Knowledge Representation Logic

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

BEGIN_FLOW

ENTITY KnowledgeRepresentationLogic {
 TYPE := FormalKnowledgeRepresentationFramework
 DEFINITION := "The use of logical languages, formal semantics and inference mechanisms to represent entities, properties, relations, rules, constraints and knowledge so that machines can interpret and reason over them."
}

ENTITY KnowledgeBase { TYPE := StructuredKnowledgeCollection }
ENTITY Fact { TYPE := AssertedProposition }
ENTITY Rule { TYPE := InferenceStructure }
ENTITY Concept { TYPE := ClassOrCategory }
ENTITY Relation { TYPE := SemanticRelationship }
ENTITY Individual { TYPE := NamedOrAnonymousEntity }

RELATIONSHIP KnowledgeRepresentationLogic --represents--> Fact
RELATIONSHIP KnowledgeRepresentationLogic --represents--> Concept
RELATIONSHIP KnowledgeRepresentationLogic --represents--> Relation
RELATIONSHIP KnowledgeRepresentationLogic --uses--> FormalSemantics
RELATIONSHIP KnowledgeRepresentationLogic --supports--> AutomatedReasoning
RELATIONSHIP KnowledgeRepresentationLogic --supports--> Ontology

KNOWLEDGE_BASE {
 KB := TBox + ABox + Rules + Constraints
 under frameworks supporting these components
}

TBOX {
 TYPE := TerminologicalKnowledge
 CONTENT := concept_and_relation_axioms
}
ABOX {
 TYPE := AssertionalKnowledge
 CONTENT := individual_facts
}
RBOX {
 TYPE := RoleAxioms
 CONTENT := relation_properties_and_hierarchies
}

ATOMIC_FACT {
 FORM := P(a)
}
RELATIONAL_FACT {
 FORM := R(a,b)
}
CLASS_MEMBERSHIP {
 a : C
}
SUBCLASS {
 C subclass_of D
}
EQUIVALENCE {
 C equivalent_to D
}
DISJOINTNESS {
 Disjoint(C,D)
}
DOMAIN_AXIOM {
 Domain(R)=C
}
RANGE_AXIOM {
 Range(R)=D
}

FIRST_ORDER_LOGIC_RELATIONSHIP {
 KnowledgeRepresentationLogic --may_use--> FirstOrderLogic
}
FOL_ATOM {
 P(t1,...,tn)
}
FOL_QUANTIFIERS := {FOR_ALL,EXISTS}
FOL_CONNECTIVES := {AND,OR,NOT,IMPLIES,IFF}

DESCRIPTION_LOGIC {
 TYPE := DecidableKnowledgeRepresentationFamily
 COMPONENTS := {Concepts,Roles,Individuals}
}
DL_CONCEPT_INCLUSION {
 C subsumed_by D
}
DL_CONCEPT_EQUIVALENCE {
 C equivalent_to D
}
DL_ASSERTION {
 C(a)
}
DL_ROLE_ASSERTION {
 R(a,b)
}

CONCEPT_CONSTRUCTORS {
 MAY_INCLUDE := {
  Intersection,
  Union,
  Complement,
  ExistentialRestriction,
  UniversalRestriction,
  CardinalityRestriction
 }
}

INTERSECTION {
 C AND D := individuals_belonging_to_both
}
UNION {
 C OR D := individuals_belonging_to_either
}
COMPLEMENT {
 NOT C := individuals_not_in_C
 under declared universe semantics
}
EXISTENTIAL_RESTRICTION {
 EXISTS R.C :=
  individuals having at_least_one R-successor in C
}
UNIVERSAL_RESTRICTION {
 FOR_ALL R.C :=
  individuals whose R-successors are all in C
}
MIN_CARDINALITY {
 >=n R.C
}
MAX_CARDINALITY {
 <=n R.C
}
EXACT_CARDINALITY {
 =n R.C
}

