Luz de escalera con tres interruptores: A XOR B XOR C
Una luz de escalera conectada a tres interruptores, A XOR B XOR C: accionar cualquier interruptor conmuta la luz. La tabla de ocho filas, las formas normales de cuatro términos y un mapa con cuatro unos aislados muestran que es la función de paridad.
Tres interruptores gobiernan una lámpara, y la regla es que cualquier accionamiento individual cambia el estado. La fórmula que lo hace es el XOR encadenado: la luz está encendida cuando un número impar de interruptores está arriba. La lectura del motor dice que light es verdadero exactamente cuando un número impar de A, B y C son verdaderos, uno o tres de ellos. En la tabla de ocho filas la salida es verdadera en las filas 1, 2, 4 y 7, Σm(1, 2, 4, 7), y cada una de esas filas difiere de cada fila falsa exactamente en un interruptor, que es lo que convierte cada accionamiento en una conmutación.
Las comprobaciones: no es tautología, contraejemplo A = FALSO, B = FALSO, C = FALSO; satisfacible, testigo A = FALSO, B = FALSO, C = VERDADERO. El simplificador no encontró ninguna ley que acorte la expresión de dos operadores. El conversor, con dos eliminaciones de XOR y dos distributividades, produce una FND de cuatro términos, (A AND B AND C) OR (A AND NOT B AND NOT C) OR (NOT A AND B AND NOT C) OR (NOT A AND NOT B AND C), y una FNC de cuatro cláusulas, doce literales cada una. El mapa de Karnaugh dibuja cuatro grupos de una celda, las celdas 1, 2, 4 y 7, cuatro implicantes primos, y deriva los mismos cuatro términos. Compuertas: 20 NAND, 41 NOR.
Los electricistas lo montan con dos conmutadores y un conmutador de cruce en medio, y el cableado es una cadena XOR física. La misma función mueve cualquier funcionalidad de «conmutar desde cualquier sitio» en software: un ajuste que varios controles pueden cambiar, donde cada control invierte el estado actual en vez de fijarlo. Lo que hay que vigilar es la posición absoluta: con XOR ninguna posición de interruptor significa encendido o apagado por sí sola, solo la paridad lo hace, cosa que sorprende a quien espera que «arriba» sea «encendido». La página de paridad es esta misma función con su nombre de hardware.
light := A XOR B XOR C · salida
Comprobaciones exactas
Tabla de verdad (8 filas)
| # | A | B | C | light |
|---|---|---|---|---|
| 0 | FALSO | FALSO | FALSO | FALSO |
| 1 | FALSO | FALSO | VERDADERO | VERDADERO |
| 2 | FALSO | VERDADERO | FALSO | VERDADERO |
| 3 | FALSO | VERDADERO | VERDADERO | FALSO |
| 4 | VERDADERO | FALSO | FALSO | VERDADERO |
| 5 | VERDADERO | FALSO | VERDADERO | FALSO |
| 6 | VERDADERO | VERDADERO | FALSO | FALSO |
| 7 | VERDADERO | VERDADERO | VERDADERO | VERDADERO |
Lectura en lenguaje claro
Principiantelight es verdadero exactamente cuando un número impar entre A, B y C son verdaderos (1 o 3 de ellos).
En todos los demás casos, light es falso.
Sobre los operadores:
- Una cadena de XOR es verdadera cuando un número impar de sus partes son verdaderas. Con tres partes, el caso en que las tres son verdaderas también cuenta: no significa «exactamente una».
Intermediolight es verdadero exactamente cuando un número impar entre A, B y C son verdaderos (1 o 3 de ellos).
En todos los demás casos, light es falso.
Avanzadolight es verdadero exactamente cuando un número impar entre A, B y C son verdaderos (1 o 3 de ellos) [paridad: disyunción exclusiva n-aria ⊕].
En todos los demás casos, light es falso.
