AND Gate Ontology

FLOW_ALGEBRA AndGateOntology v1.0

TARGET_URL := "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/boolean-logic/and-gate-ontology.html"
LANGUAGE := "en-GB"
DOMAIN := MathematicalOntologies > MathematicalLogicOntologies > BooleanLogic > AndGateOntology

BEGIN_FLOW

ENTITY ANDGate {
    TYPE := LogicGate
    LOGICAL_ROLE := Conjunction
    MIN_ARITY := 2
    INPUT_DOMAIN := {0,1} ≡ {FALSE,TRUE}
    OUTPUT_DOMAIN := {0,1} ≡ {FALSE,TRUE}
    DEFINITION := "A logic gate whose output is true exactly when all of its Boolean inputs are true."
}

RELATIONSHIP ANDGate --is_a--> LogicGate
RELATIONSHIP ANDGate --implements--> BooleanConjunction
RELATIONSHIP ANDGate --implements--> BooleanFunction
RELATIONSHIP ANDGate --corresponds_to--> ANDOperator
RELATIONSHIP ANDGate --can_be_represented_by--> BooleanExpression
RELATIONSHIP ANDGate --can_be_specified_by--> TruthTable
RELATIONSHIP ANDGate --can_be_composed_into--> LogicCircuit

ENTITY ANDOperator {
    SYMBOLS := {"AND","∧","·"}
    TYPE := BooleanOperator
}

BOOLEAN_FUNCTION {
    BINARY :=
        AND(x,y) := x ∧ y

    NARY :=
        AND(x1,...,xn) :=
            x1 ∧ x2 ∧ ... ∧ xn

    CONDITION :=
        OUTPUT=1 IFF FOR_ALL xi: xi=1
}

BINARY_TRUTH_TABLE {
    (A=0,B=0) -> Y=0
    (A=0,B=1) -> Y=0
    (A=1,B=0) -> Y=0
    (A=1,B=1) -> Y=1
}

NARY_SEMANTICS {
    INPUTS := x1,...,xn
    n >= 2

    IF FOR_ALL i IN {1,...,n}:
        xi=1
    THEN
        OUTPUT := 1
    ELSE
        OUTPUT := 0
}

EVALUATE_AND(InputVector X) {
    REQUIRE length(X) >= 2
    REQUIRE FOR_ALL x IN X: x IN {0,1}

    FOR_EACH x IN X {
        IF x=0
            RETURN 0
    }

    RETURN 1
}

ALGEBRAIC_FORM {
    FOR BooleanValues xi IN {0,1}:
        AND(x1,...,xn) =
            PRODUCT(i=1..n, xi)
}

SET_THEORETIC_ANALOGY {
    BooleanConjunction --corresponds_structurally_to--> SetIntersection
    TRUE --corresponds_to--> MembershipConditionSatisfied
    FALSE --corresponds_to--> MembershipConditionNotSatisfied
}

PROPOSITIONAL_LOGIC_MAPPING {
    INPUT_A := proposition_P
    INPUT_B := proposition_Q
    OUTPUT := P ∧ Q

    SEMANTICS :=
        P ∧ Q IS TRUE
        IFF P IS TRUE AND Q IS TRUE
}

BOOLEAN_IDENTITIES {
    IDENTITY :=
        A ∧ 1 ≡ A

    DOMINATION :=
        A ∧ 0 ≡ 0

    IDEMPOTENCE :=
        A ∧ A ≡ A

    COMPLEMENT :=
        A ∧ ¬A ≡ 0

    COMMUTATIVITY :=
        A ∧ B ≡ B ∧ A

    ASSOCIATIVITY :=
        (A ∧ B) ∧ C ≡
        A ∧ (B ∧ C)

    DISTRIBUTIVITY_OVER_OR :=
        A ∧ (B ∨ C) ≡
        (A ∧ B) ∨ (A ∧ C)

    ABSORPTION :=
        A ∧ (A ∨ B) ≡ A
}

DEMORGAN_TRANSFORMATION {
    A ∧ B ≡
        ¬(¬A ∨ ¬B)

    AND(A,B) ≡
        NOT(
            OR(
                NOT(A),
                NOT(B)
            )
        )
}

NAND_IMPLEMENTATION {
    AND(A,B) :=
        NAND(
            NAND(A,B),
            NAND(A,B)
        )
}

NOR_IMPLEMENTATION {
    NOT(A) :=
        NOR(A,A)

    NOT(B) :=
        NOR(B,B)

    AND(A,B) :=
        NOR(
            NOR(A,A),
            NOR(B,B)
        )
}

CIRCUIT_SYMBOL_SEMANTICS {
    INPUT_PORTS := {A,B,...}
    OUTPUT_PORT := Y
    LOGICAL_EQUATION :=
        Y = A ∧ B
        FOR binary_gate
}