OPEN_WORLD_ASSUMPTION {
 absence_of_assertion
 DOES_NOT imply negation
}
CLOSED_WORLD_ASSUMPTION {
 absence_or_failure_to_prove MAY imply false
 REQUIRE explicit_nonmonotonic_or_database_semantics
}
UNIQUE_NAME_ASSUMPTION {
 different_names imply different_individuals
 NOT automatic in all KR formalisms
}

ONTOLOGY {
 COMPONENTS := {
  Classes,
  Individuals,
  Properties,
  Axioms,
  Constraints
 }
}
TAXONOMY {
 hierarchical_classification_of_concepts
}
SUBSUMPTION {
 C subsumed_by D
 IFF every instance_of_C is instance_of_D
 under selected semantics
}

INFERENCE_TASKS {
 INCLUDE := {
  Satisfiability,
  Consistency,
  Subsumption,
  Classification,
  InstanceChecking,
  Realisation,
  QueryAnswering
 }
}

CONCEPT_SATISFIABILITY {
 C satisfiable
 IFF EXISTS model with nonempty interpretation_of_C
}
KB_CONSISTENCY {
 KB consistent
 IFF EXISTS model satisfying all axioms
}
SUBSUMPTION_REASONING {
 determine whether KB entails C subclass_of D
}
CLASSIFICATION {
 compute concept_hierarchy implied_by_KB
}
INSTANCE_CHECKING {
 determine whether KB entails C(a)
}
REALISATION {
 determine most_specific_concepts for individual a
}
QUERY_ANSWERING {
 INPUT := KB + Query
 OUTPUT := answers entailed under selected semantics
}

ENTAILMENT {
 KB entails phi
 IFF every model of KB satisfies phi
}
MODEL {
 Interpretation satisfying declared axioms
}
COUNTERMODEL {
 Model of KB in which queried phi is false
 -> demonstrates non_entailment
}

RULE_BASED_KR {
 Facts + Rules -> DerivedFacts
}
HORN_RULE {
 A1 AND ... AND An -> B
}
FORWARD_CHAINING {
 Facts -> ApplicableRules -> NewFacts -> FixedPoint
}
BACKWARD_CHAINING {
 Goal -> MatchingRuleConsequents -> Subgoals
}
DATALOG_RELATIONSHIP {
 function_free_Horn_rules
 + finite_database_style_facts
 -> fixed_point_reasoning
 under positive Datalog semantics
}

LOGIC_PROGRAMMING_RELATIONSHIP {
 KnowledgeRepresentationLogic --relates_to--> LogicProgramming
}
NON_MONOTONIC_RELATIONSHIP {
 Defaults + Exceptions + Priorities
 -> defeasible_knowledge_representation
}
DEFAULT_RULE {
 normally P -> Q
 REQUIRE nonmonotonic_semantics
}

FRAME_REPRESENTATION {
 ENTITY := Frame
 COMPONENTS := {Slots,Values,Defaults,Constraints}
}
SLOT {
 property_of_frame
}
INHERITANCE {
 subclass_or_frame_specialisation
 -> inherited_properties
 subject_to semantics
}

SEMANTIC_NETWORK {
 NODES := EntitiesOrConcepts
 EDGES := SemanticRelations
}
KNOWLEDGE_GRAPH {
 NODES := Entities
 EDGES := TypedRelations
 STATEMENTS := triples_or_graph_assertions
}
TRIPLE {
 FORM := (Subject,Predicate,Object)
}

RDF_RELATIONSHIP {
 TripleBasedRepresentation
 -> RDF-style graph semantics
 REQUIRE framework-specific semantics
}
RDFS_RELATIONSHIP {
 supports class/property hierarchies and domain/range reasoning
}
OWL_RELATIONSHIP {
 WebOntologyLanguage
 -> description-logic-related ontology semantics
 depending on profile/version
}

IDENTITY_REASONING {
 sameAs(a,b)
 -> treat a,b as same individual
 under equality semantics
}
DIFFERENCE_REASONING {
 differentFrom(a,b)
 -> a != b
}

