Knowledge Graph Logic Ontology

FLOW_ALGEBRA KnowledgeGraphLogicOntology v1.0
TARGET_URL := "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/knowledge-representation-logic/knowledge-graph-logic-ontology.html"
LANGUAGE := "en-GB"
DOMAIN := MathematicalOntologies > MathematicalLogicOntologies > KnowledgeRepresentationLogic > KnowledgeGraphLogicOntology

BEGIN_FLOW

ENTITY KnowledgeGraphLogic {
 TYPE := FormalGraphSemanticReasoningFramework
 DEFINITION := "A formal framework for representing entities and typed relationships as a graph while applying explicit schema, identity, logical, provenance and inference semantics."
}
ENTITY KnowledgeGraph { TYPE := SemanticEntityRelationGraph }
ENTITY Entity { TYPE := IdentifiableNode }
ENTITY Relation { TYPE := TypedDirectedEdge }
ENTITY Statement { TYPE := GraphAssertion }
ENTITY Schema { TYPE := GraphSemanticModel }

RELATIONSHIP KnowledgeGraphLogic --formalises--> KnowledgeGraph
RELATIONSHIP KnowledgeGraph --contains--> Entity
RELATIONSHIP KnowledgeGraph --contains--> Relation
RELATIONSHIP KnowledgeGraph --contains--> Statement
RELATIONSHIP KnowledgeGraph --governed_by--> Schema
RELATIONSHIP KnowledgeGraphLogic --supports--> GraphReasoning

KNOWLEDGE_GRAPH {
 KG := (V,E,S)
 V := Entities
 E := TypedRelations
 S := SchemaAndSemanticConstraints
}
TRIPLE {
 FORM := (Subject,Predicate,Object)
}
EDGE_FORM {
 Subject --Predicate--> Object
}
ENTITY_IDENTIFIER {
 SHOULD be stable_and_unambiguous
}
LITERAL {
 TYPE := TypedDataValue
}
ENTITY_TYPE {
 Entity --instance_of--> Class
}
CLASS { TYPE := SchemaConcept }
PROPERTY { TYPE := SchemaRelation }

SUBCLASS { C subclass_of D }
SUBPROPERTY { R subproperty_of S }
DOMAIN { Domain(R)=C }
RANGE { Range(R)=D }
INVERSE_PROPERTY { inverseOf(R,S) }
TRANSITIVE_PROPERTY {
 R(a,b) AND R(b,c) -> R(a,c)
 REQUIRE R declared_transitive
}
SYMMETRIC_PROPERTY {
 R(a,b) -> R(b,a)
}
FUNCTIONAL_PROPERTY {
 R(a,b) AND R(a,c) -> b=c
 under equality_semantics
}

IDENTITY {
 sameAs(a,b) -> same_entity
 under declared equality_semantics
}
DIFFERENCE {
 differentFrom(a,b) -> a!=b
}
IDENTITY_WARNING {
 labels_similarity_shared_attributes
 DO_NOT establish identity by themselves
}
ENTITY_RESOLUTION {
 MentionsOrIdentifiers -> CandidateEntities -> IdentityEvidence -> ResolvedEntity
}
ENTITY_LINKING {
 Mention -> KnowledgeGraphEntity
}
ALIAS {
 alternative_identifier_for_same_entity
}
ENTITY_MERGE {
 IF identity_established:
  merge_nodes
  redirect_relations
  preserve_provenance
}
ENTITY_SPLIT {
 IF conflated_node represents_multiple_entities:
  create_distinct_nodes
  redistribute_statements
}

SCHEMA_LEVEL {
 Classes,Properties,Hierarchies,Constraints,Axioms
}
INSTANCE_LEVEL {
 Entities,LiteralValues,RelationAssertions
}
META_LEVEL {
 Provenance,Confidence,Context,StatementMetadata
}

RDF_RELATIONSHIP {
 KnowledgeGraph MAY use RDF_triples_and_semantics
}
RDFS_RELATIONSHIP {
 MAY provide subclass_subproperty_domain_range_reasoning
}
OWL_RELATIONSHIP {
 MAY add class_constructors_identity_property_characteristics_cardinality
}
DESCRIPTION_LOGIC_RELATIONSHIP {
 ontology_layer MAY use description_logic_semantics
}

