Machine interlock: start AND guard_closed AND NOT (estop OR fault)
A machine interlock, start AND guard_closed AND NOT (estop OR fault): of sixteen rows only one lets the machine run. De Morgan turns the negated OR into two negated literals, and the Karnaugh map is a single cell.
The machine runs only when the start command is given, the guard is closed, and neither the emergency stop nor a fault is active. The last condition is written as a negated OR, which is how a safety engineer thinks of it: any of these stops the machine. The engine's reading lists the conditions, with the third expanded as “the following does not hold: at least one of estop or fault”. On the sixteen-row table the output is true on exactly one row, Σm(12): start = TRUE, guard_closed = TRUE, estop = FALSE, fault = FALSE.
The checks: not a tautology, counterexample all four inputs FALSE; satisfiable, witness start = TRUE, guard_closed = TRUE, estop = FALSE, fault = FALSE, the one running row. The simplifier found no law that shortens the four-operator rule. The normal-form converter applies De Morgan once and both forms come out as start AND guard_closed AND NOT estop AND NOT fault: as a CNF it is four one-literal clauses, as a DNF one four-literal term. The Karnaugh map draws one group on cell 12 and reports one prime implicant, deriving the same four-literal term. Gate counts: 23 NAND, 11 NOR.
Interlocks are the simplest safety logic and the most consequential: every input is a veto, and the function is true on a single row. That makes testing unusual. One positive test covers the only running state, and the important tests are the fifteen stopping states, in particular the four that differ from the running row in exactly one input, where a single stuck sensor would matter. The thing to watch is that NOT (estop OR fault) and NOT estop AND NOT fault are the same by De Morgan, while NOT estop OR NOT fault is not, and would let the machine run with one of the two active.
run := start AND guard_closed AND NOT (estop OR fault) · output
Exact checks
Truth table (16 rows)
| # | start | guard_closed | estop | fault | run |
|---|---|---|---|---|---|
| 0 | FALSE | FALSE | FALSE | FALSE | FALSE |
| 1 | FALSE | FALSE | FALSE | TRUE | FALSE |
| 2 | FALSE | FALSE | TRUE | FALSE | FALSE |
| 3 | FALSE | FALSE | TRUE | TRUE | FALSE |
| 4 | FALSE | TRUE | FALSE | FALSE | FALSE |
| 5 | FALSE | TRUE | FALSE | TRUE | FALSE |
| 6 | FALSE | TRUE | TRUE | FALSE | FALSE |
| 7 | FALSE | TRUE | TRUE | TRUE | FALSE |
| 8 | TRUE | FALSE | FALSE | FALSE | FALSE |
| 9 | TRUE | FALSE | FALSE | TRUE | FALSE |
| 10 | TRUE | FALSE | TRUE | FALSE | FALSE |
| 11 | TRUE | FALSE | TRUE | TRUE | FALSE |
| 12 | TRUE | TRUE | FALSE | FALSE | TRUE |
| 13 | TRUE | TRUE | FALSE | TRUE | FALSE |
| 14 | TRUE | TRUE | TRUE | FALSE | FALSE |
| 15 | TRUE | TRUE | TRUE | TRUE | FALSE |
Plain-language reading
Beginnerrun is true exactly when all of the following hold:
1. start is true
2. guard_closed is true
3. the following condition does not hold:
3.1. at least one of estop or fault is true (both may be true)
In every other case, run is false.
About the operators:
- AND is true only when all of its parts are true.
- NOT reverses a value: true becomes false and false becomes true.
- OR is inclusive: it is true when at least one part is true, including when several are.
Intermediaterun is true exactly when all of the following hold:
1. start is true
2. guard_closed is true
3. the following condition does not hold:
3.1. at least one of estop or fault is true (both may be true)
In every other case, run is false.
Advancedrun is true exactly when all of the following hold [conjunction ∧]:
1. start is true
2. guard_closed is true
3. the following condition does not hold [negation ¬]:
3.1. at least one of estop or fault is true (both may be true) [inclusive disjunction ∨]
In every other case, run is false.