PROPERTY_CHARACTERISTICS {
 MAY_INCLUDE := {
  Functional,
  InverseFunctional,
  Transitive,
  Symmetric,
  Asymmetric,
  Reflexive,
  Irreflexive
 }
}
TRANSITIVE_PROPERTY {
 R(a,b) AND R(b,c) -> R(a,c)
}
SYMMETRIC_PROPERTY {
 R(a,b) -> R(b,a)
}
FUNCTIONAL_PROPERTY {
 R(a,b) AND R(a,c) -> b=c
 under equality semantics
}

CONSTRAINT_REPRESENTATION {
 KnowledgeBase
 -> IntegrityConstraints
 -> ValidationOrReasoning
}
SHACL_RELATIONSHIP {
 graph_constraints MAY be expressed with SHACL-style validation
 DISTINCT_FROM := OWL_open_world_entailment
}
VALIDATION_VS_INFERENCE {
 Validation := check_data_against_constraints
 Inference := derive_semantic_consequences
 MUST_REMAIN_DISTINCT := TRUE
}

PROVENANCE {
 Statement -> Source + Derivation + Context
}
JUSTIFICATION {
 EntailedStatement -> MinimalOrRelevantSupportingAxioms
 under selected explanation method
}
EXPLANATION {
 reasoning_result
 -> supporting_facts
 -> axioms
 -> inference_steps
}

INCONSISTENCY {
 KB has no model
 under classical model-theoretic semantics
}
INCONSISTENCY_EXPLANATION {
 identify conflicting_axioms_or_assertions
}
MINIMAL_INCONSISTENT_SUBSET {
 subset inconsistent
 AND every proper subset consistent
 under subset-minimal definition
}

PARACONSISTENT_KR {
 permits reasoning from inconsistent_information
 without unrestricted explosion
 REQUIRE paraconsistent_semantics
}
PROBABILISTIC_KR {
 knowledge statements include uncertainty
 REQUIRE probability_semantics
}
FUZZY_KR {
 concepts/relations may have graded_membership
 REQUIRE fuzzy_semantics
}
TEMPORAL_KR {
 facts/relations scoped_over_time
 REQUIRE temporal_semantics
}
MODAL_KR {
 statements include necessity_possibility_or_agent_modalities
 REQUIRE modal_semantics
}

CONTEXT {
 Statement validity or meaning
 may depend_on declared_context
}
NAMED_GRAPH_RELATIONSHIP {
 graph_statements MAY be grouped/scoped
 REQUIRE RDF_dataset_semantics_if_used
}

REIFICATION {
 Statement treated_as_entity
 to attach metadata
}
N_ARY_RELATION_PATTERN {
 complex relation
 -> relation_instance_entity
 -> participant_roles
}
EVENT_REPRESENTATION {
 Event
 -> Participants
 -> Time
 -> Location
 -> Properties
}
STATE_REPRESENTATION {
 Entity + Property + TemporalOrContextualScope
}

SCHEMA_LEVEL {
 classes_properties_axioms
}
INSTANCE_LEVEL {
 individuals_and_assertions
}
META_LEVEL {
 statements_about_schema_or_knowledge
}

KNOWLEDGE_ACQUISITION {
 Sources
 -> ExtractedStatements
 -> EntityResolution
 -> SchemaAlignment
 -> Validation
 -> KnowledgeBase
}
ENTITY_RESOLUTION {
 determine whether mentions refer_to same_entity
}
SCHEMA_ALIGNMENT {
 map semantically_corresponding concepts/properties
 across schemas
}
ONTOLOGY_MAPPING {
 SourceOntology -> Correspondences -> TargetOntology
}
SEMANTIC_EQUIVALENCE_MAPPING {
 A ≈ B
 REQUIRE explicit_equivalence_criterion
}

KNOWLEDGE_UPDATE {
 ExistingKB + NewKnowledge
 -> consistency_check
 -> add_revise_or_reject
}
MONOTONIC_UPDATE {
 adding_axioms preserves prior_entailments
 in monotonic logic
}
NONMONOTONIC_UPDATE {
 new_information MAY retract prior_conclusions
}

