Access control rule: owner OR admin OR (editor AND NOT locked)
An edit-permission rule, owner OR admin OR (editor AND NOT locked): sixteen rows with thirteen permitting ones, three maxterms, a two-clause CNF, and a Karnaugh map with two eight-cell groups.
Owners and admins can always edit; editors can edit unless the document is locked. The engine's reading says can_edit holds when at least one of three conditions does: owner, admin, or editor with locked false. On the sixteen-row table the output is true on thirteen rows and false on only three, ΠM(0, 1, 3): nobody has any role, nobody has a role and the document is locked, and an editor on a locked document.
The checks: not a tautology, counterexample all four inputs FALSE; satisfiable, witness owner = FALSE, admin = FALSE, editor = TRUE, locked = FALSE. The simplifier found no law that shortens the four-operator rule, and the DNF converter returns it unchanged as three terms. The CNF converter applies distribution once and gives (owner OR admin OR editor) AND (owner OR admin OR NOT locked): some role, and either a strong role or an unlocked document. The Karnaugh map draws owner, admin and editor AND NOT locked, three prime implicants, deriving the original. Gate counts: 10 NAND, 13 NOR.
Role rules are usually written as the DNF, one term per role, and read as the CNF by the security reviewer, who wants to know what every path has in common; here the CNF says a role is always required and the lock only stops editors. The thing to watch is that the lock does not stop owners or admins, which is a policy choice the table makes visible on the rows with locked = TRUE. If a locked document should block everyone, the rule needs NOT locked at the top level, and the equivalence check would show the rows that change.
can_edit := owner OR admin OR (editor AND NOT locked) · output
Exact checks
Truth table (16 rows)
| # | owner | admin | editor | locked | can_edit |
|---|---|---|---|---|---|
| 0 | FALSE | FALSE | FALSE | FALSE | FALSE |
| 1 | FALSE | FALSE | FALSE | TRUE | FALSE |
| 2 | FALSE | FALSE | TRUE | FALSE | TRUE |
| 3 | FALSE | FALSE | TRUE | TRUE | FALSE |
| 4 | FALSE | TRUE | FALSE | FALSE | TRUE |
| 5 | FALSE | TRUE | FALSE | TRUE | TRUE |
| 6 | FALSE | TRUE | TRUE | FALSE | TRUE |
| 7 | FALSE | TRUE | TRUE | TRUE | TRUE |
| 8 | TRUE | FALSE | FALSE | FALSE | TRUE |
| 9 | TRUE | FALSE | FALSE | TRUE | TRUE |
| 10 | TRUE | FALSE | TRUE | FALSE | TRUE |
| 11 | TRUE | FALSE | TRUE | TRUE | TRUE |
| 12 | TRUE | TRUE | FALSE | FALSE | TRUE |
| 13 | TRUE | TRUE | FALSE | TRUE | TRUE |
| 14 | TRUE | TRUE | TRUE | FALSE | TRUE |
| 15 | TRUE | TRUE | TRUE | TRUE | TRUE |
Plain-language reading
Beginnercan_edit is true exactly when at least one of the following holds (several may hold):
1. owner is true
2. admin is true
3. editor is true and locked is false
In every other case, can_edit is false.
About the operators:
- OR is inclusive: it is true when at least one part is true, including when several are.
- AND is true only when all of its parts are true.
- NOT reverses a value: true becomes false and false becomes true.
Intermediatecan_edit is true exactly when at least one of the following holds (several may hold):
1. owner is true
2. admin is true
3. editor is true and locked is false
In every other case, can_edit is false.
Advancedcan_edit is true exactly when at least one of the following holds (several may hold) [inclusive disjunction ∨]:
1. owner is true
2. admin is true
3. editor is true and locked is false [conjunction ∧]
In every other case, can_edit is false.
Formal notation: owner ∨ admin ∨ (editor ∧ ¬locked)
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.
owner OR admin OR (editor AND NOT locked)
Karnaugh map
- cells m8, m9, m10, m11, m12, m13, m14, m15
owner - cells m4, m5, m6, m7, m12, m13, m14, m15
admin - cells m2, m6, m10, m14
editor AND NOT locked
Minterms and maxterms
can_edit(owner, admin, editor, locked) = Σm(2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15)can_edit(owner, admin, editor, locked) = ΠM(0, 1, 3)Variable order: owner, admin, editor, locked · 16 rows
Normal forms
Conjunctive normal form (CNF)
(owner OR admin OR editor) AND (owner OR admin OR NOT locked)2 clauses, 6 literals · complete · Distribution ×1
Disjunctive normal form (DNF)
owner OR admin OR (editor AND NOT locked)3 terms, 4 literals · complete
NAND-only and NOR-only
NAND-only form
NAND(NAND(NAND(NAND(owner, owner), NAND(admin, admin)), NAND(NAND(owner, owner), NAND(admin, admin))), NAND(editor, NAND(locked, locked)))10 gates · verified equivalent on every row
NOR-only form
NOR(NOR(NOR(NOR(owner, admin), NOR(owner, admin)), NOR(NOR(editor, editor), locked)), NOR(NOR(NOR(owner, admin), NOR(owner, admin)), NOR(NOR(editor, editor), locked)))13 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: f0140eb5d7bba0f9c642f6251ffb8927adcbc5ddde8fb142944a84fb2f2751af
Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78
Inputs (dict keys, declared order) and generated parameter names:
'owner' -> input_owner
'admin' -> input_admin
'editor' -> input_editor
'locked' -> input_locked
Outputs (declared order) and their functions:
'can_edit' -> output_can_edit(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({'owner': False, 'admin': False, 'editor': False, 'locked': False})
# -> {'can_edit': False}
model.output_can_edit({'owner': False, 'admin': False, 'editor': False, 'locked': False})
# -> False
model.evaluate({'owner': 1, 'admin': False, 'editor': False, 'locked': 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_can_edit",
]
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": "f0140eb5d7bba0f9c642f6251ffb8927adcbc5ddde8fb142944a84fb2f2751af",
"assumptions_digest": "d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78",
