Computational Topology Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of computational topology within the Kireth Research Semantic Library. Computational topology studies algorithms and data structures for extracting, representing and analysing topological information from discrete geometric, combinatorial and sampled data. It connects topology, algebraic topology, computational geometry and data analysis through simplicial complexes, cell complexes, homology, cohomology, Betti numbers, Euler characteristic, persistent homology, filtrations, boundary matrices, discrete Morse theory, triangulations and topological simplification. This ontology supports exact and approximate topological computation while preserving the distinction between combinatorial representation, algebraic invariants and underlying continuous spaces.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Computational Topology Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Advanced Geometry / Computational Topology"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/advanced-geometry/computational-topology-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ComputationalTopology
 SYMBOLIC_FORMS = [
  "K simplicial complex",
  "H_k(K)",
  "beta_k",
  "chi(K)",
  "PH_k(F)"
 ]
 ENTITY_CLASS = AlgorithmicTopologicalFramework
 PRIMARY_CONTEXT = AdvancedGeometry
 DEFINITION = "Computational topology is the study of algorithms and discrete representations for computing topological structure and invariants."
 SEMANTIC_ROLE = "Provides the algorithmic framework for simplicial and cell complexes, homology, Betti numbers, persistence, discrete Morse theory, topology extraction and topological simplification."
}

SCOPE {
 INCLUDE = [
  ComputationalTopology,
  SimplicialComplex,
  CellComplex,
  CubicalComplex,
  ChainComplex,
  BoundaryOperator,
  BoundaryMatrix,
  Cycle,
  Boundary,
  HomologyGroup,
  CohomologyGroup,
  BettiNumber,
  EulerCharacteristic,
  Filtration,
  PersistentHomology,
  PersistenceDiagram,
  Barcode,
  VietorisRipsComplex,
  CechComplex,
  AlphaComplex,
  NerveComplex,
  Triangulation,
  MeshTopology,
  DiscreteMorseTheory,
  MorseMatching,
  CriticalCell,
  TopologicalSimplification,
  ConnectedComponent,
  FundamentalGroupApproximation,
  MapperConstruction,
  ReebGraph,
  ContourTree,
  TopologicalDataAnalysis
 ]

 EXCLUDE_AS_PRIMARY = [
  PureTopologyWithoutAlgorithmicRepresentation,
  NumericalGeometryWithoutTopologicalGoal,
  GeneralGraphAlgorithmsWithoutTopologicalSemantics,
  StatisticalDataAnalysisWithoutTopologicalStructure
 ]

 DISAMBIGUATION_RULE =
  "When topology is represented or computed through complexes, boundary matrices, homology, persistence, Morse reductions, Reeb graphs or algorithmic invariants, interpret the context as computational topology."
}

CORE_CONCEPTS {

 ComputationalTopology {
  definition = "The algorithmic study of topological spaces through finite combinatorial or algebraic representations."
 }

 SimplicialComplex {
  definition = "A collection of simplices closed under taking faces and satisfying compatible intersection rules."
 }

 CellComplex {
  definition = "A space assembled from cells attached according to combinatorial incidence."
 }

 ChainComplex {
  definition = "A sequence of abelian groups or modules connected by boundary maps whose consecutive composition is zero."
 }

 BoundaryOperator {
  symbol = "partial_k"
  definition = "A map from k-chains to (k-1)-chains satisfying partial_{k-1} composed partial_k = 0."
 }

 HomologyGroup {
  symbol = "H_k"
  definition = "The quotient of k-cycles by k-boundaries."
 }

 BettiNumber {
  symbol = "beta_k"
  definition = "The rank or dimension of the k-th homology group over a chosen coefficient field."
 }

 EulerCharacteristic {
  symbol = "chi"
  definition = "An alternating sum of cell counts or Betti numbers under suitable finite conditions."
 }

 Filtration {
  definition = "A nested family of complexes indexed by scale or parameter."
 }

 PersistentHomology {
  definition = "The study of homological features across a filtration."
 }

 PersistenceDiagram {
  definition = "A multiset of birth-death points representing persistent topological features."
 }