Notación formal: A ⊕ B ⊕ C
Diagrama lógico
Simplificación paso a paso
El motor no encontró ninguna ley que acorte esta expresión; ya es tan corta como sus leyes de reescritura permiten.
A XOR B XOR C
Mapa de Karnaugh
- celdas m1
NOT A AND NOT B AND C - celdas m2
NOT A AND B AND NOT C - celdas m4
A AND NOT B AND NOT C - celdas m7
A AND B AND C
Mintérminos y maxtérminos
light(A, B, C) = Σm(1, 2, 4, 7)light(A, B, C) = ΠM(0, 3, 5, 6)Orden de variables: A, B, C · 8 filas
Formas normales
Forma normal conjuntiva (FNC)
(A OR B OR C) AND (A OR NOT B OR NOT C) AND (NOT A OR B OR NOT C) AND (NOT A OR NOT B OR C)4 cláusulas, 12 literales · complete · Eliminación de XOR ×2, Distributividad ×2
Forma normal disyuntiva (FND)
(A AND B AND C) OR (A AND NOT B AND NOT C) OR (NOT A AND B AND NOT C) OR (NOT A AND NOT B AND C)4 términos, 12 literales · complete · Eliminación de XOR ×2, Distributividad ×2
Solo NAND, solo NOR
Forma solo con NAND
NAND(NAND(NAND(NAND(A, NAND(B, B)), NAND(NAND(A, A), B)), NAND(C, C)), NAND(NAND(NAND(NAND(A, NAND(B, B)), NAND(NAND(A, A), B)), NAND(NAND(A, NAND(B, B)), NAND(NAND(A, A), B))), C))20 compuertas · equivalencia verificada en todas las filas
Forma solo con NOR
NOR(NOR(NOR(NOR(NOR(NOR(A, A), B), NOR(A, NOR(B, B))), C), NOR(NOR(NOR(NOR(NOR(A, A), B), NOR(A, NOR(B, B))), NOR(NOR(NOR(A, A), B), NOR(A, NOR(B, B)))), NOR(C, C))), NOR(NOR(NOR(NOR(NOR(A, A), B), NOR(A, NOR(B, B))), C), NOR(NOR(NOR(NOR(NOR(A, A), B), NOR(A, NOR(B, B))), NOR(NOR(NOR(A, A), B), NOR(A, NOR(B, B)))), NOR(C, C))))41 compuertas · equivalencia verificada en todas las filas
La misma regla como código
Python
# -*- coding: utf-8 -*-
# Generated by Tangibool (tangibool.python/1). Do not edit by hand; regenerate from the source revision.
# Model: not stated
# Source revision: reference
# Interpretation status: not stated
"""Tangibool evaluator (decision model).
Generated deterministically from the validated Boolean model; it evaluates exactly that
model. It does not establish that the model is a complete, legal, or safe statement of a
real-world rule, and evaluating examples does not verify the rule.
Source revision: reference
Interpretation status: not stated
Exporter: 1.0.0 (format tangibool.python/1; engine 1.0.0; semantics tangibool-semantics/1)
Logic digest: 53ee60e932ba3e779f694db0406de7228505c1c8e2c1466fe3788480ca81e0a4
Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78
Inputs (dict keys, declared order) and generated parameter names:
'A' -> input_A
'B' -> input_B
'C' -> input_C
Outputs (declared order) and their functions:
'light' -> output_light(inputs)
Assumptions (the declared scope, checked by in_scope(inputs)):
none; every complete assignment is in scope.
Input contract:
Every required input must be exactly True or False (type(value) is bool). Missing
inputs, None, strings, integers (including 0 and 1), and keys that are not declared
inputs raise InputError before anything is evaluated, including inputs that a
short-circuiting operator would never read. Results are always bool.
Scope:
Decision functions raise OutOfScopeError when the inputs violate an assumption; an
out-of-scope assignment is never turned into a False result. Pass check_scope=False
to evaluate without the scope check, and use in_scope(inputs) to test it separately.