"primary_output": "can_edit",
"unused_inputs": (),
"subcircuits": (),
}
# Stable ID -> generated identifier (reversible mapping; names are exact, escaped strings).
IDENTIFIER_MAP = {
"v_3e42a2ff4a81": {"kind": "input", "name": "owner", "identifier": "input_owner"},
"v_856339ba25f9": {"kind": "input", "name": "admin", "identifier": "input_admin"},
"v_f182dc2287ee": {"kind": "input", "name": "editor", "identifier": "input_editor"},
"v_68d28e15d536": {"kind": "input", "name": "locked", "identifier": "input_locked"},
"f_a849da994abc": {"kind": "output", "name": "can_edit", "identifier": "output_can_edit"},
}
INPUT_NAMES = ("owner", "admin", "editor", "locked")
OUTPUT_NAMES = ("can_edit",)
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_can_edit(input_owner: bool, input_admin: bool, input_editor: bool, input_locked: bool) -> bool:
"""Decision output 'can_edit': owner OR admin OR (editor AND NOT locked)"""
return input_owner or input_admin or (input_editor and (not input_locked))
# ---- Public API ----
def output_can_edit(inputs: Mapping[str, bool], *, check_scope: bool = True) -> bool:
"""Decision output 'can_edit' (stable ID f_a849da994abc).
'can_edit' := owner OR admin OR (editor AND NOT locked)
Required inputs: 'owner', 'admin', 'editor', 'locked'.
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, ("owner", "admin", "editor", "locked"), check_scope)
if check_scope:
_require_in_scope(values)
output_can_edit_value = _compute_output_can_edit(values["owner"], values["admin"], values["editor"], values["locked"])
return output_can_edit_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: 'owner', 'admin', 'editor', 'locked'.
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, ("owner", "admin", "editor", "locked"), check_scope)
if check_scope:
_require_in_scope(values)
output_can_edit_value = _compute_output_can_edit(values["owner"], values["admin"], values["editor"], values["locked"])
return {
"can_edit": output_can_edit_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: f0140eb5d7bba0f9c642f6251ffb8927adcbc5ddde8fb142944a84fb2f2751af
-- 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_3e42a2ff4a81: owner -> "in_owner"
-- v_856339ba25f9: admin -> "in_admin"
-- v_f182dc2287ee: editor -> "in_editor"
-- v_68d28e15d536: locked -> "in_locked"
-- Output columns:
-- f_a849da994abc: can_edit -> "out_can_edit"
-- 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_can_edit] output can_edit (stable ID f_a849da994abc)
-- can_edit := owner OR admin OR (editor AND NOT locked)
-- PRECONDITION: "in_owner", "in_admin", "in_editor", "in_locked" are non-NULL booleans.
-- ("in_owner" OR "in_admin" OR ("in_editor" AND (NOT "in_locked")))
--
-- ----------------------------------------------------------------------------------------------
-- 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_owner", "in_admin", "in_editor", "in_locked") 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_owner" IS NULL OR "in_admin" IS NULL OR "in_editor" IS NULL OR "in_locked" IS NULL THEN 'invalid_input'
ELSE 'ok'
END AS "status"
FROM inputs
)
SELECT
"row_id",
"in_owner",
"in_admin",
"in_editor",
"in_locked",
"status",
CASE WHEN "status" = 'ok' THEN ("in_owner" OR "in_admin" OR ("in_editor" AND (NOT "in_locked"))) END AS "out_can_edit"
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: f0140eb5d7bba0f9c642f6251ffb8927adcbc5ddde8fb142944a84fb2f2751af ; 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_3e42a2ff4a81: owner -> in_owner ; v_856339ba25f9: admin -> in_admin ; v_f182dc2287ee: editor -> in_editor ; v_68d28e15d536: locked -> in_locked ; f_a849da994abc: can_edit -> out_can_edit (set-option :produce-models true) (set-info :smt-lib-version 2.6) (set-logic QF_UF) ; ---- Inputs (declared order) (declare-const in_owner Bool) ; owner (declare-const in_admin Bool) ; admin (declare-const in_editor Bool) ; editor (declare-const in_locked Bool) ; locked ; ---- Outputs (dependencies first) ; output can_edit (stable ID f_a849da994abc): owner OR admin OR (editor AND NOT locked) (define-fun out_can_edit () Bool (or in_owner in_admin (and in_editor (not in_locked)))) ; ---- 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
can_edit = \mathit{owner} + \mathit{admin} + \left(\mathit{editor} \cdot \overline{\mathit{locked}}\right)The common slip
Assuming locked blocks everyone. On the row owner = TRUE, admin = FALSE, editor = FALSE, locked = TRUE the rule is true: the lock is only inside the editor term. The three false rows all have no owner and no admin, which is the fastest way to see who the lock actually restricts.
Questions
Why is this rule easier to read as a CNF?
The two clauses are the two questions a reviewer asks: does the user have any role at all, and if the only role is editor, is the document unlocked. The DNF answers who can edit; the CNF answers what can never be bypassed. The table proves both forms on all sixteen rows.
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
- Access control rules: roles, exceptions and the banned-user slip
- Feature flag rule: (beta_user OR internal) AND flag_on AND NOT kill_switch
- Entry rule: (member OR invited) AND NOT banned, checked every way
- CNF vs DNF: differences, examples and when to use each
- CNF and DNF converter: conjunctive and disjunctive normal forms, minterms and maxterms
- Decision table: 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.