Model Theory Ontology

KIRETH RESEARCH - SEMANTIC LIBRARY
AI CORE -> ONTOLOGIES -> MATHEMATICAL ONTOLOGIES -> MATHEMATICAL LOGIC ONTOLOGIES
MODEL THEORY -> MODEL THEORY ONTOLOGY

Canonical URL:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/model-theory/model-theory-ontology.html

FLOW_ALGEBRA ModelTheoryOntology_v1.0

BEGIN_FLOW

DEFINE DOMAIN ModelTheoryOntology := {
    ModelTheory,
    FormalLanguage,
    Signature,
    Vocabulary,
    Structure,
    Model,
    Domain,
    Universe,
    Interpretation,
    ConstantSymbol,
    FunctionSymbol,
    RelationSymbol,
    PredicateSymbol,
    Variable,
    VariableAssignment,
    Term,
    Formula,
    Sentence,
    Theory,
    Axiom,
    Satisfaction,
    TruthInStructure,
    SemanticConsequence,
    Satisfiability,
    Unsatisfiability,
    Validity,
    Substructure,
    ElementarySubstructure,
    Embedding,
    Isomorphism,
    Homomorphism,
    Automorphism,
    ElementaryEquivalence,
    Definability,
    Type,
    CompleteType,
    Realisation,
    Omission,
    Categoricity,
    Compactness,
    LowenheimSkolem,
    Ultraproduct,
    ElementaryClass
}

DEFINE ENTITY_TYPES := {
    MODEL_THEORY,
    FORMAL_LANGUAGE,
    SIGNATURE,
    SYMBOL,
    STRUCTURE,
    MODEL,
    DOMAIN,
    INTERPRETATION,
    ASSIGNMENT,
    TERM,
    FORMULA,
    SENTENCE,
    THEORY,
    SEMANTIC_RELATION,
    STRUCTURE_MAPPING,
    TYPE,
    META_PROPERTY,
    SEMANTIC_OBJECT
}

DEFINE RELATION_TYPES := {
    HAS_SIGNATURE,
    HAS_DOMAIN,
    INTERPRETS,
    ASSIGNS,
    DENOTES,
    SATISFIES,
    MODELS,
    ENTAILS,
    IS_MODEL_OF,
    IS_SUBSTRUCTURE_OF,
    IS_ELEMENTARY_SUBSTRUCTURE_OF,
    EMBEDS_INTO,
    ISOMORPHIC_TO,
    HOMOMORPHIC_TO,
    ELEMENTARILY_EQUIVALENT_TO,
    DEFINES,
    REALISES,
    OMITS,
    IS_INSTANCE_OF,
    IS_SUBTYPE_OF,
    DEPENDS_ON
}

SET ModelTheoryOntology.type = MODEL_THEORY
SET ModelTheoryOntology.parent = ModelTheory
SET ModelTheoryOntology.semantic_domain = MathematicalLogic
SET ModelTheoryOntology.role =
    "provide a typed machine-readable representation of formal structures, interpretations, models, satisfaction and model-theoretic relationships"

RULE signature_definition:
    ConstantSymbols(C)
    AND FunctionSymbols(F)
    AND RelationSymbols(R)
    -> Signature(L) = {
        constants: C,
        functions: F,
        relations: R,
        arities: AssignedArities(F,R)
    }

RULE structure_definition:
    Signature(L)
    AND NonemptySet(D)
    AND Interpretation(I)
    -> Structure(M) = <D,I>

RULE domain_assignment:
    Structure(M)
    -> HAS_DOMAIN(M,D)

RULE constant_interpretation:
    ConstantSymbol(c)
    AND Structure(M)
    -> Interpretation_M(c) MEMBER_OF Domain(M)

RULE function_interpretation:
    FunctionSymbol(f)
    AND Arity(f)=n
    AND Structure(M)
    -> Interpretation_M(f):
       Domain(M)^n -> Domain(M)

