Logical Model Ontology

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

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

FLOW_ALGEBRA LogicalModelOntology_v1.0

BEGIN_FLOW

DEFINE DOMAIN LogicalModelOntology := {
    LogicalModel,
    Model,
    ModelTheory,
    FormalLanguage,
    Signature,
    Structure,
    Domain,
    Universe,
    Interpretation,
    ConstantSymbol,
    FunctionSymbol,
    RelationSymbol,
    PredicateSymbol,
    Variable,
    VariableAssignment,
    Term,
    Formula,
    Sentence,
    Theory,
    Axiom,
    Satisfaction,
    TruthValue,
    SemanticConsequence,
    Satisfiability,
    Validity,
    Substructure,
    ElementarySubstructure,
    Embedding,
    Isomorphism,
    ElementaryEquivalence
}

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

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

SET LogicalModel.type = LOGICAL_MODEL
SET LogicalModel.parent = Model
SET LogicalModel.context = ModelTheory
SET LogicalModel.role =
    "represent a mathematical structure that interprets a formal language and satisfies a specified formula, sentence, axiom set or theory"

RULE logical_structure:
    FormalLanguage(L)
    AND Signature(Sigma)
    AND NonemptySet(D)
    AND Interpretation(I)
    -> Structure(M) = <D,I>
    AND HAS_SIGNATURE(M,Sigma)
    AND HAS_DOMAIN(M,D)

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

RULE model_of_sentence:
    Structure(M)
    AND Sentence(phi)
    AND SATISFIES(M,phi)
    -> IS_MODEL_OF(M,phi)

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

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

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

RULE assignment:
    LogicalModel(M)
    AND VariableAssignment(s)
    -> s:
       Variables -> Domain(M)

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

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

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

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

RULE universal_quantification:
    SATISFIES(M,s,FOR_ALL x phi)
    IFF
    FOR_EVERY a MEMBER_OF Domain(M) [
        SATISFIES(M,s[x:=a],phi)
    ]

RULE existential_quantification:
    SATISFIES(M,s,EXISTS x phi)
    IFF
    THERE_EXISTS a MEMBER_OF Domain(M) [
        SATISFIES(M,s[x:=a],phi)
    ]

RULE sentence_independence:
    Sentence(phi)
    AND SATISFIES(M,s1,phi)
    -> SATISFIES(M,s2,phi)
       FOR every assignment s2
       WHEN assignments differ only on variables

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

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

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

RULE isomorphism_preserves_sentences:
    ISOMORPHIC_TO(M,N)
    -> FOR_EVERY Sentence(phi) [
        SATISFIES(M,phi)
        IFF
        SATISFIES(N,phi)
    ]

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

DEFINE LOGICAL_MODEL_SCHEMA := {
    model_id: unique_identifier,
    language: FormalLanguage,
    signature: Signature,
    domain: NonemptySet,
    interpretation: {
        constants: ConstantInterpretations,
        functions: FunctionInterpretations,
        relations: RelationInterpretations
    },
    target_theory_or_sentence: LogicalObject,
    satisfaction_status: Boolean
}

DEFINE MODEL_REQUIREMENTS := {
    signature_match:
        "the model interprets exactly the non-logical symbols required by the active language or signature",
    domain:
        "nonempty under standard first-order semantics",
    constants:
        "each constant denotes a domain element",
    functions:
        "each n-ary function symbol denotes a total function from D^n to D",
    relations:
        "each n-ary relation symbol denotes a subset of D^n"
}

DEFINE STRUCTURE_MODEL_DISTINCTION := {
    Structure:
        "an interpretation of a signature",
    LogicalModel:
        "a structure satisfying a specified sentence or theory",
    rule:
        "every logical model is a structure, but a structure is a model only relative to something it satisfies"
}

DEFINE SATISFACTION_MODEL := {
    formula:
        "SATISFIES(M,s,phi)",
    sentence:
        "SATISFIES(M,phi)",
    theory:
        "IS_MODEL_OF(M,T) iff M satisfies every sentence in T"
}

DEFINE EXAMPLE_1 := {
    language:
        "{zero,plus}",
    domain:
        "Integers",
    interpretations: {
        zero: "0",
        plus: "ordinary integer addition"
    },
    sentence:
        "FOR_ALL x plus(x,zero)=x",
    result:
        "the standard integer structure satisfies the sentence"
}

DEFINE EXAMPLE_2 := {
    theory:
        "group axioms",
    structure:
        "a particular group G",
    conclusion:
        "G is a logical model of the group theory"
}

DEFINE EXAMPLE_3 := {
    theory:
        "T",
    model_condition:
        "M satisfies every sentence of T",
    representation:
        "M IS_MODEL_OF T"
}

