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Alarm rule: armed AND (door OR window OR motion) AND NOT disarm_code

An alarm rule with five inputs, armed AND (door OR window OR motion) AND NOT disarm_code: a 32-row table with seven triggering rows, a three-term DNF, 14 NOR gates against 47 NAND, and no Karnaugh map at five variables.

The alarm sounds when the system is armed, any sensor trips, and no valid disarm code has been entered. The engine's reading lists the three requirements: armed true, at least one of door, window or motion, and disarm_code false. Five inputs give a 32-row table, and the output is true on seven rows, Σm(18, 20, 22, 24, 26, 28, 30): every non-empty combination of the three sensors, with the system armed and no code entered.

The checks: not a tautology, counterexample all five inputs FALSE; satisfiable, witness armed = TRUE, door = FALSE, window = FALSE, motion = TRUE, disarm_code = FALSE, a motion trip on an armed system. The simplifier found no law that shortens the five-operator rule, and the CNF converter returns it unchanged as three clauses. The DNF converter applies distribution twice and gives three terms of three literals, (armed AND door AND NOT disarm_code) and its twins, one per sensor. The gate counts are 47 NAND and 14 NOR. The Karnaugh map is not available for five variables; the engine's map covers up to four.

Alarm logic is a safety rule, so the two negative conditions deserve the attention: the system must not sound when disarmed, and must not stay silent when armed and tripped. The 25 false rows cover the first, the seven true rows the second. The thing to watch is disarm_code, which overrides every sensor; in the formula it is ANDed at the top level, and a version that placed it inside the sensor OR would let a tripped door override the code. For a function this size the table and the exact checks, not a map, are the verification tools.

alarm := armed AND (door OR window OR motion) AND NOT disarm_code · output

Exact checks

Always true (tautology)? No Counterexample: armed = F, door = F, window = F, motion = F, disarm_code = F · Exhaustively checked and solver cross-checked · 1 rows
Always false (contradiction)? No Counterexample: armed = T, door = F, window = F, motion = T, disarm_code = F · Exhaustively checked and solver cross-checked · 19 rows
Can be true? Yes Example assignment: armed = T, door = F, window = F, motion = T, disarm_code = F · Exhaustively checked and solver cross-checked · 19 rows

Truth table (32 rows)

Alarm rule: armed AND (door OR window OR motion) AND NOT disarm_code
#armeddoorwindowmotiondisarm_codealarm
0FALSEFALSEFALSEFALSEFALSEFALSE
1FALSEFALSEFALSEFALSETRUEFALSE
2FALSEFALSEFALSETRUEFALSEFALSE
3FALSEFALSEFALSETRUETRUEFALSE
4FALSEFALSETRUEFALSEFALSEFALSE
5FALSEFALSETRUEFALSETRUEFALSE
6FALSEFALSETRUETRUEFALSEFALSE
7FALSEFALSETRUETRUETRUEFALSE
8FALSETRUEFALSEFALSEFALSEFALSE
9FALSETRUEFALSEFALSETRUEFALSE
10FALSETRUEFALSETRUEFALSEFALSE
11FALSETRUEFALSETRUETRUEFALSE
12FALSETRUETRUEFALSEFALSEFALSE
13FALSETRUETRUEFALSETRUEFALSE
14FALSETRUETRUETRUEFALSEFALSE
15FALSETRUETRUETRUETRUEFALSE
16TRUEFALSEFALSEFALSEFALSEFALSE
17TRUEFALSEFALSEFALSETRUEFALSE
18TRUEFALSEFALSETRUEFALSETRUE
19TRUEFALSEFALSETRUETRUEFALSE
20TRUEFALSETRUEFALSEFALSETRUE
21TRUEFALSETRUEFALSETRUEFALSE
22TRUEFALSETRUETRUEFALSETRUE
23TRUEFALSETRUETRUETRUEFALSE
24TRUETRUEFALSEFALSEFALSETRUE
25TRUETRUEFALSEFALSETRUEFALSE
26TRUETRUEFALSETRUEFALSETRUE
27TRUETRUEFALSETRUETRUEFALSE
28TRUETRUETRUEFALSEFALSETRUE
29TRUETRUETRUEFALSETRUEFALSE
30TRUETRUETRUETRUEFALSETRUE
31TRUETRUETRUETRUETRUEFALSE

Plain-language reading

Beginneralarm is true exactly when all of the following hold:
1. armed is true
2. at least one of door, window, or motion is true (several may be true)
3. disarm_code is false
In every other case, alarm is false.
About the operators:
- AND is true only when all of its parts are true.
- OR is inclusive: it is true when at least one part is true, including when several are.
- NOT reverses a value: true becomes false and false becomes true.
Intermediatealarm is true exactly when all of the following hold:
1. armed is true
2. at least one of door, window, or motion is true (several may be true)
3. disarm_code is false
In every other case, alarm is false.
Advancedalarm is true exactly when all of the following hold [conjunction ∧]:
1. armed is true
2. at least one of door, window, or motion is true (several may be true) [inclusive disjunction ∨]
3. disarm_code is false [negation ¬]
In every other case, alarm is false.
Formal notation: armed ∧ (door ∨ window ∨ motion) ∧ ¬disarm_code

