Volume Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of volume within the Kireth Research Semantic Library. Volume is the three-dimensional geometric measure of the amount of space occupied by a solid or bounded region. This ontology supports cubes, cuboids, prisms, cylinders, cones, pyramids, spheres, hemispheres, frustums, polyhedra, coordinate-defined solids, integral-based volume, scaling, units, dimensional analysis, exact and approximate values and relationships with area, surface area, solid geometry and calculus.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Volume Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Geometric Measurement / Volume"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/geometric-measurement/volume-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Volume
 SYMBOLIC_FORMS = [
  "V",
  "measure_3(S)",
  "triple_integral_S dV"
 ]
 ENTITY_CLASS = ThreeDimensionalGeometricMeasure
 PRIMARY_CONTEXT = GeometricMeasurement
 DEFINITION = "Volume is the nonnegative three-dimensional measure of the amount of space occupied by a solid or bounded region."
 SEMANTIC_ROLE = "Provides the quantitative framework for measuring solid regions, polyhedra, curved solids, composite solids, coordinate-defined regions and scale-dependent three-dimensional geometry."
}

SCOPE {
 INCLUDE = [
  Volume,
  SolidVolume,
  CubeVolume,
  CuboidVolume,
  PrismVolume,
  CylinderVolume,
  ConeVolume,
  PyramidVolume,
  SphereVolume,
  HemisphereVolume,
  FrustumVolume,
  PolyhedronVolume,
  CompositeVolume,
  CoordinateVolume,
  IntegralVolume,
  VolumeByCrossSection,
  VolumeOfRevolution,
  TripleIntegralVolume,
  ScaleFactor,
  Unit,
  UnitConversion,
  ExactVolume,
  ApproximateVolume,
  MeasurementPrecision,
  ErrorBound
 ]

 EXCLUDE_AS_PRIMARY = [
  Area,
  SurfaceArea,
  Perimeter,
  CapacityUnlessExplicitlyEquatedToGeometricVolume,
  PhysicalMassOrDensityWithoutVolumeContext
 ]

 DISAMBIGUATION_RULE =
  "When the task concerns the three-dimensional size or occupied space of a solid, region, prism, cylinder, cone, sphere, pyramid or integral-defined solid, interpret the quantity as volume."
}

CORE_CONCEPTS {

 Volume {
  definition = "A three-dimensional geometric measure."
  dimension = "L^3"
 }

 SolidVolume {
  definition = "The amount of three-dimensional space occupied by a solid."
 }

 CubeVolume {
  relation = "V=s^3"
 }

 CuboidVolume {
  relation = "V=l*w*h"
 }

 PrismVolume {
  relation = "V=B*h"
 }

 CylinderVolume {
  relation = "V=pi*r^2*h"
 }

 ConeVolume {
  relation = "V=(1/3)*pi*r^2*h"
 }

 PyramidVolume {
  relation = "V=(1/3)*B*h"
 }

 SphereVolume {
  relation = "V=(4/3)*pi*r^3"
 }

 HemisphereVolume {
  relation = "V=(2/3)*pi*r^3"
 }

 FrustumVolume {
  definition = "The volume of a truncated cone or pyramid, usually expressed using the two end areas and perpendicular height."
 }

 CompositeVolume {
  definition = "Volume obtained by adding and subtracting component-solid volumes."
 }

 IntegralVolume {
  definition = "Volume computed using single, double or triple integration depending on representation."
 }
}

RELATIONS {
 Volume --is_measure_of--> ThreeDimensionalExtent
 CubeVolume --is_special_case_of--> Volume
 CuboidVolume --is_special_case_of--> Volume
 PrismVolume --is_special_case_of--> Volume
 CylinderVolume --is_special_case_of--> Volume
 ConeVolume --is_special_case_of--> Volume
 PyramidVolume --is_special_case_of--> Volume
 SphereVolume --is_special_case_of--> Volume
 HemisphereVolume --is_special_case_of--> Volume
 FrustumVolume --is_special_case_of--> Volume
 CompositeVolume --is_combination_of--> ComponentVolumes
 CrossSectionArea --integrates_to--> Volume
 ScaleFactor --transforms--> Volume
 UnitConversion --preserves--> VolumeMagnitude
 TripleIntegralVolume --is_special_case_of--> IntegralVolume
}

OPERATORS {
 CUBE_VOLUME(s) :=
  s^3

 CUBOID_VOLUME(l,w,h) :=
  l*w*h

 PRISM_VOLUME(base_area,h) :=
  base_area*h

 CYLINDER_VOLUME(r,h) :=
  pi*r^2*h

 CONE_VOLUME(r,h) :=
  (1/3)*pi*r^2*h

 PYRAMID_VOLUME(base_area,h) :=
  (1/3)*base_area*h

 SPHERE_VOLUME(r) :=
  (4/3)*pi*r^3

 HEMISPHERE_VOLUME(r) :=
  (2/3)*pi*r^3

 CONICAL_FRUSTUM_VOLUME(r1,r2,h) :=
  (1/3)*pi*h*(r1^2+r1*r2+r2^2)