 DiscreteMorseTheory {
  definition = "A combinatorial analogue of Morse theory used to simplify complexes while preserving topology."
 }
}

RELATIONS {
 ComputationalTopology --is_branch_of--> AdvancedGeometry
 ComputationalTopology --uses--> AlgebraicTopology
 SimplicialComplex --is_special_case_of--> CellComplex
 CubicalComplex --is_special_case_of--> CellComplex
 ChainComplex --is_derived_from--> CellComplex
 BoundaryOperator --acts_on--> ChainGroup
 BoundaryOperator --satisfies--> BoundarySquaredZero
 HomologyGroup --is_quotient_of--> CycleGroupByBoundaryGroup
 BettiNumber --is_rank_of--> HomologyGroup
 EulerCharacteristic --can_be_computed_from--> BettiNumbers
 Filtration --induces--> PersistenceModule
 PersistentHomology --produces--> PersistenceDiagram
 PersistentHomology --produces--> Barcode
 VietorisRipsComplex --is_built_from--> MetricData
 CechComplex --is_built_from--> MetricData
 AlphaComplex --is_related_to--> DelaunayTriangulation
 DiscreteMorseTheory --simplifies--> CellComplex
 ReebGraph --summarises--> ScalarFieldTopology
}

OPERATORS {
 BOUNDARY(simplex[v_0,...,v_k]) :=
  sum_i (-1)^i simplex[v_0,...,hat(v_i),...,v_k]

 CHAIN_COMPLEX(K) :=
  sequence(C_k(K),partial_k)

 CYCLE_GROUP(K,k) :=
  kernel(partial_k)

 BOUNDARY_GROUP(K,k) :=
  image(partial_{k+1})

 HOMOLOGY(K,k) :=
  kernel(partial_k)/image(partial_{k+1})

 BETTI_NUMBER(K,k,F) :=
  dimension_F(H_k(K;F))

 EULER_CHARACTERISTIC_CELLS(K) :=
  sum_k (-1)^k * number_of_k_cells

 EULER_CHARACTERISTIC_BETTI(K,F) :=
  sum_k (-1)^k * beta_k
  under suitable finite assumptions

 VIETORIS_RIPS(X,epsilon) :=
  complex_whose_simplices_have_pairwise_distances<=epsilon

 CECH_COMPLEX(X,epsilon) :=
  nerve_of_radius_epsilon_balls_around_X

 PERSISTENT_HOMOLOGY(F,k) :=
  homology_classes_tracked_across_filtration_F

 REDUCE_BOUNDARY_MATRIX(D) :=
  standard_column_reduction_or_equivalent_persistence algorithm

 DISCRETE_MORSE_REDUCTION(K,matching) :=
  collapse_matched_cells_preserving_homotopy_type_under_valid_matching
}

DEFINITIONS {
 D1 = "Computational topology uses finite representations to infer topological structure."
 D2 = "Homology is computed from cycles modulo boundaries."
 D3 = "Boundary maps satisfy partial composed partial = 0."
 D4 = "Betti numbers count independent homological features over a chosen coefficient field."
 D5 = "Euler characteristic is a global invariant computable from cells or Betti numbers."
 D6 = "Persistent homology tracks births and deaths of homological features across a filtration."
 D7 = "Persistence diagrams and barcodes encode equivalent persistence information under standard finite-type settings."
 D8 = "Discrete Morse reductions can reduce complex size while preserving homotopy type under valid conditions."
 D9 = "Vietoris-Rips and Cech complexes provide scale-dependent combinatorial models of metric data."
 D10 = "Coefficient choice can change homology and Betti computations."
}

AXIOMS {
 A1_COMPLEX_CLOSURE:
  Every face of a simplex in a simplicial complex is also in the complex

 A2_BOUNDARY_SQUARED_ZERO:
  partial_{k-1} composed partial_k = 0

 A3_HOMOLOGY:
  H_k = Z_k / B_k

 A4_EULER:
  chi = alternating_sum_of_cell_counts
  and equals alternating_sum_of_Betti_numbers under finite conditions

 A5_FILTRATION:
  K_a subseteq K_b whenever a<=b

 A6_PERSISTENCE_FUNCTORIALITY:
  Inclusion maps induce homology maps across filtration levels

