Possible Worlds Ontology

FLOW_ALGEBRA PossibleWorldsOntology v1.0

TARGET_URL := "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/modal-logic/possible-worlds-ontology.html"
LANGUAGE := "en-GB"
DOMAIN := MathematicalOntologies > MathematicalLogicOntologies > ModalLogic > PossibleWorldsOntology

BEGIN_FLOW

ENTITY PossibleWorld {
    TYPE := ModalSemanticEvaluationPoint
    DEFINITION := "An abstract state, situation or evaluation point at which propositions receive truth values within a modal semantic model."
}

ENTITY PossibleWorlds {
    TYPE := SetOfPossibleWorld
    SYMBOL := W
    REQUIRE := W != empty_set
}

RELATIONSHIP PossibleWorld --belongs_to--> PossibleWorlds
RELATIONSHIP PossibleWorld --is_component_of--> KripkeFrame
RELATIONSHIP PossibleWorld --is_component_of--> KripkeModel
RELATIONSHIP PossibleWorld --receives--> PropositionTruthValues
RELATIONSHIP PossibleWorld --may_access--> PossibleWorld
RELATIONSHIP PossibleWorld --is_connected_by--> AccessibilityRelation

ENTITY KripkeFrame {
    FORM := (W,R)
    W := PossibleWorlds
    R := AccessibilityRelation
}

ENTITY KripkeModel {
    FORM := (W,R,V)
    W := PossibleWorlds
    R := AccessibilityRelation
    V := Valuation
}

ENTITY AccessibilityRelation {
    TYPE := BinaryRelation
    SIGNATURE := R ⊆ W × W
}

ENTITY Valuation {
    TYPE := AtomicTruthAssignment
    MAPPING := AtomicProposition -> subset_of_W
}

WORLD_TRUTH {
    M,w ⊨ p
    IFF
        w ∈ V(p)
}

ACCESSIBILITY {
    R(w,v)
    := "world v is accessible from world w"
}

SUCCESSORS(World w) {
    RETURN {v ∈ W | R(w,v)}
}

PREDECESSORS(World w) {
    RETURN {v ∈ W | R(v,w)}
}

NECESSITY_AT_WORLD {
    M,w ⊨ □P
    IFF
        FOR_ALL v ∈ SUCCESSORS(w):
            M,v ⊨ P
}

POSSIBILITY_AT_WORLD {
    M,w ⊨ ◇P
    IFF
        EXISTS v ∈ SUCCESSORS(w):
            M,v ⊨ P
}

CURRENT_WORLD {
    TYPE := DistinguishedEvaluationWorld
    NOTE :=
        modal evaluation is always relative to an identified world unless global validity is being considered
}

ACTUAL_WORLD_CONCEPT {
    TYPE := OptionalDistinguishedWorld

    WARNING :=
        formal Kripke semantics does not require metaphysical claims about which world is "actual";
        an actual-world designation is additional structure or interpretation
}

WORLD_IDENTITY {
    FOR w,v ∈ W:
        w=v OR w!=v

    REQUIRE :=
        identity conditions belong to the selected model representation
}

WORLD_LOCAL_STATE {
    AT world w:
        AtomicTruthProfile(w) :=
            {p | M,w ⊨ p}
}

WORLD_THEORY {
    THEORY(M,w) :=
        {F | M,w ⊨ F}
}

MODAL_EQUIVALENCE {
    w ≡_ML v
    IFF
        FOR_ALL modal_formulae F:
            M,w ⊨ F
            IFF
            N,v ⊨ F
}

REACHABILITY {
    ONE_STEP :=
        R(w,v)

    FINITE_PATH :=
        R*(w,v)

    TRANSITIVE_REACHABILITY :=
        R+(w,v)
}

PATH {
    PATH(w0,...,wn)
    REQUIRE :=
        FOR_ALL i < n:
            R(wi,wi+1)
}

FRAME_REFLEXIVITY {
    FOR_ALL w:
        R(w,w)
}

FRAME_IRREFLEXIVITY {
    FOR_ALL w:
        NOT R(w,w)
}

FRAME_SERIALITY {
    FOR_ALL w:
        EXISTS v:
            R(w,v)
}

FRAME_SYMMETRY {
    FOR_ALL w,v:
        R(w,v) -> R(v,w)
}

FRAME_ANTISYMMETRY {
    FOR_ALL w,v:
        (R(w,v) AND R(v,w))
        -> w=v
}