 PYRAMIDAL_FRUSTUM_VOLUME(A1,A2,h) :=
  (h/3)*(A1+A2+sqrt(A1*A2))
  WHEN corresponding_end_faces_are_similar

 COMPOSITE_VOLUME(parts) :=
  sum(included_part_volumes)-sum(excluded_part_volumes)

 VOLUME_BY_CROSS_SECTION(A,x,a,b) :=
  integral_a^b A(x) dx

 VOLUME_OF_REVOLUTION_DISK(R,r,a,b) :=
  pi*integral_a^b (R(x)^2-r(x)^2) dx

 VOLUME_OF_REVOLUTION_SHELL(radius,height,a,b) :=
  2*pi*integral_a^b radius(x)*height(x) dx

 TRIPLE_INTEGRAL_VOLUME(Omega) :=
  triple_integral_Omega 1 dV

 SCALE_VOLUME(V,k) :=
  abs(k)^3*V

 CONVERT_VOLUME(V,unit_from,unit_to) :=
  V*conversion_factor_cubed(unit_from,unit_to)
}

DEFINITIONS {
 D1 = "Volume has dimension L^3."
 D2 = "Ordinary geometric volume is nonnegative."
 D3 = "Prism volume equals base area times perpendicular height."
 D4 = "Pyramid and cone volume equal one third of corresponding prism or cylinder volume with equal base and height."
 D5 = "Sphere volume is (4/3)pi*r^3."
 D6 = "Composite volume is computed by addition and subtraction of component regions."
 D7 = "Cross-sectional area integrates to volume."
 D8 = "Similarity scale factor k changes volume by |k|^3."
 D9 = "Volume units are cubed length units."
 D10 = "Volume and surface area must not be conflated."
}

AXIOMS {
 A1_NONNEGATIVITY:
  Volume >= 0 for ordinary geometric volume

 A2_ADDIVITY:
  Volume of disjoint solid regions is additive under suitable conditions

 A3_CONGRUENCE_INVARIANCE:
  Congruent solids have equal volume

 A4_PRISM:
  Prism volume equals base_area*height

 A5_PYRAMID_RATIO:
  Pyramid volume is one third of prism volume for equal base and perpendicular height

 A6_SCALE:
  Similarity transformation with scale factor k multiplies volume by abs(k)^3

 A7_UNIT_INVARIANCE:
  Correct unit conversion preserves volume magnitude

 A8_DIMENSION:
  Volume has dimension L^3
}

REASONING {

 R1_VOLUME_CLASSIFICATION {
  INPUT = geometric_solid
  IDENTIFY = cube_cuboid_prism_cylinder_cone_pyramid_sphere_hemisphere_frustum_composite_or_integral_defined
  RETURN = volume_type
 }

 R2_FORMULA_SELECTION {
  INPUT = solid_type_and_known_parameters
  MATCH = valid_volume_formula
  VERIFY = required_dimensions_and_assumptions
  RETURN = formula
 }

 R3_PRISMATIC_SOLIDS {
  INPUT = base_area_and_perpendicular_height
  COMPUTE = B*h
  RETURN = volume
 }

 R4_CONE_OR_PYRAMID {
  INPUT = base_area_or_radius_and_height
  COMPUTE = one_third_base_area_times_height
  RETURN = volume
 }

 R5_SPHERE_OR_HEMISPHERE {
  INPUT = radius_and_shape
  IF sphere
   COMPUTE = (4/3)*pi*r^3
  ELSE IF hemisphere
   COMPUTE = (2/3)*pi*r^3
  RETURN = volume
 }

 R6_FRUSTUM {
  INPUT = end_dimensions_and_perpendicular_height
  IDENTIFY = conical_or_pyramidal
  APPLY = appropriate_frustum_formula
  RETURN = volume
 }

 R7_COMPOSITE_VOLUME {
  INPUT = component_solids
  IDENTIFY = included_and_removed_regions
  COMPUTE = component_volumes
  COMBINE = additions_minus_subtractions
  RETURN = total_volume
 }

 R8_CROSS_SECTION_INTEGRAL {
  INPUT = cross_section_area_function_A(x)
  IDENTIFY = interval
  INTEGRATE = A(x)
  RETURN = volume
 }

 R9_VOLUME_OF_REVOLUTION {
  INPUT = generating_region_axis_and_method
  SELECT = disk_washer_or_shell
  INTEGRATE = corresponding_formula
  RETURN = volume
 }

