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Password policy: length_ok AND (has_digit OR has_symbol) AND NOT common

A password policy, length_ok AND (has_digit OR has_symbol) AND NOT common: sixteen rows with three accepting ones, a two-term DNF, two Karnaugh groups, and a NOR form of 8 gates against 23 NAND.

A password is accepted when it is long enough, contains a digit or a symbol, and is not on the list of common passwords. The engine's reading lists the three requirements: length_ok true, at least one of has_digit or has_symbol, common false. On the sixteen-row table the output is true on three rows, Σm(10, 12, 14): long with a symbol, long with a digit, or long with both, none of them common.

The checks: not a tautology, counterexample all four inputs FALSE; satisfiable, witness length_ok = TRUE, has_digit = FALSE, has_symbol = TRUE, common = FALSE. The simplifier found no law that shortens the four-operator rule; the CNF converter returns it unchanged as three clauses with four literals. The DNF converter applies distribution twice and gives (length_ok AND has_digit AND NOT common) OR (length_ok AND has_symbol AND NOT common), two terms and six literals, the same two groups the Karnaugh map draws on cells 12 and 14 and on cells 10 and 14, two prime implicants. Gate counts: 23 NAND, 8 NOR.

Password validators are a classic source of inconsistent behaviour between client and server, because the rule is reimplemented in two languages. The thing to watch is the OR in the middle: a digit or a symbol, not both, and the table's three true rows include the row with only a symbol and the row with only a digit. A validator that requires both rejects two of the three accepting rows. The table, or the CSV export of it, is a language-neutral test fixture for both implementations.

accepted := length_ok AND (has_digit OR has_symbol) AND NOT common · output

Exact checks

Always true (tautology)? No Counterexample: length_ok = F, has_digit = F, has_symbol = F, common = F · Exhaustively checked and solver cross-checked · 1 rows
Always false (contradiction)? No Counterexample: length_ok = T, has_digit = F, has_symbol = T, common = F · Exhaustively checked and solver cross-checked · 11 rows
Can be true? Yes Example assignment: length_ok = T, has_digit = F, has_symbol = T, common = F · Exhaustively checked and solver cross-checked · 11 rows

Truth table (16 rows)

Password policy: length_ok AND (has_digit OR has_symbol) AND NOT common
#length_okhas_digithas_symbolcommonaccepted
0FALSEFALSEFALSEFALSEFALSE
1FALSEFALSEFALSETRUEFALSE
2FALSEFALSETRUEFALSEFALSE
3FALSEFALSETRUETRUEFALSE
4FALSETRUEFALSEFALSEFALSE
5FALSETRUEFALSETRUEFALSE
6FALSETRUETRUEFALSEFALSE
7FALSETRUETRUETRUEFALSE
8TRUEFALSEFALSEFALSEFALSE
9TRUEFALSEFALSETRUEFALSE
10TRUEFALSETRUEFALSETRUE
11TRUEFALSETRUETRUEFALSE
12TRUETRUEFALSEFALSETRUE
13TRUETRUEFALSETRUEFALSE
14TRUETRUETRUEFALSETRUE
15TRUETRUETRUETRUEFALSE

Plain-language reading

Beginneraccepted is true exactly when all of the following hold:
1. length_ok is true
2. at least one of has_digit or has_symbol is true (both may be true)
3. common is false
In every other case, accepted 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.
Intermediateaccepted is true exactly when all of the following hold:
1. length_ok is true
2. at least one of has_digit or has_symbol is true (both may be true)
3. common is false
In every other case, accepted is false.
Advancedaccepted is true exactly when all of the following hold [conjunction ∧]:
1. length_ok is true
2. at least one of has_digit or has_symbol is true (both may be true) [inclusive disjunction ∨]
3. common is false [negation ¬]
In every other case, accepted is false.
Formal notation: length_ok ∧ (has_digit ∨ has_symbol) ∧ ¬common

Logic diagram

Password policy: length_ok AND (has_digit OR has_symbol) AND NOT common Password policy: length_ok AND (has_digit OR has_symbol) AND NOT common length_ok has_digit has_symbol common accepted AND OR NOT
Password policy: length_ok AND (has_digit OR has_symbol) AND NOT common Password policy: length_ok AND (has_digit OR has_symbol) AND NOT common length_ok has_digit has_symbol common accepted 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.

