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Question 27554: The Complete Bounded Language Includes Epsilon and Terminates Epsilon Cycles

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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R0054-27554 · q-20261015-advance-20261118-r06-regular-expression-0029

Original question and givens

Use alphabet=[a, b, c] and typed regular-expression AST={"type":"concat","children":[{"type":"symbol","value":"a"},{"type":"symbol","value":"b"},{"type":"symbol","value":"c"},{"type":"symbol","value":"a"}]}. Words are arrays of alphabet symbols; distinguish empty language from the language containing the empty word. Enumerate every accepted word of length at most=4. Give (alphabet, canonical_typed_AST, max_length, complete_bounded_language, word_count, bounded_enumeration_complete, epsilon_cycle_scope).

  • G001 — Use alphabet=[a, b, c] and typed regular-expression AST={"type":"concat","children":[{"type":"symbol","value":"a"},{"type":"symbol","value":"b"},{"type":"symbol","value":"c"},{"type":"symbol","value":"a"}]}. Words are arrays of alphabet symbols; distinguish empty language from the language containing the empty word. Enumerate every accepted word of length at most=4. Give (alphabet, canonical_typed_AST, max_length, complete_bounded_language, word_count, bounded_enumeration_complete, epsilon_cycle_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-bounded-language]

After

derived = [[alphabet, canonical_typed_AST, max_length, complete_bounded_language, word_count, bounded_enumeration_complete, epsilon_cycle_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 bounded.ast.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 bounded.ast.children[1], [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 bounded.ast.children[2], [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, S003

S005 · implies

Before

source roles = [Regex ORIGINAL bounded.ast.children[3], [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, S004

S006 · implies

Before

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

After

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

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 = [[a, b, c], 4]

After

derived = [121]

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, S006

S008 · implies

Before

source roles = [Regex ORIGINAL bounded.language.concat[0], symbol, {'type': 'symbol', 'value': 'a'}, 4]

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, S007

S009 · implies

Before

source roles = [Regex ORIGINAL bounded.language, 0, 4, [[]], [[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, S008

S010 · implies

Before

source roles = [Regex ORIGINAL bounded.language.concat[1], symbol, {'type': 'symbol', 'value': 'b'}, 4]

After

derived = [[[b]]]

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, S009

S011 · implies

Before

source roles = [Regex ORIGINAL bounded.language, 1, 4, [[a]], [[b]]]

After

derived = [[[a, b]]]

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, S010

S012 · implies

Before

source roles = [Regex ORIGINAL bounded.language.concat[2], symbol, {'type': 'symbol', 'value': 'c'}, 4]

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, S011

S013 · implies

Before

source roles = [Regex ORIGINAL bounded.language, 2, 4, [[a, b]], [[c]]]

After

derived = [[[a, b, 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, S012

S014 · implies

Before

source roles = [Regex ORIGINAL bounded.language.concat[3], symbol, {'type': 'symbol', 'value': 'a'}, 4]

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, S013

S015 · implies

Before

source roles = [Regex ORIGINAL bounded.language, 3, 4, [[a, b, c]], [[a]]]

After

derived = [[[a, b, 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, S014

Reverse checks

  • 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:[{type:"symbol",value:"a"},{type:"symbol",value:"b"},{type:"symbol",value:"c"},{type:"symbol",value:"a"}],type:"concat"}, 4, [[a, b, c, a]], 1, Yes, epsilon_cycles_terminate_by_extensional_dedup)

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: 7e36b0bbb8aaabeac5a1cd894f48fc47d6e4b0cda0eb46590ae0cc0cd875f5bd
  • candidate record SHA-256: 4890f5939a464d6fad62ec4c179effd6adf06d9c64e6c2eef5c420ed1741c052
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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