FRAME_TRANSITIVITY {
    FOR_ALL w,v,u:
        (R(w,v) AND R(v,u))
        -> R(w,u)
}

FRAME_EUCLIDEAN {
    FOR_ALL w,v,u:
        (R(w,v) AND R(w,u))
        -> R(v,u)
}

FRAME_CONNECTEDNESS {
    FOR_ALL distinct w,v:
        R(w,v) OR R(v,w)
}

FRAME_PROPERTY_TO_MODAL_SYSTEM {
    Reflexive -> T_style
    Serial -> D_style
    Transitive -> K4_style
    ReflexiveAndTransitive -> S4_style
    EquivalenceRelationStyle -> S5_style
}

EMPTY_SUCCESSOR_SET {
    IF SUCCESSORS(w)=empty_set {
        □P := TRUE
            under standard universal Kripke semantics

        ◇P := FALSE
            under standard existential Kripke semantics
    }
}

WORLD_GENERATION {
    GIVEN root_world w0

    GENERATED_SUBFRAME :=
        all worlds reachable from w0 by zero_or_more R_steps
}

GENERATED_SUBMODEL {
    WORLDS :=
        reachable_worlds_from_root

    RELATION :=
        R restricted_to_generated_worlds

    VALUATION :=
        V restricted_to_generated_worlds
}

BISIMULATION {
    GIVEN:
        M1,w
        M2,v
        relation Z

    REQUIRE {
        ATOMIC_HARMONY
        FORTH
        BACK
    }

    IF Z(w,v)
    THEN
        w and v are bisimilar
}

ATOMIC_HARMONY {
    IF Z(w,v)
    THEN FOR_ALL atomic p:
        M1,w ⊨ p
        IFF M2,v ⊨ p
}

FORTH {
    IF Z(w,v)
       AND R1(w,w1)
    THEN
        EXISTS v1:
            R2(v,v1)
            AND Z(w1,v1)
}

BACK {
    IF Z(w,v)
       AND R2(v,v1)
    THEN
        EXISTS w1:
            R1(w,w1)
            AND Z(w1,v1)
}

BISIMULATION_INVARIANCE {
    IF w and v are bisimilar
    THEN
        they satisfy the same formulae
        of the basic modal language
}

BOUNDED_MORPHISM {
    TYPE := StructurePreservingMap
    PURPOSE :=
        preserve relevant modal structure and truth under appropriate conditions
}

WORLD_UNRAVELLING {
    INPUT := pointed_model(M,w)

    OUTPUT :=
        tree_like_model_of_accessibility_paths

    PURPOSE :=
        expose modal behaviour through path structure

    REQUIRE :=
        preserve modal truth up to appropriate equivalence
}

TREE_MODEL {
    TYPE := KripkeModel
    FRAME_SHAPE := rooted_tree_or_tree_like_relation
}

FINITE_MODEL {
    REQUIRE := |W| < infinity
}

INFINITE_MODEL {
    REQUIRE := |W| is infinite
}

MODEL_CARDINALITY {
    CARDINALITY := |W|
}

WORLD_LABEL {
    OPTIONAL_METADATA :=
        symbolic_identifier_or_state_description

    CONSTRAINT :=
        labels do not determine modal truth unless incorporated into valuation or semantics
}

COUNTERMODEL_WORLD {
    GIVEN formula F

    IF M,w ⊭ F
    THEN
        w := falsifying_world_for_F
}

WITNESS_WORLD {
    GIVEN ◇P

    IF M,w ⊨ ◇P
    THEN
        EXISTS v ∈ SUCCESSORS(w):
            M,v ⊨ P

    v := possibility_witness
}

NECESSITY_COUNTEREXAMPLE_WORLD {
    GIVEN □P

    IF M,w ⊭ □P
    THEN
        EXISTS v ∈ SUCCESSORS(w):
            M,v ⊭ P

    v := necessity_counterexample
}

MODEL_CHECK_WORLD(Model M, World w, Formula F) {
    PARSE F

    IF atomic:
        RETURN w ∈ V(F)