length_ok AND (has_digit OR has_symbol) AND NOT common

Karnaugh map

length_okhas_digit \ has_symbolcommon
00
01
11
10
00
0m0
0m1
0m3
0m2
01
0m4
0m5
0m7
0m6
11
1m12
0m13
0m15
1m14
10
0m8
0m9
0m11
1m10
  1. cells m12, m14length_ok AND has_digit AND NOT common
  2. cells m10, m14length_ok AND has_symbol AND NOT common
(length_ok AND has_digit AND NOT common) OR (length_ok AND has_symbol AND NOT common) · Derived (length_ok AND has_digit AND NOT common) OR (length_ok AND has_symbol AND NOT common). Exactly equivalent on every assignment.

Minterms and maxterms

accepted(length_ok, has_digit, has_symbol, common) = Σm(10, 12, 14)accepted(length_ok, has_digit, has_symbol, common) = ΠM(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15)

Variable order: length_ok, has_digit, has_symbol, common · 16 rows

Normal forms

Conjunctive normal form (CNF)

length_ok AND (has_digit OR has_symbol) AND NOT common

3 clauses, 4 literals · complete

Disjunctive normal form (DNF)

(length_ok AND has_digit AND NOT common) OR (length_ok AND has_symbol AND NOT common)

2 terms, 6 literals · complete · Distribution ×2

NAND-only and NOR-only

NAND-only form

NAND(NAND(NAND(NAND(length_ok, NAND(NAND(has_digit, has_digit), NAND(has_symbol, has_symbol))), NAND(length_ok, NAND(NAND(has_digit, has_digit), NAND(has_symbol, has_symbol)))), NAND(common, common)), NAND(NAND(NAND(length_ok, NAND(NAND(has_digit, has_digit), NAND(has_symbol, has_symbol))), NAND(length_ok, NAND(NAND(has_digit, has_digit), NAND(has_symbol, has_symbol)))), NAND(common, common)))

23 gates · verified equivalent on every row

NOR-only form

NOR(NOR(NOR(NOR(length_ok, length_ok), NOR(has_digit, has_symbol)), NOR(NOR(length_ok, length_ok), NOR(has_digit, has_symbol))), common)

8 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: 4e41d480db12655d9c91963d520ecd09c530586ef901c5da7d17c0b9d23a937c
Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78

Inputs (dict keys, declared order) and generated parameter names:
    'length_ok' -> input_length_ok
    'has_digit' -> input_has_digit
    'has_symbol' -> input_has_symbol
    'common' -> input_common

Outputs (declared order) and their functions:
    'accepted' -> output_accepted(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({'length_ok': False, 'has_digit': False, 'has_symbol': False, 'common': False})
    # -> {'accepted': False}
    model.output_accepted({'length_ok': False, 'has_digit': False, 'has_symbol': False, 'common': False})
    # -> False
    model.evaluate({'length_ok': 1, 'has_digit': False, 'has_symbol': False, 'common': 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_accepted",
]

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": "4e41d480db12655d9c91963d520ecd09c530586ef901c5da7d17c0b9d23a937c",
    "assumptions_digest": "d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78",
    "primary_output": "accepted",
    "unused_inputs": (),
    "subcircuits": (),
}

# Stable ID -> generated identifier (reversible mapping; names are exact, escaped strings).
IDENTIFIER_MAP = {
    "v_8e59f235b584": {"kind": "input", "name": "length_ok", "identifier": "input_length_ok"},
    "v_6555d4fcf55e": {"kind": "input", "name": "has_digit", "identifier": "input_has_digit"},
    "v_d6dca9d2de39": {"kind": "input", "name": "has_symbol", "identifier": "input_has_symbol"},
    "v_4a0105a57b99": {"kind": "input", "name": "common", "identifier": "input_common"},
    "f_c8d95c9b38ed": {"kind": "output", "name": "accepted", "identifier": "output_accepted"},
}

INPUT_NAMES = ("length_ok", "has_digit", "has_symbol", "common")
OUTPUT_NAMES = ("accepted",)
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_accepted(input_length_ok: bool, input_has_digit: bool, input_has_symbol: bool, input_common: bool) -> bool:
    """Decision output 'accepted': length_ok AND (has_digit OR has_symbol) AND NOT common"""
    return input_length_ok and (input_has_digit or input_has_symbol) and (not input_common)


# ---- Public API ----


def output_accepted(inputs: Mapping[str, bool], *, check_scope: bool = True) -> bool:
    """Decision output 'accepted' (stable ID f_c8d95c9b38ed).