GRAPH_RULE {
 IF A--R-->B AND B--S-->C
 THEN A--T-->C
 REQUIRE declared_rule_semantics
}
FORWARD_CHAINING {
 asserted_statements -> applicable_rules -> derived_statements -> fixed_point
}
BACKWARD_CHAINING {
 query_goal -> derivation_rules -> graph_subgoals
}
PROPERTY_PATH {
 R1/R2/.../Rn -> reachable_nodes
}
TRANSITIVE_CLOSURE {
 R_plus := one_or_more_R_edges
}
RELATION_COMPOSITION {
 R o S subrelation_of T
 REQUIRE declared_composition_axiom
}
TAXONOMIC_REASONING {
 C subclass_of D
 D subclass_of E
 -> C subclass_of E
}
INSTANCE_PROPAGATION {
 a instance_of C
 C subclass_of D
 -> a instance_of D
}
SUBPROPERTY_PROPAGATION {
 R subproperty_of S
 R(a,b) -> S(a,b)
}
DOMAIN_INFERENCE {
 Domain(R)=C
 R(a,b) -> C(a)
 under RDFS_OWL_style_semantics
}
RANGE_INFERENCE {
 Range(R)=D
 R(a,b) -> D(b)
 under RDFS_OWL_style_semantics
}
DOMAIN_RANGE_WARNING {
 domain_range_axioms are semantic_inference_rules
 in RDF_OWL_style_semantics,
 not merely validation_constraints
}

OPEN_WORLD_ASSUMPTION {
 missing_statement DOES_NOT imply false
}
CLOSED_WORLD_ASSUMPTION {
 missing_statement MAY be false
 only under explicit_semantics
}
UNIQUE_NAME_ASSUMPTION {
 different_identifiers DO_NOT automatically mean different_entities
}

ENTAILMENT {
 KG entails S
 IFF every model_of_KG satisfies S
 under declared_semantics
}
ASSERTED_STATEMENT {
 ORIGIN := SourceAssertion
}
DERIVED_STATEMENT {
 ORIGIN := Inference
}
ASSERTED_VS_DERIVED {
 MUST_REMAIN_DISTINCT := TRUE
}
CONSISTENCY {
 KG consistent IFF EXISTS model satisfying declared_axioms
}
COUNTERMODEL {
 model_of_KG falsifying query -> non_entailment
}

GRAPH_PATTERN {
 collection_of_triple_patterns
}
TRIPLE_PATTERN {
 (SubjectPattern,PredicatePattern,ObjectPattern)
}
QUERY {
 INPUT := GraphPattern
 OUTPUT := VariableBindings
}
SEMANTIC_QUERY {
 graph_matching + entailment_regime -> semantic_answers
}
SPARQL_RELATIONSHIP {
 graph_querying MAY use SPARQL
}
SPARQL_ENTAILMENT_WARNING {
 graph_matching != semantic_entailment
 unless entailment_regime specified
}
CERTAIN_ANSWER {
 tuple holds in every model_of_KG
 under selected_semantics
}
QUERY_REWRITING {
 Ontology + Query -> RewrittenQuery
 under suitable_profile
}

GRAPH_TRAVERSAL {
 METHODS := {BreadthFirstSearch,DepthFirstSearch,BidirectionalSearch}
}
SHORTEST_PATH {
 minimise path_length_or_declared_cost
}
SEMANTIC_DISTANCE {
 MAY derive_from path_taxonomy_embedding_or_information_content
 REQUIRE declared_metric
}
SEMANTIC_SIMILARITY {
 MAY use common_ancestors_neighbourhoods_embeddings_lexical_similarity
 MUST_NOT_CONFLATE_WITH := identity
}

KNOWLEDGE_GRAPH_EMBEDDING {
 entities_and_relations -> vector_representations
 PURPOSE := {LinkPrediction,Similarity,Clustering,Ranking}
}
EMBEDDING_WARNING {
 high_similarity_or_prediction DOES_NOT equal logical_entailment
}
LINK_PREDICTION {
 existing_graph -> candidate_missing_edge
 STATUS := prediction_or_hypothesis
}
GRAPH_COMPLETION {
 distinguish entailed_completion from predicted_completion
}
LOGICAL_INFERENCE {
 Axioms + Statements -> EntailedStatements
}
STATISTICAL_INFERENCE {
 GraphPatternsOrEmbeddings -> ScoredPredictions
}
LOGICAL_VS_STATISTICAL {
 entailed_statement MUST_NOT be conflated_with prediction
}