Logic diagram

Alarm rule: armed AND (door OR window OR motion) AND NOT disarm_code Alarm rule: armed AND (door OR window OR motion) AND NOT disarm_code armed door window motion disarm_code alarm AND OR NOT
Alarm rule: armed AND (door OR window OR motion) AND NOT disarm_code Alarm rule: armed AND (door OR window OR motion) AND NOT disarm_code armed door window motion disarm_code alarm AND OR NOT

Simplification, step by step

The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.

armed AND (door OR window OR motion) AND NOT disarm_code

Karnaugh map

A Karnaugh map shows at most 4 variables; this expression has more. Use the tool to fix some inputs and view a labelled slice. alarm depends on 5 inputs that are not shown to be irrelevant; a Karnaugh map shows at most 4. Fix disarm_code to view an explicitly labelled slice.

Minterms and maxterms

alarm(armed, door, window, motion, disarm_code) = Σm(18, 20, 22, 24, 26, 28, 30)alarm(armed, door, window, motion, disarm_code) = ΠM(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31)

Variable order: armed, door, window, motion, disarm_code · 32 rows

Normal forms

Conjunctive normal form (CNF)

armed AND (door OR window OR motion) AND NOT disarm_code

3 clauses, 5 literals · complete

Disjunctive normal form (DNF)

(armed AND door AND NOT disarm_code) OR (armed AND window AND NOT disarm_code) OR (armed AND motion AND NOT disarm_code)

3 terms, 9 literals · complete · Distribution ×2

NAND-only and NOR-only

NAND-only form

NAND(NAND(NAND(NAND(armed, NAND(NAND(NAND(NAND(door, door), NAND(window, window)), NAND(NAND(door, door), NAND(window, window))), NAND(motion, motion))), NAND(armed, NAND(NAND(NAND(NAND(door, door), NAND(window, window)), NAND(NAND(door, door), NAND(window, window))), NAND(motion, motion)))), NAND(disarm_code, disarm_code)), NAND(NAND(NAND(armed, NAND(NAND(NAND(NAND(door, door), NAND(window, window)), NAND(NAND(door, door), NAND(window, window))), NAND(motion, motion))), NAND(armed, NAND(NAND(NAND(NAND(door, door), NAND(window, window)), NAND(NAND(door, door), NAND(window, window))), NAND(motion, motion)))), NAND(disarm_code, disarm_code)))

47 gates · verified equivalent on every row

NOR-only form

NOR(NOR(NOR(NOR(armed, armed), NOR(NOR(NOR(door, window), NOR(door, window)), motion)), NOR(NOR(armed, armed), NOR(NOR(NOR(door, window), NOR(door, window)), motion))), disarm_code)

14 gates · verified equivalent on every row

The same rule as code

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: 500e0e0a87d4b3c8e00645a893e9d0928876e6a9aca62ea11163e1787c2e2dc7
Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78

Inputs (dict keys, declared order) and generated parameter names:
    'armed' -> input_armed
    'door' -> input_door
    'window' -> input_window
    'motion' -> input_motion
    'disarm_code' -> input_disarm_code

Outputs (declared order) and their functions:
    'alarm' -> output_alarm(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({'armed': False, 'door': False, 'window': False, 'motion': False, 'disarm_code': False})
    # -> {'alarm': False}
    model.output_alarm({'armed': False, 'door': False, 'window': False, 'motion': False, 'disarm_code': False})
    # -> False
    model.evaluate({'armed': 1, 'door': False, 'window': False, 'motion': False, 'disarm_code': 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_alarm",
]

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": "500e0e0a87d4b3c8e00645a893e9d0928876e6a9aca62ea11163e1787c2e2dc7",
    "assumptions_digest": "d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78",
    "primary_output": "alarm",
    "unused_inputs": (),
    "subcircuits": (),
}

# Stable ID -> generated identifier (reversible mapping; names are exact, escaped strings).
IDENTIFIER_MAP = {
    "v_edcc9b841713": {"kind": "input", "name": "armed", "identifier": "input_armed"},
    "v_4be2c65e6435": {"kind": "input", "name": "door", "identifier": "input_door"},
    "v_8032c9854781": {"kind": "input", "name": "window", "identifier": "input_window"},
    "v_79fcbe2fca27": {"kind": "input", "name": "motion", "identifier": "input_motion"},
    "v_262f657322b9": {"kind": "input", "name": "disarm_code", "identifier": "input_disarm_code"},
    "f_b136ce8a03d7": {"kind": "output", "name": "alarm", "identifier": "output_alarm"},
}

INPUT_NAMES = ("armed", "door", "window", "motion", "disarm_code")
OUTPUT_NAMES = ("alarm",)
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_alarm(input_armed: bool, input_door: bool, input_window: bool, input_motion: bool, input_disarm_code: bool) -> bool:
    """Decision output 'alarm': armed AND (door OR window OR motion) AND NOT disarm_code"""
    return input_armed and (input_door or input_window or input_motion) and (not input_disarm_code)


# ---- Public API ----


def output_alarm(inputs: Mapping[str, bool], *, check_scope: bool = True) -> bool:
    """Decision output 'alarm' (stable ID f_b136ce8a03d7).