    'accepted' := length_ok AND (has_digit OR has_symbol) AND NOT common

    Required inputs: 'length_ok', 'has_digit', 'has_symbol', 'common'.
    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, ("length_ok", "has_digit", "has_symbol", "common"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_accepted_value = _compute_output_accepted(values["length_ok"], values["has_digit"], values["has_symbol"], values["common"])
    return output_accepted_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: 'length_ok', 'has_digit', 'has_symbol', 'common'.
    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, ("length_ok", "has_digit", "has_symbol", "common"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_accepted_value = _compute_output_accepted(values["length_ok"], values["has_digit"], values["has_symbol"], values["common"])
    return {
        "accepted": output_accepted_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: 4e41d480db12655d9c91963d520ecd09c530586ef901c5da7d17c0b9d23a937c
-- 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_8e59f235b584: length_ok -> "in_length_ok"
--   v_6555d4fcf55e: has_digit -> "in_has_digit"
--   v_d6dca9d2de39: has_symbol -> "in_has_symbol"
--   v_4a0105a57b99: common -> "in_common"
-- Output columns:
--   f_c8d95c9b38ed: accepted -> "out_accepted"
-- 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_accepted] output accepted (stable ID f_c8d95c9b38ed)
--   accepted := length_ok AND (has_digit OR has_symbol) AND NOT common
--   PRECONDITION: "in_length_ok", "in_has_digit", "in_has_symbol", "in_common" are non-NULL booleans.
--   ("in_length_ok" AND ("in_has_digit" OR "in_has_symbol") AND (NOT "in_common"))
--

-- ----------------------------------------------------------------------------------------------
-- 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_length_ok", "in_has_digit", "in_has_symbol", "in_common") 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_length_ok" IS NULL OR "in_has_digit" IS NULL OR "in_has_symbol" IS NULL OR "in_common" IS NULL THEN 'invalid_input'
      ELSE 'ok'
    END AS "status"
  FROM inputs
)
SELECT
  "row_id",
  "in_length_ok",
  "in_has_digit",
  "in_has_symbol",
  "in_common",
  "status",
  CASE WHEN "status" = 'ok' THEN ("in_length_ok" AND ("in_has_digit" OR "in_has_symbol") AND (NOT "in_common")) END AS "out_accepted"
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: 4e41d480db12655d9c91963d520ecd09c530586ef901c5da7d17c0b9d23a937c
; 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_8e59f235b584: length_ok -> in_length_ok
;   v_6555d4fcf55e: has_digit -> in_has_digit
;   v_d6dca9d2de39: has_symbol -> in_has_symbol
;   v_4a0105a57b99: common -> in_common
;   f_c8d95c9b38ed: accepted -> out_accepted

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

; ---- Inputs (declared order)
(declare-const in_length_ok Bool) ; length_ok
(declare-const in_has_digit Bool) ; has_digit
(declare-const in_has_symbol Bool) ; has_symbol
(declare-const in_common Bool) ; common

; ---- Outputs (dependencies first)
; output accepted (stable ID f_c8d95c9b38ed): length_ok AND (has_digit OR has_symbol) AND NOT common
(define-fun out_accepted () Bool (and in_length_ok (or in_has_digit in_has_symbol) (not in_common)))

; ---- 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
accepted = \mathit{length\_ok} \cdot \left(\mathit{has\_digit} + \mathit{has\_symbol}\right) \cdot \overline{\mathit{common}}

The common slip

Requiring both a digit and a symbol, has_digit AND has_symbol, when the policy says either. The intended rule accepts three rows; the stricter version accepts one, length_ok = TRUE, has_digit = TRUE, has_symbol = TRUE, common = FALSE. Users with a long password and a digit are rejected for no stated reason.

Questions

Why do the three minterms end in an even number?

common is the least significant bit and must be FALSE for acceptance, so every accepting row is even. length_ok, the most significant bit, contributes 8, and the digit or symbol bits add 4 or 2 or both, giving 12, 10 and 14.

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.