Formal notation: start ∧ guard_closed ∧ ¬(estop ∨ fault)
Logic diagram
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.
start AND guard_closed AND NOT (estop OR fault)
Karnaugh map
- cells m12
start AND guard_closed AND NOT estop AND NOT fault
Minterms and maxterms
run(start, guard_closed, estop, fault) = Σm(12)run(start, guard_closed, estop, fault) = ΠM(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15)Variable order: start, guard_closed, estop, fault · 16 rows
Normal forms
Conjunctive normal form (CNF)
start AND guard_closed AND NOT estop AND NOT fault4 clauses, 4 literals · complete · De Morgan's law ×1
Disjunctive normal form (DNF)
start AND guard_closed AND NOT estop AND NOT fault1 terms, 4 literals · complete · De Morgan's law ×1
NAND-only and NOR-only
NAND-only form
NAND(NAND(NAND(NAND(start, guard_closed), NAND(start, guard_closed)), NAND(NAND(NAND(estop, estop), NAND(fault, fault)), NAND(NAND(estop, estop), NAND(fault, fault)))), NAND(NAND(NAND(start, guard_closed), NAND(start, guard_closed)), NAND(NAND(NAND(estop, estop), NAND(fault, fault)), NAND(NAND(estop, estop), NAND(fault, fault)))))23 gates · verified equivalent on every row
NOR-only form
NOR(NOR(NOR(NOR(start, start), NOR(guard_closed, guard_closed)), NOR(NOR(start, start), NOR(guard_closed, guard_closed))), NOR(NOR(estop, fault), NOR(estop, fault)))11 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: 262f12bd6a66a446a7620bad3b085b99177514dde18f0a970a9cee6ca008dc84
Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78
Inputs (dict keys, declared order) and generated parameter names:
'start' -> input_start
'guard_closed' -> input_guard_closed
'estop' -> input_estop
'fault' -> input_fault
Outputs (declared order) and their functions:
'run' -> output_run(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({'start': False, 'guard_closed': False, 'estop': False, 'fault': False})
# -> {'run': False}
model.output_run({'start': False, 'guard_closed': False, 'estop': False, 'fault': False})
# -> False
model.evaluate({'start': 1, 'guard_closed': False, 'estop': False, 'fault': 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_run",
]
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": "262f12bd6a66a446a7620bad3b085b99177514dde18f0a970a9cee6ca008dc84",
"assumptions_digest": "d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78",
"primary_output": "run",
"unused_inputs": (),
"subcircuits": (),
}
# Stable ID -> generated identifier (reversible mapping; names are exact, escaped strings).
IDENTIFIER_MAP = {
"v_509f4c34e634": {"kind": "input", "name": "start", "identifier": "input_start"},
"v_9368be394246": {"kind": "input", "name": "guard_closed", "identifier": "input_guard_closed"},
"v_c17babd3b38e": {"kind": "input", "name": "estop", "identifier": "input_estop"},
"v_e7961fc5c672": {"kind": "input", "name": "fault", "identifier": "input_fault"},
"f_1aee4ab18f55": {"kind": "output", "name": "run", "identifier": "output_run"},
}
INPUT_NAMES = ("start", "guard_closed", "estop", "fault")
OUTPUT_NAMES = ("run",)
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_run(input_start: bool, input_guard_closed: bool, input_estop: bool, input_fault: bool) -> bool:
"""Decision output 'run': start AND guard_closed AND NOT (estop OR fault)"""
return input_start and input_guard_closed and (not (input_estop or input_fault))
# ---- Public API ----
def output_run(inputs: Mapping[str, bool], *, check_scope: bool = True) -> bool:
"""Decision output 'run' (stable ID f_1aee4ab18f55).
'run' := start AND guard_closed AND NOT (estop OR fault)
Required inputs: 'start', 'guard_closed', 'estop', 'fault'.
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, ("start", "guard_closed", "estop", "fault"), check_scope)
if check_scope:
_require_in_scope(values)
output_run_value = _compute_output_run(values["start"], values["guard_closed"], values["estop"], values["fault"])
return output_run_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: 'start', 'guard_closed', 'estop', 'fault'.
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, ("start", "guard_closed", "estop", "fault"), check_scope)
if check_scope:
_require_in_scope(values)
output_run_value = _compute_output_run(values["start"], values["guard_closed"], values["estop"], values["fault"])
return {
"run": output_run_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: 262f12bd6a66a446a7620bad3b085b99177514dde18f0a970a9cee6ca008dc84
-- 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_509f4c34e634: start -> "in_start"
-- v_9368be394246: guard_closed -> "in_guard_closed"
-- v_c17babd3b38e: estop -> "in_estop"
-- v_e7961fc5c672: fault -> "in_fault"
-- Output columns:
-- f_1aee4ab18f55: run -> "out_run"
-- 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_run] output run (stable ID f_1aee4ab18f55)
-- run := start AND guard_closed AND NOT (estop OR fault)
-- PRECONDITION: "in_start", "in_guard_closed", "in_estop", "in_fault" are non-NULL booleans.