Usage:
import tangibool_model as model
model.evaluate({'A': False, 'B': False, 'C': False})
# -> {'light': False}
model.output_light({'A': False, 'B': False, 'C': False})
# -> False
model.evaluate({'A': 1, 'B': False, 'C': False})
# raises InputError: 1 is an int, not a bool
"""
import collections.abc as _abc
from typing import Dict, Mapping, Tuple
__all__ = [
"MODEL_INFO",
"IDENTIFIER_MAP",
"INPUT_NAMES",
"OUTPUT_NAMES",
"ASSUMPTIONS",
"InputError",
"OutOfScopeError",
"minterm_index",
"in_scope",
"evaluate",
"output_light",
]
MODEL_INFO = {
"format": "tangibool.python/1",
"mode": "decision",
"model": None,
"revision": "reference",
"interpretation": None,
"exporter": "1.0.0",
"engine_version": "1.0.0",
"semantics_version": "tangibool-semantics/1",
"logic_digest": "53ee60e932ba3e779f694db0406de7228505c1c8e2c1466fe3788480ca81e0a4",
"assumptions_digest": "d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78",
"primary_output": "light",
"unused_inputs": (),
"subcircuits": (),
}
# Stable ID -> generated identifier (reversible mapping; names are exact, escaped strings).
IDENTIFIER_MAP = {
"v_98df8f0d0a0a": {"kind": "input", "name": "A", "identifier": "input_A"},
"v_3f0bce171fa5": {"kind": "input", "name": "B", "identifier": "input_B"},
"v_886ba5b964aa": {"kind": "input", "name": "C", "identifier": "input_C"},
"f_f83bf3dae53d": {"kind": "output", "name": "light", "identifier": "output_light"},
}
INPUT_NAMES = ("A", "B", "C")
OUTPUT_NAMES = ("light",)
ASSUMPTIONS = ()
_DECLARED_INPUTS = frozenset(INPUT_NAMES)
_SCOPE_INPUTS = ()
_ASSUMPTION_FORMULAS = {entry["name"]: entry["formula"] for entry in ASSUMPTIONS}
class InputError(ValueError):
"""Raised when inputs are missing, undeclared, or not exactly True/False.
Every required input must satisfy type(value) is bool. Missing inputs, None, strings,
integers (including 0 and 1), and keys that are not declared inputs are rejected before
anything is evaluated. A missing input is never treated as False. The individual problems
are listed in the problems attribute.
"""
def __init__(self, problems):
self.problems = tuple(problems)
super().__init__("Invalid inputs: " + "; ".join(self.problems))
class OutOfScopeError(ValueError):
"""Raised when the inputs violate the model's assumptions (its declared scope).
Outside its assumptions the rule has no decision: this error is never a False result.
The names of the violated assumptions are in the violated attribute.
"""
def __init__(self, violated):
self.violated = tuple(violated)
details = ", ".join(name + " (" + _ASSUMPTION_FORMULAS.get(name, "?") + ")" for name in self.violated)
super().__init__(
"Out of scope: the inputs violate "
+ ("assumption " if len(self.violated) == 1 else "assumptions ")
+ details
+ ". The rule has no decision outside its assumptions; this is not a False result."
)
def _short_repr(value: object) -> str:
text = repr(value)
return text if len(text) <= 60 else text[:57] + "..."
def _checked_inputs(inputs: Mapping[str, bool], required: Tuple[str, ...], check_scope: bool) -> Dict[str, bool]:
"""Validate inputs strictly before anything is evaluated and return a plain copy."""