PROVENANCE {
 Statement -> Source + AcquisitionMethod + Time + Evidence + TransformationHistory
}
REIFICATION {
 Statement -> StatementEntity -> Metadata
}
RDF_STAR_RELATIONSHIP {
 embedded_triple_metadata MAY represent annotations
 under RDF-star_semantics
}
CONTEXT {
 Statement -> ContextIdentifier
}
TEMPORAL_SCOPE {
 Statement -> ValidTimeOrInterval
}
NAMED_GRAPH_RELATIONSHIP {
 statements MAY be grouped_into named_graphs
}
CONFIDENCE {
 Statement -> ConfidenceScore
 WARNING := score_not_automatically_probability_or_truth
}
DERIVATION_PROVENANCE {
 DerivedStatement -> ParentStatements -> RuleOrReasoner
}
JUSTIFICATION {
 Entailment -> SupportingAxiomAndStatementSet
}
EXPLANATION_PATH {
 QueryAnswer -> SupportingGraphPathOrRuleChain
}

CONFLICT_DETECTION {
 identify identity_type_cardinality_disjointness_temporal_conflicts
}
CONFLICT_RESOLUTION {
 MAY_USE := {SourceAuthority,EvidenceQuality,Recency,Context,ManualReview}
 REQUIRE explicit_policy
}
TRUST {
 SourceOrStatement -> TrustAssessment
 MUST_NOT automatically determine truth
}

KNOWLEDGE_UPDATE {
 KG_t + Additions + Deletions -> KG_t+1
}
CHANGE_SET {
 COMPONENTS := {
  AddedEntities,RemovedEntities,
  AddedStatements,RemovedStatements,SchemaChanges
 }
}
INCREMENTAL_REASONING {
 recompute affected_inferences where supported
}
DEPENDENCY_GRAPH {
 DerivedStatement -> SupportingStatements
}
TRUTH_MAINTENANCE {
 removed_support -> retract unsupported_derived_statements
}
GRAPH_VERSION {
 VersionIdentifier + Timestamp + ChangeSet
}
TEMPORAL_KNOWLEDGE_GRAPH {
 entities_and_relations scoped_over_time
}
EVENT_KNOWLEDGE_GRAPH {
 Event -> Participants + Roles + Time + Location
}

N_ARY_REIFICATION_PATTERN {
 RelationInstance -> ParticipantRoleEdges -> Qualifiers
}
SCHEMA_ALIGNMENT {
 SourceSchema + TargetSchema -> ClassAndPropertyMappings
}
ENTITY_ALIGNMENT {
 SourceEntities + TargetEntities -> IdentityOrCorrespondenceMappings
}
EXACT_MATCH {
 REQUIRE semantic_equivalence
}
BROADER_MATCH {
 source_concept broader_than target_concept
}
NARROWER_MATCH {
 source_concept narrower_than target_concept
}
RELATED_MATCH {
 related_without_equivalence
}
KNOWLEDGE_GRAPH_MERGE {
 KG1 + KG2 + Alignment -> IntegratedKG
}
MERGE_VALIDATION {
 CHECK {
  IdentifierCollisions,IdentityConflicts,SchemaConflicts,
  StatementConflicts,DuplicateStatements,
  ProvenancePreservation,SemanticDrift
 }
}
CANONICALISATION {
 equivalent_identifiers_or_forms -> canonical_representation
}

KNOWLEDGE_GRAPH_CONSTRUCTION {
 Sources
 -> Extraction
 -> EntityResolution
 -> SchemaMapping
 -> Validation
 -> Provenance
 -> GraphInsertion
}
EXTRACTION_WARNING {
 extracted_statement SHOULD retain source_and_confidence until validated
}
VALIDATION {
 GraphData + ConstraintSet -> ConformanceReport
}
SHACL_RELATIONSHIP {
 graph_validation MAY use SHACL_style_constraints
}
SHACL_VS_OWL {
 SHACL := validation_conformance
 OWL := ontology_semantics_and_entailment
 MUST_REMAIN_DISTINCT := TRUE
}
INTEGRITY_CONSTRAINT {
 restricts admissible_graph_state
}