 A7_NERVE:
  Under good-cover hypotheses, the nerve has the homotopy type of the union

 A8_MORSE_PRESERVATION:
  Valid discrete Morse reductions preserve homotopy type up to critical-cell structure
}

REASONING {

 R1_COMPLEX_VALIDATION {
  INPUT = simplices_or_cells
  VERIFY = face_closure
  VERIFY = intersection_compatibility
  RETURN = valid_complex_or_not
 }

 R2_BOUNDARY_MATRIX {
  INPUT = ordered_cells_by_dimension
  COMPUTE = incidence_coefficients
  BUILD = boundary_matrix
  VERIFY = D_{k-1}*D_k=0
  RETURN = boundary_matrices
 }

 R3_HOMOLOGY {
  INPUT = boundary_matrices_and_coefficient_field
  COMPUTE = kernels_and_images
  QUOTIENT = cycles_mod_boundaries
  RETURN = homology_groups_or_Betti_numbers
 }

 R4_EULER_CHARACTERISTIC {
  INPUT = finite_complex
  COMPUTE = alternating_cell_count
  OPTIONALLY_VERIFY = alternating_Betti_sum
  RETURN = chi
 }

 R5_RIPS_CECH {
  INPUT = metric_point_cloud_and_scale
  IF Rips_requested
   BUILD = clique_complex_from_threshold_graph
  ELSE IF Cech_requested
   BUILD = nerve_of_balls
  RETURN = scale_complex
 }

 R6_PERSISTENCE {
  INPUT = filtration
  ORDER = simplices_by_filtration_value
  BUILD = boundary_matrix
  REDUCE = persistence_matrix
  EXTRACT = birth_death_pairs
  RETURN = barcode_and_persistence_diagram
 }

 R7_DISCRETE_MORSE {
  INPUT = cell_complex
  FIND = acyclic_matching
  COLLAPSE = matched_pairs
  RETAIN = critical_cells
  RETURN = reduced_complex_or_Morse_complex
 }

 R8_REEB_GRAPH {
  INPUT = space_or_mesh_and_scalar_function
  TRACK = connected_components_of_level_sets
  QUOTIENT = by_same_level_component relation
  RETURN = ReebGraph
 }

 R9_TOPOLOGICAL_SIMPLIFICATION {
  INPUT = complex_or_persistence_data
  IDENTIFY = low_persistence_or_collapsible features
  REMOVE = according_to_valid_simplification rule
  VERIFY = preserved_target_invariants
  RETURN = simplified_topological_model
 }
}

CONSTRAINTS {
 C1 = "Complex validity must be checked before homology computation."
 C2 = "Homology depends on coefficient ring or field."
 C3 = "Boundary matrices require consistent orientations when coefficients have sign."
 C4 = "Betti numbers alone do not determine full homology with torsion."
 C5 = "Vietoris-Rips and Cech complexes are not identical."
 C6 = "Persistence results depend on filtration choice."
 C7 = "Finite sampling may not recover true underlying topology without sampling assumptions."
 C8 = "Discrete Morse matchings must be acyclic."
 C9 = "Euler characteristic does not uniquely determine homotopy type."
 C10 = "Numerical filtration values require stable ordering and tie conventions."
}

SEMANTIC_EQUIVALENCE {
 "ComputationalTopology" <=> "AlgorithmicTopology"
 "HomologyGroup" <=> "CyclesModuloBoundaries"
 "BettiNumber" <=> "HomologyRank"
 "PersistenceDiagram" <=> "BirthDeathMultiset"
 "Barcode" <=> "IntervalRepresentationOfPersistence"
 "DiscreteMorseReduction" <=> "HomotopyPreservingCombinatorialSimplification"
}