if type(check_scope) is not bool:
raise TypeError("check_scope must be True or False")
if not isinstance(inputs, _abc.Mapping):
raise InputError(["inputs must be a mapping from input names to True/False, got " + type(inputs).__name__])
problems = []
for key in inputs:
if type(key) is not str or key not in _DECLARED_INPUTS:
declared = ", ".join(repr(name) for name in INPUT_NAMES) or "none"
problems.append("undeclared input " + _short_repr(key) + " (declared inputs: " + declared + ")")
needed = tuple(required) + (_SCOPE_INPUTS if check_scope else ())
for name in needed:
if name not in inputs:
problems.append("missing required input " + repr(name))
for key, value in inputs.items():
if type(key) is str and key in _DECLARED_INPUTS and type(value) is not bool:
problems.append(
"input " + repr(key) + " must be True or False (type bool), got "
+ type(value).__name__ + " " + _short_repr(value)
)
if problems:
raise InputError(list(dict.fromkeys(problems)))
return dict(inputs.items())
def _require_bool_args(function: str, pairs: Tuple[Tuple[str, object], ...]) -> None:
problems = [
"argument " + repr(name) + " of " + function + " must be True or False (type bool), got "
+ type(value).__name__ + " " + _short_repr(value)
for name, value in pairs
if type(value) is not bool
]
if problems:
raise InputError(problems)
def minterm_index(*values: bool) -> int:
"""Row index of an assignment over a declared variable order.
The first value is the most significant bit, so all-False is row 0. Used for indexed
minterm (sum) and maxterm (product) forms.
"""
index = 0
for value in values:
if type(value) is not bool:
raise InputError(["minterm_index() takes True/False values, got " + type(value).__name__ + " " + _short_repr(value)])
index = index * 2 + (1 if value else 0)
return index
def _scope_values(values: Dict[str, bool]) -> Tuple[Tuple[str, bool], ...]:
"""Value of every enabled, confirmed assumption for validated inputs, in declared order."""
return ()
def _require_in_scope(values: Dict[str, bool]) -> None:
violated = tuple(name for name, holds in _scope_values(values) if not holds)
if violated:
raise OutOfScopeError(violated)
def in_scope(inputs: Mapping[str, bool]) -> bool:
"""True when the inputs satisfy every assumption (the model's declared scope).
Requires the inputs the assumptions read; raises InputError for invalid inputs.
"""
values = _checked_inputs(inputs, (), True)
return all(holds for _, holds in _scope_values(values))
# ---- Formula bodies (inputs are validated by the public functions below) ----
def _compute_output_light(input_A: bool, input_B: bool, input_C: bool) -> bool:
"""Decision output 'light': A XOR B XOR C"""
return (input_A != input_B) != input_C
# ---- Public API ----
def output_light(inputs: Mapping[str, bool], *, check_scope: bool = True) -> bool:
"""Decision output 'light' (stable ID f_f83bf3dae53d).
'light' := A XOR B XOR C
Required inputs: 'A', 'B', 'C'.
Raises InputError for missing, undeclared, or non-bool inputs, and OutOfScopeError
when check_scope is True and the inputs violate an assumption.
"""
values = _checked_inputs(inputs, ("A", "B", "C"), check_scope)
if check_scope:
_require_in_scope(values)
output_light_value = _compute_output_light(values["A"], values["B"], values["C"])
return output_light_value
def evaluate(inputs: Mapping[str, bool], *, check_scope: bool = True) -> Dict[str, bool]:
"""Evaluate every output (declared order); each output is computed once.
Required inputs: 'A', 'B', 'C'.
Raises InputError for missing, undeclared, or non-bool inputs, and OutOfScopeError
when check_scope is True and the inputs violate an assumption.
"""
values = _checked_inputs(inputs, ("A", "B", "C"), check_scope)
if check_scope:
_require_in_scope(values)
output_light_value = _compute_output_light(values["A"], values["B"], values["C"])
return {
"light": output_light_value,
}
PostgreSQL
-- Tangibool PostgreSQL export (tangibool.postgresql/1)
-- Target: PostgreSQL 16 or later with typed boolean inputs; tested with PostgreSQL 18.3 (PGlite 0.5.8).
-- Not a claim of compatibility with other SQL dialects.