CASCADE_AND_GATES {
    GIVEN inputs x1,...,xn

    y1 := AND(x1,x2)
    y2 := AND(y1,x3)
    ...
    y_final := AND(y_previous,xn)

    BY_ASSOCIATIVITY:
        y_final ≡ AND(x1,...,xn)
}

EXPRESSION_TRANSLATION {
    BooleanExpression:
        A ∧ B

    MAP_TO:
        ANDGate(
            input1=A,
            input2=B,
            output=Y
        )
}

REVERSE_TRANSLATION {
    ANDGate(A,B)->Y

    MAP_TO BooleanExpression:
        Y := A ∧ B
}

COMPOSITION_EXAMPLE {
    EXPRESSION :=
        (A ∧ B) ∧ C

    NETWORK :=
        G1 := ANDGate(A,B)
        G2 := ANDGate(output(G1),C)

    OUTPUT :=
        A ∧ B ∧ C
}

TRUTH_TABLE_GENERATION(Arity n) {
    REQUIRE n >= 2

    FOR_EACH X IN {0,1}^n {
        Y := EVALUATE_AND(X)
        EMIT ROW(X,Y)
    }

    PROPERTY :=
        exactly_one_input_vector_has_output_1
        NAMELY (1,1,...,1)
}

LOGICAL_PROPERTIES {
    MONOTONE := TRUE
    ZERO_PRESERVING := TRUE
    ONE_PRESERVING := TRUE
    COMMUTATIVE := TRUE
    ASSOCIATIVE := TRUE
    IDEMPOTENT := TRUE
    SELF_DUAL := FALSE
}

MONOTONICITY {
    FOR BooleanVectors X,Y:

    IF X <= Y componentwise
    THEN
        AND(X) <= AND(Y)
}

SENSITIVITY {
    AT INPUT_VECTOR (1,1,...,1):
        changing_any_single_input_to_0
        changes_output_from_1_to_0

    IF at_least_one_input_already_0:
        changing_other_inputs_may_not_change_output
}

CONTROLLING_VALUE {
    CONTROLLING_INPUT_VALUE := 0
    REASON :=
        any FALSE input forces output FALSE
}

IDENTITY_VALUE {
    IDENTITY_INPUT_VALUE := 1
    REASON :=
        A ∧ 1 = A
}

DUAL_GATE {
    ANDGate --dual_of--> ORGate

    DUALITY_TRANSFORMATION {
        AND <-> OR
        0 <-> 1
    }
}

AND_VS_NAND {
    NAND(A,B) :=
        NOT(AND(A,B))

    RELATIONSHIP ANDGate --negated_output_yields--> NANDGate
}

AND_VS_OR {
    AND :=
        TRUE only_if_all_inputs_true

    OR :=
        TRUE if_at_least_one_input_true
}

NORMAL_FORM_ROLE {
    DNF :=
        ANDGate combines literals inside product/minterm terms

    CNF :=
        ANDGate combines clauses at outer conjunction level

    DIGITAL_IMPLEMENTATION :=
        AND_OR_networks may implement sum-of-products expressions
}

MINTERM_RELATIONSHIP {
    MINTERM :=
        conjunction containing each variable exactly once,
        either complemented or uncomplemented

    ANDGate --can_implement--> Minterm
}

COMBINATIONAL_APPLICATIONS {
    ANDGate --used_in--> EnableLogic
    ANDGate --used_in--> MaskingLogic
    ANDGate --used_in--> AddressDecoding
    ANDGate --used_in--> ArithmeticCircuits
    ANDGate --used_in--> MultiplexingStructures
    ANDGate --used_in--> ComparatorLogic
    ANDGate --used_in--> ControlLogic
    ANDGate --used_in--> BooleanConditionEvaluation
}

ENTITY EnableLogic {
    INPUTS := {Data,Enable}
    OUTPUT := Data ∧ Enable

    IF Enable=0 THEN OUTPUT=0
    IF Enable=1 THEN OUTPUT=Data
}

ENTITY HalfAdderCarry {
    INPUTS := {A,B}
    CARRY := A ∧ B
}

RELATIONSHIP ANDGate --implements--> HalfAdderCarryCondition

MULTI_INPUT_DECOMPOSITION {
    AND(x1,x2,...,xn) ≡
        AND(
            AND(
                ...AND(x1,x2),
                x3
            ),
            ...,
            xn
        )

    VALID_BY := Associativity
}

EQUIVALENCE_TEST(Expression E, ANDNetwork N) {
    VARIABLES := union(inputs(E),inputs(N))

    FOR_ALL assignments A {
        yE := EVALUATE(E,A)
        yN := EVALUATE(N,A)