BELIEF_REVISION_RELATIONSHIP {
 inconsistent_or_new_information
 -> revision_operator
 -> updated_belief_set
}
TRUTH_MAINTENANCE_RELATIONSHIP {
 track_dependencies
 -> retract unsupported_derived_facts
}

REASONING_PIPELINE {
 INPUT := KnowledgeBase + Query
 IDENTIFY representation_language
 PARSE axioms
 NORMALISE where required
 CHECK consistency
 SELECT reasoning_procedure
 DERIVE consequences
 BUILD justification
 RETURN answer
}

VERIFY_KNOWLEDGE_BASE(KB) {
 CHECK syntax
 CHECK entity_references
 CHECK relation_domains_ranges
 CHECK logical_consistency_if_required
 CHECK constraint_conformance_if_validation_requested
 RETURN validation_report
}

QUERY_REASONING(KB,Q) {
 IF KB inconsistent_under_classical_semantics:
  HANDLE according_to declared_policy

 IF Q is subsumption:
  perform_subsumption_reasoning

 IF Q is instance_query:
  perform_instance_checking

 IF Q is rule_query:
  perform_rule_inference

 IF Q is graph_query:
  evaluate under declared entailment_regime

 RETURN entailed_not_entailed_unknown_or_framework_specific_status
}

KNOWLEDGE_REPRESENTATION_TRADEOFFS {
 EXPRESSIVITY
  vs
 DECIDABILITY
  vs
 COMPUTATIONAL_COMPLEXITY
  vs
 INTERPRETABILITY
}
EXPRESSIVITY {
 richer_language -> potentially_more_representational_power
}
DECIDABILITY {
 reasoning_problem has terminating decision_procedure
}
COMPLEXITY {
 computational_cost depends_on logic_fragment_and_task
}

APPLICATION_RELATIONSHIPS {
 KnowledgeRepresentationLogic --supports--> Ontologies
 KnowledgeRepresentationLogic --supports--> KnowledgeGraphs
 KnowledgeRepresentationLogic --supports--> ExpertSystems
 KnowledgeRepresentationLogic --supports--> SemanticWeb
 KnowledgeRepresentationLogic --supports--> ArtificialIntelligence
 KnowledgeRepresentationLogic --supports--> AutomatedReasoning
 KnowledgeRepresentationLogic --supports--> NaturalLanguageUnderstanding
 KnowledgeRepresentationLogic --supports--> DecisionSupport
 KnowledgeRepresentationLogic --supports--> DataIntegration
}

ONTOLOGY_LINKS {
 PARENT := MathematicalLogicOntologies
 RELATED := {
  LogicalReasoningOntology,
  PredicateLogicOntology,
  DescriptionLogicOntology,
  RuleBasedReasoningOntology,
  NonMonotonicReasoningOntology,
  OntologyOntology,
  KnowledgeGraphOntology,
  SemanticNetworkOntology,
  LogicProgrammingOntology,
  ModelTheoryOntology,
  AutomatedReasoningOntology
 }
}

SEMANTIC_EQUIVALENCE {
 KnowledgeRepresentationLogic ≈ LogicBasedKnowledgeRepresentation
 KnowledgeBase ≈ FormalKnowledgeRepository WHEN logical_semantics_explicit
 Concept ≈ Class WHEN ontology_context
 Role ≈ Relation WHEN description_logic_context
 Individual ≈ Instance WHEN class_membership_context
}

CONSTRAINTS {
 C1 := every represented statement MUST have semantics in declared representation language
 C2 := schema-level and instance-level knowledge MUST remain distinguishable
 C3 := open-world absence MUST NOT automatically be interpreted as false
 C4 := unique-name assumptions MUST NOT be assumed unless declared
 C5 := syntactic graph structure MUST NOT be conflated with semantic entailment
 C6 := validation constraints MUST remain distinct from open-world logical inference
 C7 := nonmonotonic, fuzzy, probabilistic and paraconsistent extensions MUST identify their semantics
 C8 := equivalence mappings MUST NOT be asserted without appropriate semantic justification
 C9 := inferred statements SHOULD preserve provenance/justification where explanation is required
 C10 := inconsistency handling MUST follow declared logic
 C11 := reasoning claims MUST remain scoped to selected entailment regime
 C12 := transformations between representation languages MUST preserve intended semantics
}