-- Model: not stated
-- Source revision: reference
-- Interpretation status: not stated
-- Exporter: 1.0.0 (engine 1.0.0; semantics tangibool-semantics/1)
-- Logic digest: 53ee60e932ba3e779f694db0406de7228505c1c8e2c1466fe3788480ca81e0a4
-- Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78
-- Mode: decision
--
-- Read-only: this file contains only WITH / VALUES / SELECT statements. No DDL, DML,
-- procedures, credentials, network access, or database changes. The expressions are pure
-- Boolean expressions, so their result does not depend on SQL evaluation order.
--
-- NON-NULL INPUT PRECONDITION: NULL is not a truth value of this rule. Every fragment below
-- is defined only when every input column it reads is a non-NULL boolean. The guard template
-- (section 2) marks any NULL input as 'invalid_input' and any row that violates the
-- assumptions as 'out_of_scope', and fills output columns only for 'ok' rows; a NULL in an
-- output column of another row is an envelope marker, not a third truth value. Never wrap
-- the fragments in COALESCE(..., FALSE) or otherwise choose a meaning for unknown inputs.
-- This evaluates the supplied model; it does not show that the model is a complete, legal,
-- or safe statement of a real-world rule.
--
-- Input columns (stable ID: name -> column):
-- v_98df8f0d0a0a: A -> "in_A"
-- v_3f0bce171fa5: B -> "in_B"
-- v_886ba5b964aa: C -> "in_C"
-- Output columns:
-- f_f83bf3dae53d: light -> "out_light"
-- Assumptions (scope K; exported separately and never folded into an output):
-- none; every complete input row is in scope.
-- ----------------------------------------------------------------------------------------------
-- 1. Expression fragments (parenthesised; non-NULL boolean inputs required)
-- ----------------------------------------------------------------------------------------------
--
-- [out_light] output light (stable ID f_f83bf3dae53d)
-- light := A XOR B XOR C
-- PRECONDITION: "in_A", "in_B", "in_C" are non-NULL booleans.
-- (("in_A" <> "in_B") <> "in_C")
--
-- ----------------------------------------------------------------------------------------------
-- 2. Guard template (read-only SELECT over an inputs list)
-- ----------------------------------------------------------------------------------------------
-- Replace the example rows between the markers with your own rows, or replace the inputs
-- CTE with a SELECT that provides the boolean columns listed in the header. status is
-- 'invalid_input' when any input is NULL, 'out_of_scope' when the assumptions are false, and
-- 'ok' otherwise; output columns are filled only for 'ok' rows (NULL elsewhere is an
-- envelope marker). Assumption columns are shown for rows with complete inputs.
WITH inputs ("row_id", "in_A", "in_B", "in_C") AS (
-- tangibool:example-rows:begin
VALUES
(1, FALSE, FALSE, FALSE),
(2, FALSE, FALSE, TRUE),
(3, FALSE, TRUE, FALSE),
(4, FALSE, TRUE, TRUE),
(5, TRUE, FALSE, FALSE),
(6, TRUE, FALSE, TRUE),
(7, TRUE, TRUE, FALSE),
(8, TRUE, TRUE, TRUE),
(9, NULL::boolean, FALSE, FALSE)
-- tangibool:example-rows:end
),
checked AS (
SELECT
inputs.*,
CASE
WHEN "in_A" IS NULL OR "in_B" IS NULL OR "in_C" IS NULL THEN 'invalid_input'
ELSE 'ok'
END AS "status"
FROM inputs
)
SELECT
"row_id",
"in_A",
"in_B",
"in_C",
"status",
CASE WHEN "status" = 'ok' THEN (("in_A" <> "in_B") <> "in_C") END AS "out_light"
FROM checked
ORDER BY "row_id";
SMT-LIB