 R10_TRIPLE_INTEGRAL {
  INPUT = region_Omega
  CHOOSE = coordinate_system
  SET = integration_bounds
  INTEGRATE = 1 over Omega
  RETURN = volume
 }

 R11_SCALE_AND_UNITS {
  INPUT = volume_scale_or_unit_conversion
  APPLY = abs(k)^3_or_cubed_conversion_factor
  RETURN = transformed_volume
 }
}

CONSTRAINTS {
 C1 = "Volume and surface area must not be conflated."
 C2 = "Volume units are cubed."
 C3 = "Prism, cone and pyramid heights are perpendicular heights."
 C4 = "Composite volumes must avoid overlap double-counting."
 C5 = "Frustum formulas require correct end geometry and height."
 C6 = "Volume-of-revolution formulas depend on the selected axis and method."
 C7 = "Triple integrals require correct bounds and Jacobians under coordinate changes."
 C8 = "Scale factors affect volume cubically."
 C9 = "Approximate dimensions produce approximate volume."
 C10 = "Signed volume and ordinary volume must be distinguished when orientation is introduced."
}

SEMANTIC_EQUIVALENCE {
 "Volume" <=> "ThreeDimensionalMeasure"
 "PrismVolume" <=> "BaseAreaTimesHeight"
 "CylinderVolume" <=> "CircularPrismVolume"
 "ConeVolume" <=> "OneThirdCircularPrismVolume"
 "PyramidVolume" <=> "OneThirdPrismVolume"
 "SphereVolume" <=> "FourThirdsPiRadiusCubed"
 "IntegralVolume" <=> "VolumeComputedByIntegration"
}

VALIDATION {
 V1 = "Result is nonnegative unless signed volume is explicitly requested."
 V2 = "Output unit has dimension L^3."
 V3 = "Formula assumptions match the solid."
 V4 = "Perpendicular height is used where required."
 V5 = "Composite components do not overlap unintentionally."
 V6 = "Frustum end dimensions are interpreted correctly."
 V7 = "Integral bounds cover exactly the target region."
 V8 = "Scale factor uses third power."
 V9 = "Approximation status is preserved."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 VOLUME_DEFINITION = 0.99
 PRISM_REASONING = 0.99
 CONE_PYRAMID_REASONING = 0.99
 SPHERE_REASONING = 0.99
 FRUSTUM_REASONING = 0.98
 COMPOSITE_VOLUME_REASONING = 0.98
 INTEGRAL_VOLUME_REASONING = 0.98
 SCALE_REASONING = 0.99
 UNIT_REASONING = 0.99

 POLICY =
  "Reduce confidence when solid type, perpendicular height, overlap structure, integration bounds, axis of revolution, scale factor, units or approximation precision are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  VolumeReasoning,
  GeometricMeasurementReasoning,
  SolidGeometryReasoning,
  PrismReasoning,
  CylinderReasoning,
  ConeReasoning,
  PyramidReasoning,
  SphereReasoning,
  FrustumReasoning,
  CompositeVolumeReasoning,
  IntegralVolumeReasoning,
  VolumeOfRevolutionReasoning,
  ScaleReasoning,
  UnitConversionReasoning,
  DimensionalAnalysis,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "volume" -> Volume WHEN geometry_context
  "cube volume" -> CubeVolume
  "cuboid volume" -> CuboidVolume
  "prism volume" -> PrismVolume
  "cylinder volume" -> CylinderVolume
  "cone volume" -> ConeVolume
  "pyramid volume" -> PyramidVolume
  "sphere volume" -> SphereVolume
  "hemisphere volume" -> HemisphereVolume
  "frustum volume" -> FrustumVolume
  "volume of revolution" -> VolumeOfRevolution
 }

 INTERPRETATION_ORDER = [
  detect_volume_context,
  identify solid_type_dimensions_base_area_radius_height_units_scale_and_precision,
  classify formula_based_composite_cross_section_revolution_or_triple_integral goal,
  select valid_formula_decomposition_or_integration pathway,
  compute volume,
  verify dimensional_nonnegativity_height_overlap_and_formula assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("volume")
  AND geometric_measurement_context
  THEN activate(VolumeOntology)

 E2:
  IF prism_or_cylinder
  THEN use base_area_times_perpendicular_height

 E3:
  IF cone_or_pyramid
  THEN use one_third_base_area_times_height

 E4:
  IF sphere
  THEN use (4/3)*pi*r^3

 E5:
  IF hemisphere
  THEN use (2/3)*pi*r^3

 E6:
  IF composite_solid
  THEN add_and_subtract component volumes without overlap duplication

 E7:
  IF cross_section_area_function_given
  THEN integrate area_over_length

 E8:
  IF volume_of_revolution_requested
  THEN choose disk_washer_or_shell method based_on_geometry