        IF yE != yN
            RETURN FALSE
    }

    RETURN TRUE
}

SIMPLIFICATION_RULES {
    AND(A,1) -> A
    AND(A,0) -> 0
    AND(A,A) -> A
    AND(A,NOT(A)) -> 0
    AND(A,OR(A,B)) -> A
}

ONTOLOGY_LINKS {
    PARENT := LogicGateOntology

    ANCESTORS := {
        BooleanLogicOntology,
        MathematicalLogicOntology,
        MathematicalOntology
    }

    RELATED := {
        ORGateOntology,
        NOTGateOntology,
        NANDGateOntology,
        NORGateOntology,
        XORGateOntology,
        BooleanFunctionOntology,
        BooleanExpressionOntology,
        BooleanAlgebraOntology,
        TruthTableOntology,
        PropositionalLogicOntology
    }
}

SEMANTIC_EQUIVALENCE {
    ANDGate ≈ ConjunctionGate
    ANDOperator ≈ LogicalConjunction
    ANDOperator ≈ BooleanMultiplication
        WHEN operands_are_boolean
    A AND B ≈ A ∧ B
    TRUE ≈ 1
    FALSE ≈ 0
}

CONSTRAINTS {
    C1 :=
        every abstract ANDGate input MUST belong_to {0,1}

    C2 :=
        ANDGate output MUST belong_to {0,1}

    C3 :=
        output MUST equal 1 exactly when all inputs equal 1

    C4 :=
        binary ANDGate MUST match the canonical four-row truth table

    C5 :=
        decomposition into multiple AND gates MUST preserve conjunction semantics

    C6 :=
        transformations into NAND/NOR implementations MUST preserve the complete truth table

    C7 :=
        abstract logical semantics MUST remain distinct from implementation-specific electronic behaviour
}

REASONING_CHAIN Query Q {
    IDENTIFY ANDGate concept
    IDENTIFY requested representation
    DETERMINE arity
    RESOLVE Boolean aliases
    RESOLVE input_values

    IF evaluation_requested
        THEN EVALUATE_AND

    IF truth_table_requested
        THEN TRUTH_TABLE_GENERATION

    IF expression_requested
        THEN RETURN conjunction_expression

    IF BooleanExpression_provided
        THEN MAP conjunction_nodes_to_ANDGates

    IF NAND_only_requested
        THEN APPLY NAND_IMPLEMENTATION

    IF NOR_only_requested
        THEN APPLY NOR_IMPLEMENTATION

    IF simplification_requested
        THEN APPLY SIMPLIFICATION_RULES

    IF equivalence_requested
        THEN EQUIVALENCE_TEST

    VALIDATE canonical_AND_semantics
    RETURN result WITH derivation
}

CONFIDENCE {
    CANONICAL_AND_DEFINITION := 1.00
    BINARY_TRUTH_TABLE := 1.00
    NARY_BOOLEAN_SEMANTICS := 1.00
    BOOLEAN_IDENTITY_DERIVATION := 0.99
    NAND_NOR_TRANSFORMATION :=
        IF truth_table_verified THEN 1.00
        ELSE 0.99
    PHYSICAL_IMPLEMENTATION_CLAIM :=
        REQUIRE implementation_specific_source
}

UPDATE_MODEL {
    IMMUTABLE_CORE := {
        BooleanDomain,
        ConjunctionTruthFunction,
        CanonicalBinaryTruthTable,
        IdentityAndDominationRules
    }

    EXTENSIBLE := {
        CircuitApplications,
        AlternativeImplementations,
        OptimisationMethods,
        HardwareRealisation,
        RelatedOntologies
    }

    ON_NEW_KNOWLEDGE {
        VALIDATE against ANDTruthFunction
        VALIDATE arity
        VALIDATE BooleanDomain
        CHECK semantic_consistency
        CHECK truth_table_equivalence
        ADD relationship IF non_contradictory
        UPDATE confidence
    }
}

AI_CONSUMPTION {
    MACHINE_READABLE := TRUE
    HUMAN_INTERPRETABLE := TRUE
    RECURSIVE_REASONING := TRUE
    ENTITY_RESOLUTION := TRUE
    SYMBOL_RESOLUTION := TRUE
    RELATIONSHIP_TRAVERSAL := TRUE
    BOOLEAN_EVALUATION := TRUE
    TRUTH_TABLE_REASONING := TRUE
    CIRCUIT_TRANSLATION := TRUE
    EQUIVALENCE_VALIDATION := TRUE
}

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

OUTPUT {
    PRIMARY_ENTITY := ANDGate

    INTERPRETATION :=
        "An AND gate implements Boolean conjunction: its output is true exactly when every input is true. Its truth-functional semantics connect Boolean algebra, propositional conjunction, Boolean expressions and compositional digital logic."
}

END_FLOW