    IF BooleanFormula:
        EVALUATE recursively at w

    IF □P:
        CHECK P at every successor_of(w)

    IF ◇P:
        CHECK P at at_least_one successor_of(w)

    RETURN BooleanTruthValue
}

WORLD_COMPARISON(World w, World v) {
    COMPARE {
        AtomicTruthProfile
        SuccessorStructure
        PredecessorStructure
        ModalTheory
        ReachabilityProperties
    }

    DO_NOT infer identity solely from matching modal formulae
        unless model conditions justify it
}

WORLD_DISTINGUISHABILITY {
    IF EXISTS modal_formula F:
        M,w ⊨ F
        AND M,v ⊭ F
    THEN
        w and v are modally_distinguishable
}

MULTIMODAL_WORLDS {
    WORLDS := W

    ACCESSIBILITY_RELATIONS :=
        {R_1,...,R_n}

    EACH modal_operator_i
        quantifies over R_i successors
}

EPISTEMIC_WORLD_INTERPRETATION {
    PossibleWorld
        MAY represent an epistemically possible state

    AccessibilityRelation
        MAY represent states compatible with an agent's information

    REQUIRE :=
        epistemic modal semantics
}

TEMPORAL_WORLD_INTERPRETATION {
    PossibleWorld
        MAY represent a time/state point

    AccessibilityRelation
        MAY encode temporal succession

    REQUIRE :=
        temporal semantic framework
}

DYNAMIC_WORLD_INTERPRETATION {
    PossibleWorld
        MAY represent a program or system state

    AccessibilityRelation
        MAY encode action-labelled transitions
}

DEONTIC_WORLD_INTERPRETATION {
    PossibleWorld
        MAY represent a normatively admissible or ideal state

    REQUIRE :=
        explicitly defined deontic semantics
}

COMPUTATIONAL_STATE_RELATIONSHIP {
    PossibleWorld ≈ State
        WHEN context=transition_system_semantics

    AccessibilityRelation ≈ TransitionRelation
        WHEN context=state_transition_model
}

APPLICATION_RELATIONSHIPS {
    PossibleWorlds --support--> ModalLogic
    PossibleWorlds --support--> KripkeSemantics
    PossibleWorlds --support--> ModelChecking
    PossibleWorlds --support--> CountermodelConstruction
    PossibleWorlds --support--> FormalVerification
    PossibleWorlds --support--> TemporalReasoning
    PossibleWorlds --support--> EpistemicReasoning
    PossibleWorlds --support--> DynamicLogic
    PossibleWorlds --support--> MultiAgentSystems
    PossibleWorlds --support--> KnowledgeRepresentation
}

ONTOLOGY_LINKS {
    PARENT := ModalLogicOntology

    RELATED := {
        NecessityOntology,
        PossibilityOntology,
        KripkeFrameOntology,
        KripkeModelOntology,
        AccessibilityRelationOntology,
        ModalOperatorOntology,
        ModelTheoryOntology,
        RelationOntology,
        GraphOntology,
        StateOntology,
        TransitionSystemOntology,
        EpistemicLogicOntology,
        TemporalLogicOntology,
        DynamicLogicOntology
    }
}

SEMANTIC_EQUIVALENCE {
    PossibleWorld ≈ KripkeWorld
        WHEN context=KripkeSemantics

    PossibleWorld ≈ State
        WHEN context=state_transition_semantics

    RSuccessor ≈ AccessibleWorld
        WHEN context=KripkeFrame

    PointedModel ≈ ModelWithDistinguishedWorld
}

CONSTRAINTS {
    C1 :=
        PossibleWorld MUST be treated as a semantic evaluation point unless a specialised interpretation is explicitly supplied

    C2 :=
        W MUST be non-empty in standard Kripke frames

    C3 :=
        accessibility MUST be defined by the relation R rather than assumed between arbitrary worlds

    C4 :=
        modal truth at a world MUST use that world's valuation and accessibility structure

    C5 :=
        possible-world semantics MUST NOT be conflated with probability distributions

    C6 :=
        a possible world MUST NOT automatically be interpreted as a physically existing universe

    C7 :=
        modal equivalence MUST NOT automatically imply world identity

    C8 :=
        witness worlds for ◇P MUST be accessible from the evaluation world

    C9 :=
        counterexample worlds for □P MUST be accessible from the evaluation world