RULE relation_interpretation:
    RelationSymbol(R)
    AND Arity(R)=n
    AND Structure(M)
    -> Interpretation_M(R)
       SUBSET_OF Domain(M)^n

RULE variable_assignment:
    VariableAssignment(s)
    AND Structure(M)
    -> s:
       Variables -> Domain(M)

RULE variable_term_evaluation:
    Variable(x)
    -> Value(M,s,x)=s(x)

RULE constant_term_evaluation:
    ConstantSymbol(c)
    -> Value(M,s,c)=Interpretation_M(c)

RULE function_term_evaluation:
    Term(f(t1,...,tn))
    -> Value(M,s,f(t1,...,tn))
       = Interpretation_M(f)(
           Value(M,s,t1),
           ...,
           Value(M,s,tn)
         )

RULE atomic_relation_satisfaction:
    Formula(R(t1,...,tn))
    -> SATISFIES(M,s,R(t1,...,tn))
       IFF
       (
         Value(M,s,t1),
         ...,
         Value(M,s,tn)
       ) MEMBER_OF Interpretation_M(R)

RULE equality_satisfaction:
    Formula(t1=t2)
    -> SATISFIES(M,s,t1=t2)
       IFF
       Value(M,s,t1)=Value(M,s,t2)

RULE negation_satisfaction:
    SATISFIES(M,s,NOT(phi))
    IFF
    NOT SATISFIES(M,s,phi)

RULE conjunction_satisfaction:
    SATISFIES(M,s,AND(phi,psi))
    IFF
    (
        SATISFIES(M,s,phi)
        AND
        SATISFIES(M,s,psi)
    )

RULE disjunction_satisfaction:
    SATISFIES(M,s,OR(phi,psi))
    IFF
    (
        SATISFIES(M,s,phi)
        OR
        SATISFIES(M,s,psi)
    )

RULE implication_satisfaction:
    SATISFIES(M,s,IMPLIES(phi,psi))
    IFF
    (
        NOT SATISFIES(M,s,phi)
        OR
        SATISFIES(M,s,psi)
    )

RULE universal_satisfaction:
    Formula(FOR_ALL x phi)
    -> SATISFIES(M,s,FOR_ALL x phi)
       IFF
       FOR_EVERY a MEMBER_OF Domain(M) [
           SATISFIES(M,s[x:=a],phi)
       ]

RULE existential_satisfaction:
    Formula(EXISTS x phi)
    -> SATISFIES(M,s,EXISTS x phi)
       IFF
       THERE_EXISTS a MEMBER_OF Domain(M) [
           SATISFIES(M,s[x:=a],phi)
       ]

RULE sentence_truth:
    Sentence(phi)
    AND Structure(M)
    -> SATISFIES(M,phi)
       independent_of_variable_assignment

RULE model_of_theory:
    Theory(T)
    AND Structure(M)
    AND FOR_EVERY phi MEMBER_OF T [
        SATISFIES(M,phi)
    ]
    -> IS_MODEL_OF(M,T)

RULE satisfiable_theory:
    Theory(T)
    AND THERE_EXISTS Structure(M) [
        IS_MODEL_OF(M,T)
    ]
    -> Satisfiable(T)

RULE unsatisfiable_theory:
    Theory(T)
    AND NOT THERE_EXISTS Structure(M) [
        IS_MODEL_OF(M,T)
    ]
    -> Unsatisfiable(T)

RULE semantic_consequence:
    Theory(T)
    AND Sentence(phi)
    AND FOR_EVERY Structure(M) [
        IS_MODEL_OF(M,T)
        -> SATISFIES(M,phi)
    ]
    -> ENTAILS(T,phi)

RULE validity:
    Sentence(phi)
    AND FOR_EVERY Structure(M) [
        SATISFIES(M,phi)
    ]
    -> Valid(phi)

