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Question 27591: Whole-Word Membership Uses a Sufficient Complete Length Slice

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0054-27591 · q-20261015-advance-20261118-r06-regular-expression-0066

Original question and givens

Use alphabet=[a, b, c] and typed regular-expression AST={"type":"union","children":[{"type":"union","children":[{"type":"union","children":[{"type":"symbol","value":"a"},{"type":"star","child":{"type":"symbol","value":"b"}}]},{"type":"star","child":{"type":"symbol","value":"c"}}]},{"type":"star","child":{"type":"symbol","value":"a"}}]}. Words are arrays of alphabet symbols; distinguish empty language from the language containing the empty word. Decide word=[]. Give (alphabet, canonical_typed_AST, word, member, accepted_words_at_same_length, membership_scope).

  • G001 — Use alphabet=[a, b, c] and typed regular-expression AST={"type":"union","children":[{"type":"union","children":[{"type":"union","children":[{"type":"symbol","value":"a"},{"type":"star","child":{"type":"symbol","value":"b"}}]},{"type":"star","child":{"type":"symbol","value":"c"}}]},{"type":"star","child":{"type":"symbol","value":"a"}}]}. Words are arrays of alphabet symbols; distinguish empty language from the language containing the empty word. Decide word=[]. Give (alphabet, canonical_typed_AST, word, member, accepted_words_at_same_length, membership_scope).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [regex-word-membership]

After

derived = [[alphabet, canonical_typed_AST, word, member, accepted_words_at_same_length, membership_scope]]

Conditions: none

List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[0], [a, b, c], {'type': 'symbol', 'value': 'a'}]

After

derived = [{'type': 'symbol', 'value': 'a'}, {type:"symbol",value:"a"}]

Conditions: none

For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S001

S003 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[1].child, [a, b, c], {'type': 'symbol', 'value': 'b'}]

After

derived = [{'type': 'symbol', 'value': 'b'}, {type:"symbol",value:"b"}]

Conditions: none

For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S002

S004 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[1], [a, b, c], {'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}]

After

derived = [{'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}, {child:{type:"symbol",value:"b"},type:"star"}]

Conditions: none

For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S003

S005 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0], [a, b, c], {'type': 'union', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}]}]

After

derived = [{'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}, {'type': 'symbol', 'value': 'a'}]}, {children:[{child:{type:"symbol",value:"b"},type:"star"},{type:"symbol",value:"a"}],type:"union"}]

Conditions: none

For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S004

S006 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[1].child, [a, b, c], {'type': 'symbol', 'value': 'c'}]

After

derived = [{'type': 'symbol', 'value': 'c'}, {type:"symbol",value:"c"}]

Conditions: none

For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S005

S007 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[1], [a, b, c], {'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}]

After

derived = [{'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}, {child:{type:"symbol",value:"c"},type:"star"}]

Conditions: none

For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S006

S008 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0], [a, b, c], {'type': 'union', 'children': [{'type': 'union', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}]}]

After

derived = [{'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}, {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}, {'type': 'symbol', 'value': 'a'}]}]}, {children:[{child:{type:"symbol",value:"c"},type:"star"},{children:[{child:{type:"symbol",value:"b"},type:"star"},{type:"symbol",value:"a"}],type:"union"}],type:"union"}]

Conditions: none

For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S007

S009 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[1].child, [a, b, c], {'type': 'symbol', 'value': 'a'}]

After

derived = [{'type': 'symbol', 'value': 'a'}, {type:"symbol",value:"a"}]

Conditions: none

For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S008

S010 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[1], [a, b, c], {'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}]

After

derived = [{'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}, {child:{type:"symbol",value:"a"},type:"star"}]

Conditions: none

For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S009

S011 · implies

Before

source roles = [Regex ORIGINAL membership.ast, [a, b, c], {'type': 'union', 'children': [{'type': 'union', 'children': [{'type': 'union', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}]}]

After

derived = [{'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}, {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}, {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}, {'type': 'symbol', 'value': 'a'}]}]}]}, {children:[{child:{type:"symbol",value:"a"},type:"star"},{children:[{child:{type:"symbol",value:"c"},type:"star"},{children:[{child:{type:"symbol",value:"b"},type:"star"},{type:"symbol",value:"a"}],type:"union"}],type:"union"}],type:"union"}]

Conditions: none

For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S010

S012 · implies

Before

source roles = [[a, b, c], 0]

After

derived = [1]

Conditions: none

The bounded universe has sum |alphabet|^k words,including epsilon,and source budget<=4096.This defines complete bounded enumeration,not unbounded equivalence.

