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Entry rule: (member OR invited) AND NOT banned, checked every way

One small access rule, (member OR invited) AND NOT banned, run through every tool on the site: the eight-row truth table, the three exact checks, the simplifier, both normal forms, the NAND and NOR gate counts and a Karnaugh map.

The rule has three inputs. member means the person is a member, invited means they hold an invitation, and banned means they are banned. Entry is allowed when at least one of the first two is true and the third is false. The engine's plain-language reading says the same thing: allow_entry is true exactly when at least one of member or invited holds and banned is false. With three inputs the truth table has eight rows, and the output is true on three of them: Σm(2, 4, 6) in minterm notation.

The tautology check is violated, with the counterexample member = FALSE, invited = FALSE, banned = FALSE; the contradiction check is violated too, and satisfiability holds, both with the witness member = FALSE, invited = TRUE, banned = FALSE. The simplifier found no law that shortens the expression: it stays at three operators. The CNF is the rule itself, two clauses; the DNF, reached by one distribution step, is (member AND NOT banned) OR (invited AND NOT banned), which is also what the Karnaugh map derives from its two prime implicants. The NOR-only form needs 2 gates, NOR(NOR(member, invited), banned); the NAND-only form needs 11.

This shape, an OR of ways in followed by an AND with a blocker, is the most common access rule in software: a guard clause, a feature gate, a membership check. The point to watch is the grouping. The OR must be evaluated before the NOT banned condition is applied, otherwise banned members get in, which is exactly what the companion slip page shows row by row. When the rule is exported to Python or SQL the parentheses travel with it, and the eight-row table is the test oracle for any reimplementation.

member: the person is a member; invited: the person has an invitation; banned: the person is banned;

allow_entry := (member OR invited) AND NOT banned · output

Exact checks

Always true (tautology)? No Counterexample: member = F, invited = F, banned = F · Exhaustively checked and solver cross-checked · 1 rows
Always false (contradiction)? No Counterexample: member = F, invited = T, banned = F · Exhaustively checked and solver cross-checked · 3 rows
Can be true? Yes Example assignment: member = F, invited = T, banned = F · Exhaustively checked and solver cross-checked · 3 rows

Truth table (8 rows)

Entry rule: (member OR invited) AND NOT banned, checked every way
#memberinvitedbannedallow_entry
0FALSEFALSEFALSEFALSE
1FALSEFALSETRUEFALSE
2FALSETRUEFALSETRUE
3FALSETRUETRUEFALSE
4TRUEFALSEFALSETRUE
5TRUEFALSETRUEFALSE
6TRUETRUEFALSETRUE
7TRUETRUETRUEFALSE

Plain-language reading

Beginnerallow_entry is true exactly when both of the following hold:
1. at least one of member or invited is true (both may be true)
2. banned is false
In every other case, allow_entry 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.
Intermediateallow_entry is true exactly when both of the following hold:
1. at least one of member or invited is true (both may be true)
2. banned is false
In every other case, allow_entry is false.
Advancedallow_entry is true exactly when both of the following hold [conjunction ∧]:
1. at least one of member or invited is true (both may be true) [inclusive disjunction ∨]
2. banned is false [negation ¬]
In every other case, allow_entry is false.
Formal notation: (member ∨ invited) ∧ ¬banned

Logic diagram

Entry rule: (member OR invited) AND NOT banned, checked every way Entry rule: (member OR invited) AND NOT banned, checked every way member invited banned allow_entry AND OR NOT
Entry rule: (member OR invited) AND NOT banned, checked every way Entry rule: (member OR invited) AND NOT banned, checked every way member invited banned allow_entry 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.

(member OR invited) AND NOT banned

Karnaugh map

member \ invitedbanned
00
01
11
10
0
0m0
0m1
0m3
1m2
1
1m4
0m5
0m7
1m6
  1. cells m4, m6member AND NOT banned
  2. cells m2, m6invited AND NOT banned
(member AND NOT banned) OR (invited AND NOT banned) · Derived (member AND NOT banned) OR (invited AND NOT banned). Exactly equivalent on every assignment.

Minterms and maxterms

allow_entry(member, invited, banned) = Σm(2, 4, 6)allow_entry(member, invited, banned) = ΠM(0, 1, 3, 5, 7)

Variable order: member, invited, banned · 8 rows

Normal forms

Conjunctive normal form (CNF)

(member OR invited) AND NOT banned

2 clauses, 3 literals · complete

Disjunctive normal form (DNF)

(member AND NOT banned) OR (invited AND NOT banned)

2 terms, 4 literals · complete · Distribution ×1

NAND-only and NOR-only

NAND-only form

NAND(NAND(NAND(NAND(member, member), NAND(invited, invited)), NAND(banned, banned)), NAND(NAND(NAND(member, member), NAND(invited, invited)), NAND(banned, banned)))

11 gates · verified equivalent on every row

NOR-only form

NOR(NOR(member, invited), banned)

2 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: f0daa82174670c788f59eb058d07f7601cd0ace4ed94d61a747cc26c41fd4e0d
Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78

Inputs (dict keys, declared order) and generated parameter names:
    'member' -> input_member
    'invited' -> input_invited
    'banned' -> input_banned