RULE isomorphism:
    Structure(M)
    AND Structure(N)
    AND Bijection(f:Domain(M)->Domain(N))
    AND PreservesConstantsFunctionsRelations(f)
    -> ISOMORPHIC_TO(M,N)

RULE isomorphism_invariance:
    ISOMORPHIC_TO(M,N)
    -> ELEMENTARILY_EQUIVALENT_TO(M,N)

RULE elementary_equivalence:
    Structure(M)
    AND Structure(N)
    AND FOR_EVERY Sentence(phi) [
        SATISFIES(M,phi)
        IFF
        SATISFIES(N,phi)
    ]
    -> ELEMENTARILY_EQUIVALENT_TO(M,N)

RULE substructure:
    Structure(M)
    AND Structure(N)
    AND Domain(M) SUBSET_OF Domain(N)
    AND InterpretationsInherited(M,N)
    -> IS_SUBSTRUCTURE_OF(M,N)

RULE elementary_substructure:
    IS_SUBSTRUCTURE_OF(M,N)
    AND FOR_EVERY Formula(phi(x1,...,xn))
    AND FOR_EVERY parameters a1,...,an MEMBER_OF Domain(M) [
        SATISFIES(M,phi(a1,...,an))
        IFF
        SATISFIES(N,phi(a1,...,an))
    ]
    -> IS_ELEMENTARY_SUBSTRUCTURE_OF(M,N)

RULE compactness:
    Theory(T)
    AND FOR_EVERY finite_subset T0 SUBSET_OF T [
        Satisfiable(T0)
    ]
    -> Satisfiable(T)

RULE definability:
    Structure(M)
    AND Formula(phi(x,p1,...,pn))
    AND Parameters(p1,...,pn)
    -> DefinedSet(phi,M)
       = {
           a MEMBER_OF Domain(M)
           | SATISFIES(M,phi(a,p1,...,pn))
       }

DEFINE MODEL_SCHEMA := {
    model_id: unique_identifier,
    language: FormalLanguage,
    signature: Signature,
    domain: NonemptySet,
    interpretation: {
        constants: ConstantInterpretations,
        functions: FunctionInterpretations,
        relations: RelationInterpretations
    },
    satisfied_theories: SetOfTheories
}

DEFINE SIGNATURE_SCHEMA := {
    constants: SetOfConstantSymbols,
    functions: {
        FunctionSymbol -> Arity
    },
    relations: {
        RelationSymbol -> Arity
    }
}

DEFINE SATISFACTION_MODEL := {
    formula_form:
        "SATISFIES(M,s,phi)",
    sentence_form:
        "SATISFIES(M,phi)",
    interpretation:
        "formula phi is true in structure M under assignment s"
}

DEFINE THEORY_MODEL := {
    theory: T,
    model_class:
        "Mod(T) = {M | IS_MODEL_OF(M,T)}",
    satisfiable:
        "Mod(T) is nonempty",
    semantic_consequence:
        "T ENTAILS phi iff every model of T satisfies phi"
}

DEFINE STRUCTURE_RELATIONS := {
    Isomorphism:
        "bijective structure-preserving correspondence",
    Homomorphism:
        "mapping preserving specified operations or relations",
    Embedding:
        "injective structure-preserving mapping",
    Substructure:
        "smaller structure with inherited interpretations",
    ElementarySubstructure:
        "substructure agreeing with the larger structure on all first-order formulas with parameters from the smaller domain",
    ElementaryEquivalence:
        "structures satisfy exactly the same first-order sentences"
}

DEFINE META_THEOREMS := {
    CompactnessTheorem,
    DownwardLowenheimSkolemTheorem,
    UpwardLowenheimSkolemTheorem,
    IsomorphismInvariance
}

