 C9 = "Non-dimensionalisation depends on appropriate characteristic scales."
 C10 = "Automated dimension inference requires unambiguous quantity and operator semantics."
}

SEMANTIC_EQUIVALENCE {
 "DimensionalFormula" <=> "DimensionalSignature"
 "InternationalSystemOfUnits" <=> "SI"
 "InternationalSystemOfQuantities" <=> "ISQ"
 "DimensionOne" <=> "DimensionlessDimension"
}

VALIDATION {
 V1 = "Quantity system and base dimensions are identified."
 V2 = "Dimensional signatures are documented or derivable."
 V3 = "Products, quotients and powers propagate dimensions algebraically."
 V4 = "Addition and subtraction occur only between compatible dimensions."
 V5 = "Both sides of quantity equations are dimensionally homogeneous."
 V6 = "Dimensionless quantities retain quantity-kind semantics."
 V7 = "Unit compatibility is not inferred from dimension alone when quantity kinds differ."
 V8 = "Non-dimensionalisation records characteristic scales."
 V9 = "Dimension inference preserves equation constraints."
 V10 = "Dimensional definitions and mappings are traceable to documented standards or ontology versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DIMENSION_DEFINITION = 0.99
 BASE_DIMENSION_REASONING = 0.99
 DERIVED_DIMENSION_REASONING = 0.99
 HOMOGENEITY_REASONING = 0.99
 DIMENSIONLESS_REASONING = 0.98
 BUCKINGHAM_PI_REASONING = 0.98
 DIMENSION_INFERENCE_REASONING = 0.98
 POLICY = "Reduce confidence when quantity system, base dimensions, quantity kinds, operator semantics or characteristic scales are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
 SemanticSearch,OntologyGrounding,DimensionReasoning,
 DimensionalSignatureReasoning,BaseDimensionReasoning,
 DerivedDimensionReasoning,DimensionalAlgebraReasoning,
 DimensionalHomogeneityReasoning,DimensionlessReasoning,
 BuckinghamPiReasoning,NonDimensionalisationReasoning,
 DimensionInferenceReasoning,UnitCompatibilityReasoning,
 QuantityEquationReasoning,DimensionalValidationReasoning,
 SymbolicReasoning,KnowledgeGraphConstruction,RetrievalAugmentation,
 RecursiveInference
 ]
 NORMALISE {
  "dimension" -> Dimension
  "dimensional formula" -> DimensionalSignature
  "dimension one" -> DimensionlessDimension
  "SI" -> InternationalSystemOfUnits
  "ISQ" -> InternationalSystemOfQuantities
 }
}

EXECUTION_RULES {
 E1 = "If dimension refers to quantity dependence on base quantities, activate DimensionOntology."
 E2 = "Identify the quantity system and base dimensions before deriving signatures."
 E3 = "Propagate dimensions algebraically through multiplication, division and powers."
 E4 = "Require compatible dimensions for addition, subtraction and equation equality."
 E5 = "Distinguish dimensional equivalence from quantity-kind equivalence."
 E6 = "Require dimensionless arguments for standard transcendental functions unless explicitly defined otherwise."
 E7 = "Preserve semantic identity of dimensionless quantities."
 E8 = "Record characteristic scales when non-dimensionalising equations."
 E9 = "Record dimensional definitions, base systems and provenance."
}