VALIDATION {
 V1 = "Simplicial complexes are closed under faces."
 V2 = "Boundary-square-zero identities hold."
 V3 = "Coefficient field or ring is stated."
 V4 = "Betti number calculation matches homology rank."
 V5 = "Euler characteristic agrees across valid formulations."
 V6 = "Filtration order is monotone."
 V7 = "Persistence pairing respects filtration values."
 V8 = "Morse matching is acyclic."
 V9 = "Sampling-based topology claims state reconstruction assumptions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COMPUTATIONAL_TOPOLOGY_DEFINITION = 0.99
 COMPLEX_REASONING = 0.99
 HOMOLOGY_REASONING = 0.99
 EULER_REASONING = 0.99
 PERSISTENCE_REASONING = 0.98
 DISCRETE_MORSE_REASONING = 0.98
 SAMPLING_TOPOLOGY_REASONING = 0.96
 REEB_GRAPH_REASONING = 0.97

 POLICY =
  "Reduce confidence when coefficient choice, orientation, filtration, sampling density, reconstruction assumptions, tie handling or discrete-versus-continuous interpretation are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ComputationalTopologyReasoning,
  SimplicialComplexReasoning,
  CellComplexReasoning,
  HomologyReasoning,
  BettiNumberReasoning,
  EulerCharacteristicReasoning,
  PersistentHomologyReasoning,
  FiltrationReasoning,
  DiscreteMorseReasoning,
  ReebGraphReasoning,
  TopologicalSimplificationReasoning,
  TopologicalDataAnalysis,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "computational topology" -> ComputationalTopology
  "algorithmic topology" -> ComputationalTopology
  "simplicial complex" -> SimplicialComplex
  "cell complex" -> CellComplex
  "homology" -> HomologyGroup
  "Betti number" -> BettiNumber
  "Euler characteristic" -> EulerCharacteristic
  "persistent homology" -> PersistentHomology
  "persistence diagram" -> PersistenceDiagram
  "barcode" -> Barcode
  "discrete Morse theory" -> DiscreteMorseTheory
  "Reeb graph" -> ReebGraph
 }

 INTERPRETATION_ORDER = [
  detect_computational_topology_context,
  identify complex_type_dimension_coefficients_and_data_source,
  identify homology_Euler_persistence_Morse_Reeb_or_simplification goal,
  validate combinatorial representation,
  construct algebraic_topological operators,
  compute requested invariant_or_structure,
  verify coefficient_filtration_orientation_sampling_and_stability constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("computational topology")
  OR algorithmic_homology_complex_context
  THEN activate(ComputationalTopologyOntology)

 E2:
  IF simplicial_complex_given
  THEN verify face_closure before computation

 E3:
  IF homology_requested
  THEN identify coefficient_ring_or_field

 E4:
  IF Betti_numbers_requested
  THEN compute ranks_of_homology groups

 E5:
  IF persistence_requested
  THEN require a monotone filtration

 E6:
  IF Rips_or_Cech_requested
  THEN identify metric_and_scale parameter

 E7:
  IF discrete_Morse_reduction_requested
  THEN verify acyclic_matching

 E8:
  IF topology_from_point_cloud_requested
  THEN state reconstruction_and_sampling assumptions

 E9:
  IF simplification_requested
  THEN specify which invariants_or_features must be preserved
}

RECURSION {
 ENTRY = ComputationalTopology

 EXPAND(ComputationalTopology) -> [
  SimplicialComplex,
  CellComplex,
  CubicalComplex,
  ChainComplex,
  BoundaryOperator,
  BoundaryMatrix,
  Cycle,
  Boundary,
  HomologyGroup,
  CohomologyGroup,
  BettiNumber,
  EulerCharacteristic,
  Filtration,
  PersistentHomology,
  PersistenceDiagram,
  Barcode,
  VietorisRipsComplex,
  CechComplex,
  AlphaComplex,
  NerveComplex,
  Triangulation,
  DiscreteMorseTheory,
  ReebGraph,
  ContourTree,
  TopologicalSimplification,
  TopologicalDataAnalysis
 ]

 EXPAND(HomologyComputation) -> [
  ChainGroup,
  BoundaryMatrix,
  Kernel,
  Image,
  Quotient,
  SmithNormalForm,
  MatrixReduction
 ]

 EXPAND(PersistentHomology) -> [
  Filtration,
  PersistenceModule,
  Birth,
  Death,
  Barcode,
  PersistenceDiagram,
  Stability
 ]