Outputs (declared order) and their functions:
    'allow_entry' -> output_allow_entry(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({'member': False, 'invited': False, 'banned': False})
    # -> {'allow_entry': False}
    model.output_allow_entry({'member': False, 'invited': False, 'banned': False})
    # -> False
    model.evaluate({'member': 1, 'invited': False, 'banned': 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_allow_entry",
]

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": "f0daa82174670c788f59eb058d07f7601cd0ace4ed94d61a747cc26c41fd4e0d",
    "assumptions_digest": "d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78",
    "primary_output": "allow_entry",
    "unused_inputs": (),
    "subcircuits": (),
}

# Stable ID -> generated identifier (reversible mapping; names are exact, escaped strings).
IDENTIFIER_MAP = {
    "v_79df34978043": {"kind": "input", "name": "member", "identifier": "input_member"},
    "v_8afde06f51e3": {"kind": "input", "name": "invited", "identifier": "input_invited"},
    "v_60b2536f5dc8": {"kind": "input", "name": "banned", "identifier": "input_banned"},
    "f_ae8686d89791": {"kind": "output", "name": "allow_entry", "identifier": "output_allow_entry"},
}

INPUT_NAMES = ("member", "invited", "banned")
OUTPUT_NAMES = ("allow_entry",)
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_allow_entry(input_member: bool, input_invited: bool, input_banned: bool) -> bool:
    """Decision output 'allow_entry': (member OR invited) AND NOT banned"""
    return (input_member or input_invited) and (not input_banned)


# ---- Public API ----


def output_allow_entry(inputs: Mapping[str, bool], *, check_scope: bool = True) -> bool:
    """Decision output 'allow_entry' (stable ID f_ae8686d89791).

    'allow_entry' := (member OR invited) AND NOT banned

    Required inputs: 'member', 'invited', 'banned'.
    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, ("member", "invited", "banned"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_allow_entry_value = _compute_output_allow_entry(values["member"], values["invited"], values["banned"])
    return output_allow_entry_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: 'member', 'invited', 'banned'.
    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, ("member", "invited", "banned"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_allow_entry_value = _compute_output_allow_entry(values["member"], values["invited"], values["banned"])
    return {
        "allow_entry": output_allow_entry_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: f0daa82174670c788f59eb058d07f7601cd0ace4ed94d61a747cc26c41fd4e0d
-- 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_79df34978043: member -> "in_member"
--   v_8afde06f51e3: invited -> "in_invited"
--   v_60b2536f5dc8: banned -> "in_banned"
-- Output columns:
--   f_ae8686d89791: allow_entry -> "out_allow_entry"
-- 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_allow_entry] output allow_entry (stable ID f_ae8686d89791)
--   allow_entry := (member OR invited) AND NOT banned
--   PRECONDITION: "in_member", "in_invited", "in_banned" are non-NULL booleans.
--   (("in_member" OR "in_invited") AND (NOT "in_banned"))
--

-- ----------------------------------------------------------------------------------------------
-- 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_member", "in_invited", "in_banned") 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_member" IS NULL OR "in_invited" IS NULL OR "in_banned" IS NULL THEN 'invalid_input'
      ELSE 'ok'
    END AS "status"
  FROM inputs
)
SELECT
  "row_id",
  "in_member",
  "in_invited",
  "in_banned",
  "status",
  CASE WHEN "status" = 'ok' THEN (("in_member" OR "in_invited") AND (NOT "in_banned")) END AS "out_allow_entry"
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: f0daa82174670c788f59eb058d07f7601cd0ace4ed94d61a747cc26c41fd4e0d
; 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_79df34978043: member -> in_member
;   v_8afde06f51e3: invited -> in_invited
;   v_60b2536f5dc8: banned -> in_banned
;   f_ae8686d89791: allow_entry -> out_allow_entry

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

; ---- Inputs (declared order)
(declare-const in_member Bool) ; member
(declare-const in_invited Bool) ; invited
(declare-const in_banned Bool) ; banned

; ---- Outputs (dependencies first)
; output allow_entry (stable ID f_ae8686d89791): (member OR invited) AND NOT banned
(define-fun out_allow_entry () Bool (and (or in_member in_invited) (not in_banned)))

; ---- 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
allow_entry = \left(\mathit{member} + \mathit{invited}\right) \cdot \overline{\mathit{banned}}

The common slip

Writing the rule from memory as member OR invited AND NOT banned and trusting the reader to group it. The truth table shows what grouping does: the intended rule is true on three rows, member OR (invited AND NOT banned) on five, and the two extra rows both have banned = TRUE.

Questions

Why does the NOR form need only 2 gates when the NAND form needs 11?

The rule is an OR feeding an AND with one inverted input, which is the natural shape of a NOR of a NOR: NOR(NOR(member, invited), banned). NAND gates have to build the OR and the inversion from inverters, so the mechanical construction grows to 11.

Is (member AND NOT banned) OR (invited AND NOT banned) a better form of the rule?

It is the same function, proved equal on all eight rows, and it is the sum-of-products form the Karnaugh map derives. It has one more literal than the original, so neither form is shorter in every sense; pick the one your readers parse fastest.

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.