    'alarm' := armed AND (door OR window OR motion) AND NOT disarm_code

    Required inputs: 'armed', 'door', 'window', 'motion', 'disarm_code'.
    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, ("armed", "door", "window", "motion", "disarm_code"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_alarm_value = _compute_output_alarm(values["armed"], values["door"], values["window"], values["motion"], values["disarm_code"])
    return output_alarm_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: 'armed', 'door', 'window', 'motion', 'disarm_code'.
    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, ("armed", "door", "window", "motion", "disarm_code"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_alarm_value = _compute_output_alarm(values["armed"], values["door"], values["window"], values["motion"], values["disarm_code"])
    return {
        "alarm": output_alarm_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: 500e0e0a87d4b3c8e00645a893e9d0928876e6a9aca62ea11163e1787c2e2dc7
-- 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_edcc9b841713: armed -> "in_armed"
--   v_4be2c65e6435: door -> "in_door"
--   v_8032c9854781: window -> "in_window"
--   v_79fcbe2fca27: motion -> "in_motion"
--   v_262f657322b9: disarm_code -> "in_disarm_code"
-- Output columns:
--   f_b136ce8a03d7: alarm -> "out_alarm"
-- 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_alarm] output alarm (stable ID f_b136ce8a03d7)
--   alarm := armed AND (door OR window OR motion) AND NOT disarm_code
--   PRECONDITION: "in_armed", "in_door", "in_window", "in_motion", "in_disarm_code" are non-NULL booleans.
--   ("in_armed" AND ("in_door" OR "in_window" OR "in_motion") AND (NOT "in_disarm_code"))
--

-- ----------------------------------------------------------------------------------------------
-- 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_armed", "in_door", "in_window", "in_motion", "in_disarm_code") AS (
  -- tangibool:example-rows:begin
  VALUES
    (1, FALSE, FALSE, FALSE, FALSE, FALSE),
    (2, TRUE, TRUE, TRUE, TRUE, TRUE),
    (3, NULL::boolean, FALSE, FALSE, FALSE, FALSE)
  -- tangibool:example-rows:end
),
checked AS (
  SELECT
    inputs.*,
    CASE
      WHEN "in_armed" IS NULL OR "in_door" IS NULL OR "in_window" IS NULL OR "in_motion" IS NULL OR "in_disarm_code" IS NULL THEN 'invalid_input'
      ELSE 'ok'
    END AS "status"
  FROM inputs
)
SELECT
  "row_id",
  "in_armed",
  "in_door",
  "in_window",
  "in_motion",
  "in_disarm_code",
  "status",
  CASE WHEN "status" = 'ok' THEN ("in_armed" AND ("in_door" OR "in_window" OR "in_motion") AND (NOT "in_disarm_code")) END AS "out_alarm"
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: 500e0e0a87d4b3c8e00645a893e9d0928876e6a9aca62ea11163e1787c2e2dc7
; 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_edcc9b841713: armed -> in_armed
;   v_4be2c65e6435: door -> in_door
;   v_8032c9854781: window -> in_window
;   v_79fcbe2fca27: motion -> in_motion
;   v_262f657322b9: disarm_code -> in_disarm_code
;   f_b136ce8a03d7: alarm -> out_alarm

(set-option :produce-models true)
(set-info :smt-lib-version 2.6)
(set-logic QF_UF)

; ---- Inputs (declared order)
(declare-const in_armed Bool) ; armed
(declare-const in_door Bool) ; door
(declare-const in_window Bool) ; window
(declare-const in_motion Bool) ; motion
(declare-const in_disarm_code Bool) ; disarm_code

; ---- Outputs (dependencies first)
; output alarm (stable ID f_b136ce8a03d7): armed AND (door OR window OR motion) AND NOT disarm_code
(define-fun out_alarm () Bool (and in_armed (or in_door in_window in_motion) (not in_disarm_code)))

; ---- 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
alarm = \mathit{armed} \cdot \left(\mathit{door} + \mathit{window} + \mathit{motion}\right) \cdot \overline{\mathit{disarm\_code}}

The common slip

Writing the sensors as an AND, door AND window AND motion, so the alarm needs every sensor to trip at once. The intended rule is true on seven rows; the AND version on one. The minterm list on the page, with its seven entries, is the fastest way to see that any single sensor must suffice.

Questions

Why do all seven minterms have even numbers between 18 and 30?

armed is the most significant bit, contributing 16, and disarm_code is the least significant, which must be 0, so every triggering row is even and at least 16. The three sensor bits add 2, 4 or 8 in any non-empty combination, giving the seven values 18 to 30.

Related

Operators and gates

Tools

See also

Worked examples

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build . Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.