Step identifiers and dependencies

Rule: Regex_COMPLETE_bounded_word_universe_not_unbounded_equivalence · Uses: G001, S011

S013 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[0].star_base, symbol, {'type': 'symbol', 'value': 'a'}, 0]

After

derived = [[]]

Conditions: none

The empty language has no words;epsilon has the single empty word;a symbol has its one token within the queried length bound.Words are token arrays,not reinterpreted character splits.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S012

S014 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[0], 0, 0, [[]], []]

After

derived = [[], [[]]]

Conditions: none

Start with epsilon and add ALL base concatenations within the length bound.Only genuinely new words enter the next frontier.Epsilon cycles add none.The finite alphabet/length bound and4096 budget give a finite universe;completion requires an empty frontier,not an arbitrary iteration cutoff.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S013

S015 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[0].star_base, symbol, {'type': 'symbol', 'value': 'c'}, 0]

After

derived = [[]]

Conditions: none

The empty language has no words;epsilon has the single empty word;a symbol has its one token within the queried length bound.Words are token arrays,not reinterpreted character splits.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S014

S016 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[0], 0, 0, [[]], []]

After

derived = [[], [[]]]

Conditions: none

Start with epsilon and add ALL base concatenations within the length bound.Only genuinely new words enter the next frontier.Epsilon cycles add none.The finite alphabet/length bound and4096 budget give a finite universe;completion requires an empty frontier,not an arbitrary iteration cutoff.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S015

S017 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[0].star_base, symbol, {'type': 'symbol', 'value': 'b'}, 0]

After

derived = [[]]

Conditions: none

The empty language has no words;epsilon has the single empty word;a symbol has its one token within the queried length bound.Words are token arrays,not reinterpreted character splits.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S016

S018 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[0], 0, 0, [[]], []]

After

derived = [[], [[]]]

Conditions: none

Start with epsilon and add ALL base concatenations within the length bound.Only genuinely new words enter the next frontier.Epsilon cycles add none.The finite alphabet/length bound and4096 budget give a finite universe;completion requires an empty frontier,not an arbitrary iteration cutoff.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S017

S019 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1], symbol, {'type': 'symbol', 'value': 'a'}, 0]

After

derived = [[]]

Conditions: none

The empty language has no words;epsilon has the single empty word;a symbol has its one token within the queried length bound.Words are token arrays,not reinterpreted character splits.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S018

S020 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1], 0, [[[]], []]]

After

derived = [[[]]]

Conditions: none

Union includes ALL child languages,retaining each word once without reordering its symbols.Structural induction accounts for inclusion/exclusion of every word within the bound.

Step identifiers and dependencies

Rule: Regex_union_ALL_child_languages_extensional_bound · Uses: G001, S019

S021 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1], 0, [[[]], [[]]]]

After

derived = [[[]]]

Conditions: none

Union includes ALL child languages,retaining each word once without reordering its symbols.Structural induction accounts for inclusion/exclusion of every word within the bound.

Step identifiers and dependencies

Rule: Regex_union_ALL_child_languages_extensional_bound · Uses: G001, S020

S022 · implies

Before

source roles = [Regex ORIGINAL membership.language, 0, [[[]], [[]]]]

After

derived = [[[]]]

Conditions: none

Union includes ALL child languages,retaining each word once without reordering its symbols.Structural induction accounts for inclusion/exclusion of every word within the bound.

Step identifiers and dependencies

Rule: Regex_union_ALL_child_languages_extensional_bound · Uses: G001, S021

S023 · implies

Before

source roles = [[a, b, c], {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}, {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}, {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}, {'type': 'symbol', 'value': 'a'}]}]}]}, [], 0]

After

derived = [Yes, [[]]]

Conditions: none

Recompute through the FULL queried length and retain every accepted word at that length.The empty word differs from a symbol;absence from an insufficient shorter bound is not nonmembership.

Step identifiers and dependencies

Rule: Regex_WHOLE_queried_word_exact_membership_complete_same_length_slice · Uses: G001, S022

Reverse checks

  • S023 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S022 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S021 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S020 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S019 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S018 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S016 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S015 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S014 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S013 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S012 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S011 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S010 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S008 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S006 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S005 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S004 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S003 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S002 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.

Result

([a, b, c], {children:[{child:{type:"symbol",value:"a"},type:"star"},{children:[{child:{type:"symbol",value:"c"},type:"star"},{children:[{child:{type:"symbol",value:"b"},type:"star"},{type:"symbol",value:"a"}],type:"union"}],type:"union"}],type:"union"}, [], Yes, [[]], complete_membership_not_absence_from_an_insufficient_bound)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 58e1e45b0ada4900a3e04c6eb36ca0e1ccaa767c61cc7ad781cb21a72a218af8
  • candidate record SHA-256: 064a07e19963ae4fac8fc291bd869213ebf900c0588db8f022d2c116cad5d9ef
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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