-- ("in_start" AND "in_guard_closed" AND (NOT ("in_estop" OR "in_fault")))
--
-- ----------------------------------------------------------------------------------------------
-- 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_start", "in_guard_closed", "in_estop", "in_fault") AS (
-- tangibool:example-rows:begin
VALUES
(1, FALSE, FALSE, FALSE, FALSE),
(2, FALSE, FALSE, FALSE, TRUE),
(3, FALSE, FALSE, TRUE, FALSE),
(4, FALSE, FALSE, TRUE, TRUE),
(5, FALSE, TRUE, FALSE, FALSE),
(6, FALSE, TRUE, FALSE, TRUE),
(7, FALSE, TRUE, TRUE, FALSE),
(8, FALSE, TRUE, TRUE, TRUE),
(9, TRUE, FALSE, FALSE, FALSE),
(10, TRUE, FALSE, FALSE, TRUE),
(11, TRUE, FALSE, TRUE, FALSE),
(12, TRUE, FALSE, TRUE, TRUE),
(13, TRUE, TRUE, FALSE, FALSE),
(14, TRUE, TRUE, FALSE, TRUE),
(15, TRUE, TRUE, TRUE, FALSE),
(16, TRUE, TRUE, TRUE, TRUE),
(17, NULL::boolean, FALSE, FALSE, FALSE)
-- tangibool:example-rows:end
),
checked AS (
SELECT
inputs.*,
CASE
WHEN "in_start" IS NULL OR "in_guard_closed" IS NULL OR "in_estop" IS NULL OR "in_fault" IS NULL THEN 'invalid_input'
ELSE 'ok'
END AS "status"
FROM inputs
)
SELECT
"row_id",
"in_start",
"in_guard_closed",
"in_estop",
"in_fault",
"status",
CASE WHEN "status" = 'ok' THEN ("in_start" AND "in_guard_closed" AND (NOT ("in_estop" OR "in_fault"))) END AS "out_run"
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: 262f12bd6a66a446a7620bad3b085b99177514dde18f0a970a9cee6ca008dc84 ; 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_509f4c34e634: start -> in_start ; v_9368be394246: guard_closed -> in_guard_closed ; v_c17babd3b38e: estop -> in_estop ; v_e7961fc5c672: fault -> in_fault ; f_1aee4ab18f55: run -> out_run (set-option :produce-models true) (set-info :smt-lib-version 2.6) (set-logic QF_UF) ; ---- Inputs (declared order) (declare-const in_start Bool) ; start (declare-const in_guard_closed Bool) ; guard_closed (declare-const in_estop Bool) ; estop (declare-const in_fault Bool) ; fault ; ---- Outputs (dependencies first) ; output run (stable ID f_1aee4ab18f55): start AND guard_closed AND NOT (estop OR fault) (define-fun out_run () Bool (and in_start in_guard_closed (not (or in_estop in_fault)))) ; ---- 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
run = \mathit{start} \cdot \mathit{guard\_closed} \cdot \overline{\mathit{estop} + \mathit{fault}}The common slip
Distributing the NOT without flipping the OR: NOT estop OR NOT fault. That version runs the machine when only one of the two is active, adding rows to the single true row of the correct interlock. The CNF on the page, with its two separate negated literals, is the correct expansion.
Questions
Why does the CNF have four clauses when the DNF has one term?
The function is a pure conjunction of literals once De Morgan is applied. A conjunction of four literals is a CNF with four single-literal clauses and a DNF with one four-literal term; both describe the single true row, cell 12 on the map.
Related
Operators and gates
- NOT gate (inverter): truth table, symbol and the NOT operator
- AND gate: truth table for 2, 3 and 4 inputs, symbol and expression
- OR gate: truth table for 2, 3 and 4 inputs, inclusive or explained
Tools
- Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns
- Boolean algebra calculator and simplifier with steps (every law named and verified)
- Karnaugh map solver: 2 to 4 variables, groups, SOP and POS, don’t-cares from assumptions
See also
- Alarm rule: armed AND (door OR window OR motion) AND NOT disarm_code
- Traffic-light safety property: never both greens, checked as “always true”
- De Morgan's laws: NOT (A AND B) = NOT A OR NOT B, with proof and examples
- NOT (A AND B) = NOT A OR NOT B, not NOT A AND NOT B
- MC/DC independence pairs, worked example with the engine’s coverage report
- Independence pair in MC/DC: meaning and examples
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.