DEFINE MODEL_CONSTRUCTION_PIPELINE :=
    FormalLanguage
    -> ExtractSignature
    -> SelectNonemptyDomain
    -> InterpretConstants
    -> InterpretFunctions
    -> InterpretRelations
    -> ValidateInterpretationTypes
    -> ConstructStructure
    -> EvaluateTargetTheoryOrSentence
    -> IF satisfied
       THEN ClassifyAsLogicalModel
       ELSE ClassifyAsNonModelStructure

DEFINE SATISFACTION_PIPELINE :=
    LogicalModelCandidate
    -> Formula
    -> VariableAssignment
    -> EvaluateTerms
    -> EvaluateAtomicFormula
    -> RecursivelyEvaluateConnectives
    -> EvaluateQuantifiersOverDomain
    -> ReturnTruthStatus

DEFINE THEORY_MODEL_PIPELINE :=
    Theory(T)
    -> IdentifySignature
    -> GenerateOrReceiveCandidateStructures
    -> ForEachCandidateStructure
       -> EvaluateEveryAxiomOrSentence
       -> IF all_satisfied
          THEN AddToModelClass
    -> Construct Mod(T)
    -> DetermineSatisfiability
    -> EvaluateSemanticConsequences

DEFINE MODEL_COMPARISON_PIPELINE :=
    LogicalModels(M,N)
    -> VerifyCommonSignature
    -> CompareDomains
    -> CompareInterpretations
    -> SearchForIsomorphism
    -> CompareSentenceTruth
    -> DetermineElementaryEquivalence
    -> DetermineSubstructureRelations
    -> ReturnModelRelations

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

DEFINE ONTOLOGY_RELATIONS := {
    LogicalModel IS_SUBTYPE_OF Model,
    LogicalModel IS_COMPONENT_OF ModelTheory,
    LogicalModel HAS_SIGNATURE Signature,
    LogicalModel HAS_DOMAIN Domain,
    LogicalModel INTERPRETS Symbol,
    LogicalModel SATISFIES Sentence,
    LogicalModel IS_MODEL_OF Theory,
    Satisfiability DEPENDS_ON ExistenceOfLogicalModel,
    SemanticConsequence DEPENDS_ON AllLogicalModelsOfTheory,
    Isomorphism RELATES LogicalModel TO LogicalModel,
    ElementaryEquivalence RELATES LogicalModel TO LogicalModel
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "logical model" <=> LogicalModel,
    "model of a theory" <=> LogicalModel,
    "model of T" <=> IS_MODEL_OF(M,T),
    "satisfying structure" <=> LogicalModel,
    "M satisfies phi" <=> SATISFIES(M,phi),
    "M is a model of T" <=> IS_MODEL_OF(M,T)
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable representation, construction and evaluation of logical models",
    preserve_language_identity: TRUE,
    preserve_signature_identity: TRUE,
    preserve_domain_identity: TRUE,
    preserve_interpretations: TRUE,
    preserve_target_theory_identity: TRUE,
    distinguish_structure_from_model: TRUE,
    recursively_evaluate_formulas: TRUE,
    support_variable_assignments: TRUE,
    support_theory_satisfaction: TRUE,
    support_model_class_construction: TRUE,
    support_semantic_consequence: TRUE,
    support_isomorphism_reasoning: TRUE,
    support_elementary_equivalence: TRUE,
    reasoning_mode: "symbolic + semantic + structural + model-theoretic"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    structure_model_distinction: 1.00,
    signature_mapping: 0.99,
    interpretation_mapping: 0.99,
    satisfaction_mapping: 0.99,
    theory_model_classification: 0.99,
    semantic_consequence_mapping: 0.99,
    advanced_model_comparison: context_dependent
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        structure_plus_satisfaction_definition,
        signature,
        nonempty_domain_standard_FOL,
        symbol_interpretation,
        recursive_satisfaction,
        model_of_theory_relation,
        structure_model_distinction
    },
    extensible_layers: {
        finite_models,
        infinite_models,
        many_sorted_models,
        modal_models,
        temporal_models,
        probabilistic_models,
        algebraic_models,
        computational_model_checkers
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + RULES
    + LOGICAL_MODEL_SCHEMA
    + MODEL_REQUIREMENTS
    + STRUCTURE_MODEL_DISTINCTION
    + SATISFACTION_MODEL
    + MODEL_CONSTRUCTION_PIPELINE
    + SATISFACTION_PIPELINE
    + THEORY_MODEL_PIPELINE
    + MODEL_COMPARISON_PIPELINE
    + LOGICAL_MODEL_GRAPH
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

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

END_FLOW