REASONING_CHAIN Query Q {
 IDENTIFY representation_language
 IDENTIFY knowledge_base KB
 IDENTIFY schema_axioms
 IDENTIFY instance_assertions
 IDENTIFY rules
 IDENTIFY constraints
 IDENTIFY query_type
 IDENTIFY entailment_regime

 VALIDATE syntax
 RESOLVE entities
 CHECK schema_alignment_if_needed

 IF consistency_required:
  CHECK KB_consistency

 IF description_logic_context {
  CLASSIFY concepts
  REALISE individuals
  ANSWER subsumption_or_instance_queries
 }

 IF rule_context {
  APPLY forward_or_backward_inference
 }

 IF graph_context {
  APPLY declared graph_entailment_semantics
 }

 IF nonmonotonic_context {
  CHECK defaults_exceptions_priorities
 }

 IF validation_context {
  APPLY constraint_validation
  KEEP validation_distinct_from_entailment
 }

 BUILD provenance
 BUILD justification
 VALIDATE constraints
 RETURN answer WITH semantic_trace
}

CONFIDENCE {
 CORE_KR_LOGIC_DEFINITION := 1.00
 FOL_RELATIONSHIP := 1.00
 DESCRIPTION_LOGIC_CORE := 1.00
 OPEN_WORLD_DISTINCTION := 1.00
 RULE_REASONING_RELATIONSHIP := 1.00
 KNOWLEDGE_GRAPH_RELATIONSHIP := 0.99
 VALIDATION_INFERENCE_DISTINCTION := 1.00

 SPECIFIC_RESULT :=
  IF language_kb_query_and_semantics_complete THEN 1.00
  ELSE REQUIRE missing_information
}

UPDATE_MODEL {
 IMMUTABLE_CORE := {
  FormalRepresentation,
  ExplicitSemantics,
  KnowledgeBaseStructure,
  Entailment,
  InferenceTaskDistinction
 }
 EXTENSIBLE := {
  LogicalLanguages,
  OntologyConstructors,
  RuleLanguages,
  KnowledgeGraphSemantics,
  NonMonotonicExtensions,
  UncertaintyModels,
  ValidationLanguages,
  ApplicationDomains,
  RelatedOntologies
 }
 ON_NEW_KNOWLEDGE {
  IDENTIFY new_statement
  RESOLVE entities
  VALIDATE syntax
  CHECK schema_semantics
  CHECK consistency
  UPDATE provenance
  DERIVE new_consequences_if_required
  UPDATE confidence
 }
}

AI_CONSUMPTION {
 MACHINE_READABLE := TRUE
 HUMAN_INTERPRETABLE := TRUE
 RECURSIVE_REASONING := TRUE
 ENTITY_RESOLUTION := TRUE
 CONCEPT_RESOLUTION := TRUE
 RELATION_RESOLUTION := TRUE
 ONTOLOGY_REASONING := TRUE
 DESCRIPTION_LOGIC_REASONING := TRUE
 RULE_REASONING := TRUE
 KNOWLEDGE_GRAPH_REASONING := TRUE
 SUBSUMPTION_REASONING := TRUE
 INSTANCE_REASONING := TRUE
 CONSISTENCY_REASONING := TRUE
 PROVENANCE_TRACKING := TRUE
 JUSTIFICATION_GENERATION := TRUE
 VALIDATION_REQUIRED := TRUE
}

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

OUTPUT {
 PRIMARY_ENTITY := KnowledgeRepresentationLogic
 INTERPRETATION := "Knowledge representation logic provides formal languages and semantics for encoding concepts, entities, relations, facts, rules and constraints so that machines can reason over structured knowledge. It connects first-order and description logics, ontologies, rule systems and knowledge graphs while making assumptions about entailment, identity, openness, inconsistency and validation explicit."
}

END_FLOW