 EXPAND(DiscreteMorseTheory) -> [
  MorseMatching,
  AcyclicMatching,
  CriticalCell,
  Collapse,
  MorseComplex
 ]

 STOP_CONDITION =
  "Stop when the complex, homology, Betti numbers, Euler characteristic, persistence structure, Morse reduction, Reeb graph or topological simplification is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Advanced Geometry"

 IMMEDIATE_CONTEXTS = [
  "Advanced Geometry",
  "Topology Ontology",
  "Computational Geometry Ontology",
  "Discrete Geometry Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Advanced Geometry",
  "Topology Ontology",
  "Homeomorphism Ontology",
  "Connectedness Ontology",
  "Compactness Ontology",
  "Computational Geometry Ontology",
  "Discrete Geometry Ontology",
  "Mesh Ontology",
  "Delaunay Triangulation Ontology",
  "Algebraic Topology Ontologies",
  "Linear Algebra Ontologies",
  "Graph Theory Ontologies",
  "Data Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Computational Topology -> Advanced Geometry for parent structural semantics",
  "Computational Topology -> Topology for continuous-space semantics",
  "Computational Topology -> Algebraic Topology for homology and cohomology",
  "Computational Topology -> Computational Geometry for triangulations and geometric complexes",
  "Computational Topology -> Discrete Geometry for finite combinatorial structures",
  "Computational Topology -> Mesh for cell incidence and topology extraction",
  "Computational Topology -> Linear Algebra for matrix reduction and rank computation",
  "Computational Topology -> Graph Theory for 1-skeletons, Reeb graphs and contour trees",
  "Computational Topology -> Data Analysis for topological data analysis workflows"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible complexes, invariants, persistence methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve finite-representation algorithmic topology semantics.",
  "Preserve boundary-squared-zero structure.",
  "Preserve cycles-modulo-boundaries homology semantics.",
  "Preserve coefficient dependence.",
  "Preserve filtration monotonicity.",
  "Preserve persistent birth-death semantics.",
  "Preserve valid discrete-Morse reduction conditions.",
  "Preserve machine-readable computational-topology semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Computational topology studies algorithms and discrete representations for computing topological structure and invariants."

 COMPLEX_SUMMARY =
  "Simplicial and cell complexes provide finite combinatorial models of topological spaces."

 HOMOLOGY_SUMMARY =
  "H_k=ker(partial_k)/im(partial_{k+1})."

 BETTI_SUMMARY =
  "The k-th Betti number is the rank or field dimension of H_k."

 EULER_SUMMARY =
  "Euler characteristic is the alternating sum of cell counts and, under finite conditions, of Betti numbers."

 PERSISTENCE_SUMMARY =
  "Persistent homology tracks homological features across nested complexes and records birth-death intervals."

 RIPS_CECH_SUMMARY =
  "Vietoris-Rips and Cech complexes construct scale-dependent topology from metric data."

 MORSE_SUMMARY =
  "Discrete Morse theory simplifies complexes using acyclic matchings while preserving homotopy-relevant structure."

 REEB_SUMMARY =
  "A Reeb graph summarises the connected-component evolution of level sets of a scalar function."
}

FLOW {
 INPUT
 -> detect_computational_topology_context
 -> identify complex_or_data_representation_dimension_coefficients_filtration_and_sampling assumptions
 -> classify homology_Betti_Euler_persistence_Rips_Cech_Morse_Reeb_or_simplification goal
 -> validate combinatorial_complex
 -> construct boundary_chain_or_filtration operators
 -> compute requested topological invariant_or_structure
 -> verify coefficient_orientation_filtration_sampling_stability_and_reduction constraints
 -> route to advanced_geometry topology algebraic_topology computational_geometry discrete_geometry mesh linear_algebra graph_theory_or_data_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Computational Topology Ontology defines algorithmic topology through finite complexes and algebraic invariants. It supports simplicial, cell and cubical complexes, chain complexes, boundary matrices, homology, cohomology, Betti numbers, Euler characteristic, persistent homology, Vietoris-Rips, Cech and alpha complexes, discrete Morse theory, Reeb graphs, contour trees and topology-preserving simplification."
}

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

}

END_FLOW