DEFINE CRITICAL_DISTINCTIONS := {
    structure_vs_model: {
        structure:
            "an interpretation of a formal signature",
        model:
            "a structure satisfying a specified sentence or theory"
    },
    syntax_vs_semantics: {
        syntax:
            "formal terms and formulas",
        semantics:
            "their interpretation and satisfaction in structures"
    },
    assignment_vs_interpretation: {
        assignment:
            "maps variables to domain elements",
        interpretation:
            "assigns meanings to non-logical symbols"
    },
    isomorphism_vs_elementary_equivalence: {
        isomorphism:
            "strong structure-preserving bijection",
        elementary_equivalence:
            "agreement on all first-order sentences",
        implication:
            "isomorphism implies elementary equivalence, but not conversely in general"
    }
}

DEFINE EXAMPLE_1 := {
    language:
        "{zero,plus,less_than}",
    domain:
        "Integers",
    interpretations: {
        zero: "integer zero",
        plus: "ordinary integer addition",
        less_than: "ordinary integer ordering"
    },
    sentence:
        "FOR_ALL x EXISTS y less_than(x,y)",
    result:
        "satisfied in the standard integer structure"
}

DEFINE EXAMPLE_2 := {
    theory:
        "group axioms",
    candidate_structure:
        "a group G with its multiplication and identity interpreted",
    relation:
        "G IS_MODEL_OF GroupTheory"
}

DEFINE LANGUAGE_TO_STRUCTURE_PIPELINE :=
    FormalLanguage
    -> ExtractSignature
    -> SelectDomain
    -> InterpretConstants
    -> InterpretFunctions
    -> InterpretRelations
    -> ValidateArities
    -> ValidateFunctionClosure
    -> ConstructStructure

DEFINE FORMULA_EVALUATION_PIPELINE :=
    Formula
    -> ParseFormulaTree
    -> ResolveVariableAssignment
    -> EvaluateTerms
    -> EvaluateAtomicFormulas
    -> RecursivelyEvaluateLogicalOperators
    -> EvaluateQuantifiersOverDomain
    -> ReturnSatisfactionStatus

DEFINE THEORY_ANALYSIS_PIPELINE :=
    Theory
    -> ResolveSignature
    -> ConstructOrDiscoverCandidateStructures
    -> TestEveryAxiom
    -> IdentifyModels
    -> DetermineSatisfiability
    -> EvaluateSemanticConsequences
    -> CompareModels
    -> ApplyRelevantMetaTheorems
    -> ReturnModelTheoreticStructure

DEFINE MODEL_COMPARISON_PIPELINE :=
    Structures(M,N)
    -> VerifyCommonSignature
    -> SearchForHomomorphisms
    -> SearchForEmbeddings
    -> SearchForIsomorphism
    -> CompareFirstOrderSentenceTruth
    -> DetermineElementaryEquivalence
    -> TestSubstructureRelations
    -> TestElementarySubstructureRelations
    -> ReturnComparisonRelations

DEFINE MODEL_GRAPH := {
    node_types: {
        LanguageNode,
        SignatureNode,
        StructureNode,
        DomainNode,
        SymbolNode,
        FormulaNode,
        TheoryNode,
        ModelNode
    },
    edge_types: {
        HAS_SIGNATURE,
        HAS_DOMAIN,
        INTERPRETS,
        SATISFIES,
        IS_MODEL_OF,
        ENTAILS,
        ISOMORPHIC_TO,
        ELEMENTARILY_EQUIVALENT_TO,
        IS_SUBSTRUCTURE_OF
    },
    recursive_formula_semantics: TRUE
}