GRAPH_AUDIT {
 CHECK {
  MissingIdentifiers,
  DanglingEdges,
  DuplicateEntities,
  DuplicateStatements,
  UndefinedProperties,
  DomainRangeAnomalies,
  IdentityConflicts,
  ProvenanceGaps,
  OrphanEntities,
  SchemaViolations
 }
}
QUALITY_DIMENSIONS := {
 Accuracy,Completeness,Consistency,Timeliness,
 Provenance,Connectivity,Uniqueness,SemanticSpecificity
}
COMPLETENESS_WARNING {
 open_world_graph cannot infer falsity from missing_statements
}

SCALABILITY {
 MAY require {
  Indexing,Partitioning,IncrementalInference,
  QueryOptimisation,DistributedProcessing
 }
}
MATERIALISATION {
 precompute_entailments -> store_derived_statements
}
QUERY_TIME_REASONING {
 derive_entailments_during_query
}
MATERIALISATION_TRADEOFF {
 storage_and_update_cost vs query_speed
}

KNOWLEDGE_GRAPH_LOGIC_VS_SEMANTIC_NETWORK {
 SemanticNetwork := general_semantic_relation_graph
 KnowledgeGraphLogic := graph + explicit_identity_schema_provenance_reasoning
}
KNOWLEDGE_GRAPH_LOGIC_VS_ONTOLOGY {
 Ontology := conceptual_schema_and_axioms
 KnowledgeGraph := entity_assertions + schema + provenance
 MAY_INTEGRATE := TRUE
}
KNOWLEDGE_GRAPH_LOGIC_VS_DATABASE {
 Database := structured_data_under_schema
 KnowledgeGraph := entity_centric_semantic_relation_structure
}

APPLICATION_RELATIONSHIPS {
 KnowledgeGraphLogic --supports--> KnowledgeRepresentation
 KnowledgeGraphLogic --supports--> SemanticSearch
 KnowledgeGraphLogic --supports--> ArtificialIntelligence
 KnowledgeGraphLogic --supports--> DataIntegration
 KnowledgeGraphLogic --supports--> QuestionAnswering
 KnowledgeGraphLogic --supports--> Recommendation
 KnowledgeGraphLogic --supports--> NaturalLanguageProcessing
 KnowledgeGraphLogic --supports--> EnterpriseKnowledge
 KnowledgeGraphLogic --supports--> ScientificKnowledge
 KnowledgeGraphLogic --supports--> AutomatedReasoning
}

ONTOLOGY_LINKS {
 PARENT := KnowledgeRepresentationLogicOntology
 RELATED := {
  KnowledgeRepresentationOntology,
  OntologyLogicOntology,
  DescriptionLogicOntology,
  SemanticNetworkOntology,
  GraphOntology,
  EntityOntology,
  RelationOntology,
  RDFOntology,
  OWLOntology,
  ProvenanceOntology,
  SemanticSearchOntology
 }
}

SEMANTIC_EQUIVALENCE {
 KnowledgeGraphLogic ≈ FormalKnowledgeGraphReasoning
 KnowledgeGraph ≈ TypedEntityRelationGraph WHEN schema_and_semantics_explicit
 Triple ≈ SubjectPredicateObjectStatement
 EntityNode ≈ IdentifiedKnowledgeEntity
 RelationEdge ≈ TypedKnowledgeRelation
}

CONSTRAINTS {
 C1 := every graph statement MUST identify subject predicate and object
 C2 := entity identity MUST_NOT be inferred from labels or similarity alone
 C3 := schema-level and instance-level statements MUST remain distinguishable
 C4 := relation directionality and property semantics MUST be preserved
 C5 := open-world missing statements MUST_NOT automatically be false
 C6 := asserted entailed and statistically predicted statements MUST remain distinguishable
 C7 := domain_range semantics MUST_NOT be conflated with validation constraints
 C8 := provenance SHOULD be preserved through extraction inference merge and update
 C9 := equivalence mappings MUST_NOT be conflated with broader narrower or related mappings
 C10 := SHACL-style validation MUST remain distinct from OWL-style entailment
 C11 := entity merges MUST require identity justification and preserve provenance
 C12 := graph transformations MUST preserve meaning or record semantic loss
}

REASONING_CHAIN Query Q {
 IDENTIFY knowledge_graph KG
 IDENTIFY schema
 IDENTIFY entities
 IDENTIFY relations
 IDENTIFY asserted_statements
 IDENTIFY query
 IDENTIFY entailment_regime
 IDENTIFY provenance_policy