 E9:
  IF similarity_scale_given
  THEN multiply volume by abs(k)^3

 E10:
  IF unit_conversion_requested
  THEN apply cubed_length_conversion
}

RECURSION {
 ENTRY = Volume

 EXPAND(Volume) -> [
  SolidVolume,
  CubeVolume,
  CuboidVolume,
  PrismVolume,
  CylinderVolume,
  ConeVolume,
  PyramidVolume,
  SphereVolume,
  HemisphereVolume,
  FrustumVolume,
  PolyhedronVolume,
  CompositeVolume,
  CoordinateVolume,
  IntegralVolume,
  VolumeByCrossSection,
  VolumeOfRevolution,
  TripleIntegralVolume,
  ScaleFactor,
  Unit,
  UnitConversion,
  ErrorBound
 ]

 EXPAND(IntegralVolume) -> [
  CrossSectionMethod,
  DiskMethod,
  WasherMethod,
  ShellMethod,
  DoubleIntegral,
  TripleIntegral,
  Jacobian
 ]

 EXPAND(CompositeVolume) -> [
  IncludedSolid,
  ExcludedSolid,
  Overlap,
  Cavity,
  BooleanCombination
 ]

 EXPAND(FrustumVolume) -> [
  ConicalFrustum,
  PyramidalFrustum,
  SimilarEndFaces,
  PerpendicularHeight
 ]

 STOP_CONDITION =
  "Stop when the requested solid, volume formula, units, scale, integration method, exactness and precision status are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometric Measurement"

 IMMEDIATE_CONTEXTS = [
  "Geometric Measurement",
  "Surface Area Ontology",
  "Solid Geometry",
  "Calculus & Analysis Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Geometric Measurement",
  "Area Ontology",
  "Surface Area Ontology",
  "Solid Geometry",
  "Cube Ontology",
  "Cuboid Ontology",
  "Prism Ontology",
  "Cylinder Ontology",
  "Cone Ontology",
  "Pyramid Ontology",
  "Sphere Ontology",
  "Hemisphere Ontology",
  "Polyhedron Ontology",
  "Calculus & Analysis Ontologies",
  "Measurement & Units Ontologies"
 ]

 ROUTE_RULES = [
  "Volume -> Geometric Measurement for parent quantitative semantics",
  "Volume -> Area for base and cross-sectional area relationships",
  "Volume -> Surface Area for related but dimensionally distinct solid measurement",
  "Volume -> Solid Geometry for solid-specific formulas",
  "Volume -> Calculus for cross-sections, revolution and multiple integrals",
  "Volume -> Polyhedron for face-cell solid structures",
  "Volume -> Measurement & Units for cubed-unit conversion"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible solid classes, formulas or integration pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve three-dimensional measure semantics.",
  "Preserve nonnegativity for ordinary volume.",
  "Preserve cubed dimensionality.",
  "Preserve additivity for disjoint regions.",
  "Preserve perpendicular-height requirements.",
  "Preserve cubic scaling.",
  "Preserve unit compatibility.",
  "Preserve machine-readable volume semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Volume is the nonnegative three-dimensional measure of the space occupied by a solid or bounded region."

 DIMENSION_SUMMARY =
  "Volume has dimension L^3."

 CUBE_SUMMARY =
  "Cube volume V=s^3."

 CUBOID_SUMMARY =
  "Cuboid volume V=l*w*h."

 PRISM_SUMMARY =
  "Prism volume V=B*h."

 CYLINDER_SUMMARY =
  "Cylinder volume V=pi*r^2*h."

 CONE_SUMMARY =
  "Cone volume V=(1/3)*pi*r^2*h."

 PYRAMID_SUMMARY =
  "Pyramid volume V=(1/3)*B*h."

 SPHERE_SUMMARY =
  "Sphere volume V=(4/3)*pi*r^3."

 SCALE_SUMMARY =
  "A similarity scale factor k changes volume by |k|^3."

 UNIT_SUMMARY =
  "Volume units are cubed length units."
}

FLOW {
 INPUT
 -> detect_volume_context
 -> identify solid_geometry_dimensions_base_area_radius_height_units_scale_and_precision
 -> classify direct_formula_composite_cross_section_revolution_or_multiple_integral volume
 -> select valid_formula_decomposition_or_integration pathway
 -> compute volume
 -> apply scaling_or_unit_conversion when required
 -> verify dimension_nonnegativity_perpendicular_height_overlap_bounds_and_precision constraints
 -> route to geometric_measurement area surface_area solid_geometry calculus polyhedron_or_units ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Volume Ontology defines three-dimensional geometric measurement and supports cubes, cuboids, prisms, cylinders, cones, pyramids, spheres, hemispheres, frustums, polyhedra, composite solids, cross-sectional and revolution methods, multiple integrals, scaling, units, dimensional analysis and approximation-aware reasoning."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-VOLUME-ONTOLOGY-V1"
}

}

END_FLOW