DEFINE ONTOLOGY_RELATIONS := {
    ModelTheoryOntology IS_COMPONENT_OF ModelTheory,
    FormalLanguage HAS_SIGNATURE Signature,
    Structure HAS_DOMAIN Domain,
    Structure INTERPRETS Symbol,
    Model IS_SUBTYPE_OF Structure,
    Model IS_MODEL_OF Theory,
    Theory HAS_MODEL Model,
    Formula EVALUATES_IN Structure,
    Sentence HAS_TRUTH_VALUE_IN Structure,
    Isomorphism RELATES Structure TO Structure,
    ElementaryEquivalence RELATES Structure TO Structure,
    ElementarySubstructure IS_SUBTYPE_OF Substructure,
    SemanticConsequence DEPENDS_ON ModelClass,
    Satisfiability DEPENDS_ON ExistenceOfModel,
    Definability DEPENDS_ON FormulaAndStructure
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "model theory ontology" <=> ModelTheoryOntology,
    "model theory" <=> ModelTheory,
    "structure" <=> Structure WHEN formal_logic_context=TRUE,
    "model" <=> Model WHEN theory_context=TRUE,
    "satisfies" <=> SATISFIES,
    "is a model of" <=> IS_MODEL_OF,
    "semantic consequence" <=> SemanticConsequence,
    "elementarily equivalent" <=> ElementaryEquivalence,
    "elementary substructure" <=> ElementarySubstructure
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable ontology for model-theoretic structures, satisfaction and semantic reasoning",
    preserve_language_identity: TRUE,
    preserve_signature_identity: TRUE,
    preserve_domain_identity: TRUE,
    preserve_symbol_identity: TRUE,
    preserve_symbol_interpretations: TRUE,
    preserve_variable_assignments: TRUE,
    recursively_evaluate_formulas: TRUE,
    distinguish_structure_from_model: TRUE,
    distinguish_syntax_from_semantics: TRUE,
    distinguish_assignment_from_interpretation: TRUE,
    support_theory_model_reasoning: TRUE,
    support_semantic_consequence: TRUE,
    support_model_comparison: TRUE,
    support_isomorphism_reasoning: TRUE,
    support_elementary_equivalence: TRUE,
    support_definability_reasoning: TRUE,
    support_meta_theorems_when_conditions_hold: TRUE,
    reasoning_mode: "symbolic + semantic + structural + model-theoretic"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    signature_mapping: 0.99,
    structure_mapping: 0.99,
    interpretation_mapping: 0.99,
    first_order_satisfaction: 0.99,
    theory_model_mapping: 0.99,
    isomorphism_mapping: 0.99,
    elementary_equivalence_mapping: 0.99,
    advanced_model_theoretic_analysis: theorem_and_context_dependent
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        language_signature_structure,
        domain,
        symbol_interpretation,
        variable_assignment,
        recursive_satisfaction,
        model_of_theory,
        semantic_consequence,
        structure_model_distinction
    },
    extensible_layers: {
        types,
        saturation,
        definability,
        categoricity,
        stability_theory,
        ultraproducts,
        compactness_applications,
        Lowenheim_Skolem_theory,
        finite_model_theory,
        o_minimality,
        geometric_model_theory
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + RULES
    + MODEL_SCHEMA
    + SIGNATURE_SCHEMA
    + SATISFACTION_MODEL
    + THEORY_MODEL
    + STRUCTURE_RELATIONS
    + META_THEOREMS
    + CRITICAL_DISTINCTIONS
    + LANGUAGE_TO_STRUCTURE_PIPELINE
    + FORMULA_EVALUATION_PIPELINE
    + THEORY_ANALYSIS_PIPELINE
    + MODEL_COMPARISON_PIPELINE
    + MODEL_GRAPH
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

RETURN {
    ontology: ModelTheoryOntology,
    canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/model-theory/model-theory-ontology.html",
    parent_domain: ModelTheory,
    semantic_domain: MathematicalLogic,
    primary_entity: Model,
    satisfaction_reasoning_enabled: TRUE,
    theory_model_reasoning_enabled: TRUE,
    model_comparison_enabled: TRUE,
    definability_reasoning_enabled: TRUE,
    meta_theorem_reasoning_enabled: TRUE,
    machine_consumable: TRUE,
    recursive_reasoning_enabled: TRUE,
    database_safe_unicode: TRUE,
    flow_complete: TRUE
}

END_FLOW