 VALIDATE graph_structure
 RESOLVE entity_identity
 CHECK schema_consistency

 IF ontology_reasoning {
  APPLY subclass_subproperty_domain_range_rules
  APPLY property_characteristics
 }

 IF rule_reasoning {
  MATCH graph_rules
  DERIVE statements
  RECORD provenance
 }

 IF query_requested {
  MATCH graph_pattern
  APPLY declared_entailment_regime
  RETURN certain_or_framework_specific_answers
 }

 IF path_requested {
  COMPUTE permitted_property_paths
 }

 IF statistical_prediction_requested {
  GENERATE scored_candidates
  LABEL as predictions_not_entailments
 }

 IF merge_requested {
  ALIGN schemas
  ALIGN entities
  MERGE graphs
  VALIDATE conflicts_and_semantic_drift
 }

 IF update_requested {
  APPLY change_set
  UPDATE dependencies
  RETRACT unsupported_derivations
  RECOMPUTE affected_entailments
 }

 IF validation_requested {
  APPLY graph_constraints
  KEEP validation_distinct_from_entailment
 }

 BUILD explanation_path
 BUILD provenance_trace
 VALIDATE constraints
 RETURN knowledge_graph_result WITH semantic_trace
}

CONFIDENCE {
 CORE_KNOWLEDGE_GRAPH_LOGIC := 1.00
 TRIPLE_ENTITY_RELATION_MODEL := 1.00
 RDF_RDFS_RELATIONSHIP := 1.00
 OWL_DESCRIPTION_LOGIC_RELATIONSHIP := 0.99
 OPEN_WORLD_DISTINCTION := 1.00
 LOGICAL_STATISTICAL_DISTINCTION := 1.00
 PROVENANCE_ROLE := 1.00
 SPECIFIC_RESULT :=
  IF graph_schema_query_and_semantics_complete THEN 1.00
  ELSE REQUIRE missing_information
}

UPDATE_MODEL {
 IMMUTABLE_CORE := {
  EntityRelationStatementStructure,
  TypedGraphSemantics,
  IdentityDistinction,
  AssertedDerivedPredictedDistinction,
  Provenance
 }
 EXTENSIBLE := {
  GraphSchemas,
  OntologyAxioms,
  RuleSystems,
  QueryLanguages,
  EmbeddingModels,
  AlignmentMethods,
  ValidationMethods,
  TemporalModels,
  ApplicationDomains,
  RelatedOntologies
 }
 ON_NEW_KNOWLEDGE {
  IDENTIFY new_entity_relation_or_statement
  VALIDATE identifiers
  VALIDATE schema_semantics
  RESOLVE identity
  CHECK conflicts
  UPDATE provenance
  DERIVE new_entailments_if_required
  UPDATE confidence
 }
}

AI_CONSUMPTION {
 MACHINE_READABLE := TRUE
 HUMAN_INTERPRETABLE := TRUE
 RECURSIVE_REASONING := TRUE
 ENTITY_RESOLUTION := TRUE
 RELATION_RESOLUTION := TRUE
 GRAPH_REASONING := TRUE
 ONTOLOGY_REASONING := TRUE
 RULE_REASONING := TRUE
 PROPERTY_PATH_REASONING := TRUE
 QUERY_REASONING := TRUE
 ALIGNMENT_REASONING := TRUE
 PROVENANCE_TRACKING := TRUE
 TRUTH_MAINTENANCE := TRUE
 GRAPH_VALIDATION := TRUE
 LOGICAL_STATISTICAL_SEPARATION := TRUE
 VALIDATION_REQUIRED := TRUE
}

FLOW_HASH {
 TARGET := TARGET_URL
 SUBJECT := "Knowledge Graph Logic Ontology"
 VERSION := "1.0"
 CANONICAL_SIGNATURE := HASH(TARGET + SUBJECT + DOMAIN + VERSION)
}

OUTPUT {
 PRIMARY_ENTITY := KnowledgeGraphLogic
 INTERPRETATION := "Knowledge graph logic combines an entity-relation graph with explicit schema, identity, provenance and reasoning semantics. It supports taxonomic and rule inference, property paths, semantic querying, graph integration and updates while keeping asserted facts, logical entailments, validation results and statistically predicted links semantically distinct."
}

END_FLOW