; Tangibool SMT-LIB export (tangibool.smtlib/1) ; Profile: SMT-LIB 2.6 Boolean subset (QF_UF): Bool constants, define-fun, let, push/pop, assert, check-sat, echo. ; Tested with Z3 5.1.0 (z3-solver 5.2.0, WebAssembly build). ; Model: not stated ; Source revision: reference ; Interpretation status: not stated ; Exporter: 1.0.0 (engine 1.0.0; semantics tangibool-semantics/1) ; Logic digest: 53ee60e932ba3e779f694db0406de7228505c1c8e2c1466fe3788480ca81e0a4 ; Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78 ; Mode: decision ; ; Reading results: each query block states what sat and unsat mean for its question. A sat ; answer comes with a witness or counterexample (ask for it with (get-model) after sat only); ; unsat settles that scoped query and is solver-checked, not an independently checked proof ; certificate; unknown settles nothing. Blocks are scoped with push/pop, so assertions never ; accumulate across questions. This checks the supplied model, not whether it is a complete, ; legal, or safe statement of a real-world rule. ; ; Symbols (stable ID: name -> symbol): ; v_98df8f0d0a0a: A -> in_A ; v_3f0bce171fa5: B -> in_B ; v_886ba5b964aa: C -> in_C ; f_f83bf3dae53d: light -> out_light (set-option :produce-models true) (set-info :smt-lib-version 2.6) (set-logic QF_UF) ; ---- Inputs (declared order) (declare-const in_A Bool) ; A (declare-const in_B Bool) ; B (declare-const in_C Bool) ; C ; ---- Outputs (dependencies first) ; output light (stable ID f_f83bf3dae53d): A XOR B XOR C (define-fun out_light () Bool (xor (xor in_A in_B) in_C)) ; ---- Assumptions (the scope K; never folded into an output) ; K: conjunction of the enabled, confirmed assumptions (true when there are none) (define-fun K () Bool true) ; ---- Definitions only: nothing is asserted and no check is run. ; Add (assert ...) and (check-sat) inside (push 1) ... (pop 1) to ask a question.
LaTeX
light = A \oplus B \oplus C
El error habitual
Cablear los interruptores como un OR, de modo que la luz esté encendida cuando cualquier interruptor esté arriba. Entonces subir un segundo interruptor no cambia nada, y la luz no se puede apagar desde lo alto de la escalera. La tabla muestra la diferencia en A = VERDADERO, B = VERDADERO, C = FALSO: XOR da apagado, OR da encendido.
Preguntas
¿Importa qué interruptor es cuál?
No. XOR es conmutativo y asociativo, así que los tres interruptores son intercambiables y la agrupación de la cadena no cambia la columna. La tabla es simétrica ante cualquier renombrado de A, B y C, y por eso las cuatro filas verdaderas son las que tienen uno o tres interruptores arriba.
Relacionado
Operadores y compuertas
Herramientas
- Generador de tablas de verdad: cualquier expresión booleana, hasta 12 variables, con columnas intermedias
- Calculadora y simplificador de álgebra de Boole paso a paso (cada ley nombrada y verificada)
- Solucionador de mapas de Karnaugh: de 2 a 4 variables, agrupaciones, SOP y POS
Véase también
- Paridad impar de tres entradas: A XOR B XOR C
- Sumador completo, suma: A XOR B XOR Cin
- Compuerta XOR: tabla de verdad de 2, 3 y 4 entradas (paridad impar)
- Lección 2: XOR: O exclusivo y paridad
- XOR de tres entradas es paridad impar, no «exactamente uno»
- Ley asociativa: (A·B)·C = A·(B·C)
Ejemplos resueltos
Generado a partir del motor de Tangibool, semántica tangibool-semantics/1, motor 1.0.0, compilación . Cada tabla, mapa, simplificación y veredicto de estas páginas lo calculó el motor de Tangibool al compilar el sitio, y cada afirmación la volvió a comprobar el mismo verificador exacto que usa la app. Los límites se indican donde aplican. © 2026 Foreranger LTD.