Question 27540: Empty Language, Epsilon, Wildcards and Bounded Equality Are Not Interchangeable

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0054-27540 · q-20261015-advance-20261118-r06-regular-expression-0090

Original question and givens

Use mode=concat_order, complete model={"alphabet":["a","b","c"],"left":{"type":"concat","children":[{"type":"symbol","value":"c"},{"type":"symbol","value":"c"},{"type":"symbol","value":"a"}]},"right":{"type":"concat","children":[{"type":"symbol","value":"a"},{"type":"symbol","value":"c"},{"type":"symbol","value":"c"}]},"max_length":3}. Give (mode, left_AST, right_AST, left_bounded_language, right_bounded_language, bounded_languages_equal, order_irrelevance_claim, boundary).

G001 — Use mode=concat_order, complete model={"alphabet":["a","b","c"],"left":{"type":"concat","children":[{"type":"symbol","value":"c"},{"type":"symbol","value":"c"},{"type":"symbol","value":"a"}]},"right":{"type":"concat","children":[{"type":"symbol","value":"a"},{"type":"symbol","value":"c"},{"type":"symbol","value":"c"}]},"max_length":3}. Give (mode, left_AST, right_AST, left_bounded_language, right_bounded_language, bounded_languages_equal, order_irrelevance_claim, boundary).

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.

mko-regular-expression

Step-by-step derivation

S001 · implies

Before

source roles = [regex-boundary]

After

derived = [[mode, left_AST, right_AST, left_bounded_language, right_bounded_language, bounded_languages_equal, order_irrelevance_claim, boundary]]

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 boundary LEFT.ast.children[0], [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, S001

S003 · implies

Before

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

After

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

Conditions: none

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 boundary LEFT.ast.children[2], [a, b, c], {'type': 'symbol', 'value': 'a'}]

After

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

Conditions: none

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 boundary LEFT.ast, [a, b, c], {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}]}]

After

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

Conditions: none

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 = [[a, b, c], 3]

After

derived = [40]

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, S005

S007 · implies

Before

source roles = [Regex boundary LEFT.language.concat[0], symbol, {'type': 'symbol', 'value': 'c'}, 3]

After

derived = [[[c]]]

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, S006

S008 · implies

Before

source roles = [Regex boundary LEFT.language, 0, 3, [[]], [[c]]]

After

derived = [[[c]]]

Conditions: none

In ORIGINAL concatenation order take EVERY prefix/next-child word concatenation and discard only excessive length.If total length<=m,every factor length<=m,so truncation loses no allowed combination.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S007

S009 · implies

Before

source roles = [Regex boundary LEFT.language.concat[1], symbol, {'type': 'symbol', 'value': 'c'}, 3]

After

derived = [[[c]]]

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, S008

S010 · implies

Before

source roles = [Regex boundary LEFT.language, 1, 3, [[c]], [[c]]]

After

derived = [[[c, c]]]

Conditions: none

In ORIGINAL concatenation order take EVERY prefix/next-child word concatenation and discard only excessive length.If total length<=m,every factor length<=m,so truncation loses no allowed combination.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S009

S011 · implies

Before

source roles = [Regex boundary LEFT.language.concat[2], symbol, {'type': 'symbol', 'value': 'a'}, 3]

After

derived = [[[a]]]

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, S010

S012 · implies

Before

source roles = [Regex boundary LEFT.language, 2, 3, [[c, c]], [[a]]]

After

derived = [[[c, c, a]]]

Conditions: none

In ORIGINAL concatenation order take EVERY prefix/next-child word concatenation and discard only excessive length.If total length<=m,every factor length<=m,so truncation loses no allowed combination.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S011

S013 · implies

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S012

S014 · implies

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S013

S015 · implies

Before

source roles = [Regex boundary RIGHT.ast.children[2], [a, b, c], {'type': 'symbol', 'value': 'c'}]

After

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

Conditions: none

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S014

S016 · implies

Before

source roles = [Regex boundary RIGHT.ast, [a, b, c], {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'c'}]}]

After

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

Conditions: none

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S015

S017 · implies

Before

source roles = [[a, b, c], 3]

After

derived = [40]

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, S016

S018 · implies

Before

source roles = [Regex boundary RIGHT.language.concat[0], symbol, {'type': 'symbol', 'value': 'a'}, 3]

After

derived = [[[a]]]

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, S017

S019 · implies

Before

source roles = [Regex boundary RIGHT.language, 0, 3, [[]], [[a]]]

After

derived = [[[a]]]

Conditions: none

In ORIGINAL concatenation order take EVERY prefix/next-child word concatenation and discard only excessive length.If total length<=m,every factor length<=m,so truncation loses no allowed combination.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S018

S020 · implies

Before

source roles = [Regex boundary RIGHT.language.concat[1], symbol, {'type': 'symbol', 'value': 'c'}, 3]

After

derived = [[[c]]]

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, S019

S021 · implies

Before

source roles = [Regex boundary RIGHT.language, 1, 3, [[a]], [[c]]]

After

derived = [[[a, c]]]

Conditions: none

In ORIGINAL concatenation order take EVERY prefix/next-child word concatenation and discard only excessive length.If total length<=m,every factor length<=m,so truncation loses no allowed combination.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S020

S022 · implies

Before

source roles = [Regex boundary RIGHT.language.concat[2], symbol, {'type': 'symbol', 'value': 'c'}, 3]

After

derived = [[[c]]]

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, S021

S023 · implies

Before

source roles = [Regex boundary RIGHT.language, 2, 3, [[a, c]], [[c]]]

After

derived = [[[a, c, c]]]

Conditions: none

In ORIGINAL concatenation order take EVERY prefix/next-child word concatenation and discard only excessive length.If total length<=m,every factor length<=m,so truncation loses no allowed combination.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S022

S024 · implies

Before

source roles = [{'type': 'concat', 'children': [{'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}]}, {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'c'}]}, 3]

After

derived = [[[c, c, a]], [[a, c, c]], No, No]

Conditions: none

Concatenation order is semantic.These complete bounded languages actually differ;concatenation is not a commutative union.

Step identifiers and dependencies

Rule: Regex_ACTUAL_ordered_concat_language_difference · Uses: G001, S023

Reverse checks

S024 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
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

(concat_order, {children:[{type:"symbol",value:"c"},{type:"symbol",value:"c"},{type:"symbol",value:"a"}],type:"concat"}, {children:[{type:"symbol",value:"a"},{type:"symbol",value:"c"},{type:"symbol",value:"c"}],type:"concat"}, [[c, c, a]], [[a, c, c]], No, No, concatenation_order_is_semantic)

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: fecf39b93cf3e8121263ec2a3f3f52225424a37c4e22c62833f112a980aa84c8
candidate record SHA-256: 29a165eaaaaf7c84c711806f3e09b05524c530beeaa5bebb827f7baff3c54984
semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
Lean: NOT_RUN · human review: NOT_PERFORMED

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