RECURSION {
 ENTRY = Dimension
 EXPAND(Dimension) -> [
 PhysicalDimension,QuantityDimension,DimensionalSignature,
 DimensionalFormula,DimensionVector,DimensionExpression,BaseDimension,
 DerivedDimension,DimensionlessDimension,DimensionalExponent,BaseQuantity,
 DerivedQuantity,PhysicalQuantity,Quantity,QuantityKind,Unit,
 MeasurementUnit,DimensionalCompatibility,DimensionalEquivalence,
 DimensionEquality,DimensionInference,DimensionalAnalysis,
 DimensionalAlgebra,DimensionalEquation,DimensionalConstraint,
 DimensionalConsistency,DimensionalHomogeneity,DimensionalValidation,
 DimensionMismatch,ProductDimension,QuotientDimension,PowerDimension,
 InverseDimension,SystemOfQuantities,BaseQuantitySet,
 InternationalSystemOfQuantities,ISQ,InternationalSystemOfUnits,SI,
 LengthDimension,MassDimension,TimeDimension,ElectricCurrentDimension,
 ThermodynamicTemperatureDimension,AmountOfSubstanceDimension,
 LuminousIntensityDimension,AreaDimension,VolumeDimension,
 VelocityDimension,AccelerationDimension,FrequencyDimension,
 ForceDimension,PressureDimension,EnergyDimension,PowerDimension,
 DensityDimension,ChargeDimension,VoltageDimension,ResistanceDimension,
 DimensionlessNumber,BuckinghamPiTheorem,BuckinghamPiGroup,
 NonDimensionalisation,CharacteristicScale,ScalingLaw,DimensionMatrix,
 SymbolicDimensionInference,Measurement,Measurand,MeasurementUncertainty,
 ScientificMeasurement,Metrology,QuantityEquation,ConservationLaw,
 PhysicalConstant,UnitConversion,DerivedUnit,CompoundUnit,
 DimensionProvenance
 ]
 EXPAND(DimensionFamilies) -> [
 BaseDerivedAndDimensionlessDimensions,
 DimensionalSignaturesExponentsAndAlgebra,
 DimensionalHomogeneityAndEquationValidation,
 QuantityAndUnitCompatibility,
 BuckinghamPiAndDimensionlessGroups,
 ScalingNonDimensionalisationAndSimilarity,
 SymbolicDimensionInferenceAndTypeSafety,
 StandardsValidationAndProvenance
 ]
 STOP_CONDITION = "Stop when quantity system, base dimensions, dimensional signature, algebra, homogeneity, compatibility and provenance are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Quantity Ontology"
 RELATED_ONTOLOGIES = [
 "Quantity Ontology","Physical Quantity Ontology",
 "Foundations of Measurement","Measurement Ontology",
 "Scientific Measurement Ontology 2","Metrology Ontology",
 "Measurement & Units Ontologies","Units Ontology",
 "Dimensional Analysis Ontology","Algebra Ontologies",
 "Linear Algebra Ontology","Verification & Validation",
 "Symbolic Computation"
 ]
 ROUTE_RULES = [
 "Dimension -> Quantity Ontology for parent quantity semantics",
 "Dimension -> Physical Quantity Ontology for physical dimensional interpretation",
 "Dimension -> Foundations of Measurement for measurement theory",
 "Dimension -> Measurement and Scientific Measurement for measured-quantity relationships",
 "Dimension -> Metrology for standards and traceability",
 "Dimension -> Units Ontology for unit-dimension compatibility",
 "Dimension -> Dimensional Analysis for equation and scaling reasoning",
 "Dimension -> Algebra and Linear Algebra for exponent-vector and null-space methods",
 "Dimension -> Verification & Validation for dimensional consistency checks",
 "Dimension -> Symbolic Computation for automated dimensional inference"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible dimensional classes, inference methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
 "Preserve dimension-versus-unit distinction.",
 "Preserve dimension-versus-quantity-kind distinction.",
 "Preserve base-versus-derived dimension distinction.",
 "Preserve dimensionless-versus-semantically-untyped distinction.",
 "Preserve dimensional-equivalence-versus-quantity-equivalence distinction.",
 "Preserve homogeneity-versus-physical-correctness distinction.",
 "Preserve spatial-dimension-versus-quantity-dimension distinction.",
 "Preserve machine-readable dimension semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A dimension expresses the dependence of a quantity on selected base quantities independently of the units used to represent it."
 BASE_SUMMARY = "Base dimensions correspond to the base quantities selected for a system of quantities."
 DERIVED_SUMMARY = "Derived dimensions are products of powers of base dimensions and can be represented as exponent vectors."
 HOMOGENEITY_SUMMARY = "Valid quantity equations require dimensional compatibility between additive terms and corresponding equation sides."
 DIMENSIONLESS_SUMMARY = "Dimensionless quantities have zero base-dimensional exponents but can retain distinct semantic meanings."
 ANALYSIS_SUMMARY = "Dimensional analysis supports validation, scaling, non-dimensionalisation, similarity reasoning and inference of unknown dimensional structure."
 VALIDATION_SUMMARY = "Dimensional reasoning should preserve quantity-system context, quantity-kind distinctions, algebraic consistency and provenance."
}

FLOW {
 INPUT
 -> detect_quantity_dimension_context
 -> identify_quantity_or_expression_quantity_system_base_dimensions_units_and_intended_analysis
 -> identify_selected_base_quantities_and_corresponding_dimensions
 -> derive_exponent_vector_for_each_quantity_or_expression
 -> propagate_dimensions_through_products_quotients_powers_and_supported_functions
 -> check_additive_terms_equation_sides_and_constraints_for_dimensional_homogeneity
 -> distinguish_dimensional_equivalence_from_quantity_kind_equivalence
 -> if_required_construct_characteristic_scales_and_dimensionless_groups
 -> infer_unknown_dimensions_from_valid_quantity_equations
 -> detect_dimension_mismatch_unit_mismatch_and_invalid_operations
 -> record_quantity_system_base_set_definitions_equations_and_standard_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-FOUNDATIONS-DIMENSION-ONTOLOGY-2-V2"
}

}

END_FLOW