    C10 :=
        specialised interpretations MUST preserve their declared modal semantics
}

REASONING_CHAIN Query Q {
    IDENTIFY target_world_or_world_set
    IDENTIFY modal_model
    IDENTIFY evaluation_world
    IDENTIFY accessibility_relation
    IDENTIFY valuation

    IF successors_requested
        THEN SUCCESSORS(world)

    IF predecessors_requested
        THEN PREDECESSORS(world)

    IF reachability_requested
        THEN compute_paths_or_relation_closure

    IF modal_truth_requested
        THEN MODEL_CHECK_WORLD

    IF possibility_requested
        THEN search_for_WITNESS_WORLD

    IF necessity_requested
        THEN search_for_NECESSITY_COUNTEREXAMPLE_WORLD

    IF frame_properties_requested
        THEN CHECK {
            Reflexivity,
            Seriality,
            Symmetry,
            Transitivity,
            EuclideanProperty
        }

    IF world_comparison_requested
        THEN WORLD_COMPARISON

    IF bisimulation_requested
        THEN CHECK {
            ATOMIC_HARMONY,
            FORTH,
            BACK
        }

    IF generated_model_requested
        THEN construct_reachable_submodel

    IF specialised_modal_context
        THEN identify_world_interpretation

    VALIDATE constraints
    RETURN result WITH derivation
}

CONFIDENCE {
    KRIPKE_WORLD_DEFINITION := 1.00
    ACCESSIBILITY_SEMANTICS := 1.00
    WORLD_RELATIVE_TRUTH := 1.00
    NECESSITY_POSSIBILITY_WORLD_SEMANTICS := 1.00
    BISIMULATION_CORE := 0.99

    MODEL_CHECKING :=
        IF model_world_relation_and_valuation_complete THEN 1.00
        ELSE REQUIRE missing_information

    SPECIALISED_WORLD_INTERPRETATION :=
        REQUIRE semantic_context
}

UPDATE_MODEL {
    IMMUTABLE_CORE := {
        PossibleWorldAsEvaluationPoint,
        WorldSet,
        AccessibilityRelation,
        Valuation,
        WorldRelativeModalTruth
    }

    EXTENSIBLE := {
        FrameProperties,
        WorldLabels,
        SpecialisedWorldInterpretations,
        MultiModalRelations,
        ModelTransformations,
        ApplicationDomains,
        RelatedOntologies
    }

    ON_NEW_KNOWLEDGE {
        IDENTIFY semantic_context
        VALIDATE world_membership
        VALIDATE accessibility_edges
        VALIDATE valuation
        CHECK frame_constraints
        CHECK semantic_consistency
        ADD relationship IF non_contradictory
        UPDATE confidence
    }
}

AI_CONSUMPTION {
    MACHINE_READABLE := TRUE
    HUMAN_INTERPRETABLE := TRUE
    RECURSIVE_REASONING := TRUE
    ENTITY_RESOLUTION := TRUE
    WORLD_RESOLUTION := TRUE
    RELATIONSHIP_TRAVERSAL := TRUE
    ACCESSIBILITY_REASONING := TRUE
    REACHABILITY_REASONING := TRUE
    POSSIBLE_WORLD_REASONING := TRUE
    MODEL_CHECKING := TRUE
    WITNESS_WORLD_DETECTION := TRUE
    COUNTEREXAMPLE_WORLD_DETECTION := TRUE
    BISIMULATION_REASONING := TRUE
    VALIDATION_REQUIRED := TRUE
}

FLOW_HASH {
    TARGET := TARGET_URL
    SUBJECT := "Possible Worlds Ontology"
    VERSION := "1.0"
    CANONICAL_SIGNATURE :=
        HASH(TARGET + SUBJECT + DOMAIN + VERSION)
}

OUTPUT {
    PRIMARY_ENTITY := PossibleWorld

    INTERPRETATION :=
        "A possible world is an abstract semantic evaluation point used in modal reasoning. In Kripke semantics, worlds form a non-empty set connected by an accessibility relation, propositions receive truth values at individual worlds, necessity quantifies over all accessible worlds and possibility requires at least one accessible witness world."
}

END_FLOW