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Question 17221: Construct a Complete Prefix-Trie DFA for an Exact Finite Language

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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R0033-17221 · q-20261014-advance-20261102-r07-regular-language-0071

Original question and givens

Use alphabet=Fin(2)={0,…,1} and the complete nonempty duplicate-free word family=[[0], [1, 0, 0, 1, 0, 1]]. The family may include the empty word; this constructor does not handle an empty family. Probe=[1, 0, 0, 1, 0, 1]. Give (prefix_states,dead_state,accepting,transition,probe,trace,accepted?,explicit_DFA_recognizes_exact_finite_language).

  • G001 — Use alphabet=Fin(2)={0,…,1} and the complete nonempty duplicate-free word family=[[0], [1, 0, 0, 1, 0, 1]]. The family may include the empty word; this constructor does not handle an empty family. Probe=[1, 0, 0, 1, 0, 1]. Give (prefix_states,dead_state,accepting,transition,probe,trace,accepted?,explicit_DFA_recognizes_exact_finite_language).

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

DECLARED complete nonempty duplicate-free word FAMILY=[[0], [1, 0, 0, 1, 0, 1]];alphabet Fin(2)

After

all prefixes including empty sorted by length then source JSON order=[[], [0], [1], [1, 0], [1, 0, 0], [1, 0, 0, 1], [1, 0, 0, 1, 0], [1, 0, 0, 1, 0, 1]];dead state appended index=8

Conditions: none

This family is the entire finite language,not test samples.It may contain the empty word but the family is nonempty.Sort prefixes by length and the source JSON order;symbols0..9 make same-length ordering compatible.Do not inherit another family's30-state cap.

Step identifiers and dependencies

Rule: regular_complete_finite_language_prefix_states_not_sampled_tests · Uses: G001

S002 · implies

Before

prefix state=0;represented prefix=[];next symbol=0;extended word=[0]

After

extension is declared prefix=Yes;next state=1

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S001

S003 · implies

Before

prefix state=0;represented prefix=[];next symbol=1;extended word=[1]

After

extension is declared prefix=Yes;next state=2

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S002

S004 · implies

Before

prefix state=1;represented prefix=[0];next symbol=0;extended word=[0, 0]

After

extension is declared prefix=No;next state=8

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S003

S005 · implies

Before

prefix state=1;represented prefix=[0];next symbol=1;extended word=[0, 1]

After

extension is declared prefix=No;next state=8

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S004

S006 · implies

Before

prefix state=2;represented prefix=[1];next symbol=0;extended word=[1, 0]

After

extension is declared prefix=Yes;next state=3

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S005

S007 · implies

Before

prefix state=2;represented prefix=[1];next symbol=1;extended word=[1, 1]

After

extension is declared prefix=No;next state=8

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S006

S008 · implies

Before

prefix state=3;represented prefix=[1, 0];next symbol=0;extended word=[1, 0, 0]

After

extension is declared prefix=Yes;next state=4

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S007

S009 · implies

Before

prefix state=3;represented prefix=[1, 0];next symbol=1;extended word=[1, 0, 1]

After

extension is declared prefix=No;next state=8

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S008

S010 · implies

Before

prefix state=4;represented prefix=[1, 0, 0];next symbol=0;extended word=[1, 0, 0, 0]

After

extension is declared prefix=No;next state=8

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S009

S011 · implies

Before

prefix state=4;represented prefix=[1, 0, 0];next symbol=1;extended word=[1, 0, 0, 1]

After

extension is declared prefix=Yes;next state=5

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S010

S012 · implies

Before

prefix state=5;represented prefix=[1, 0, 0, 1];next symbol=0;extended word=[1, 0, 0, 1, 0]

After

extension is declared prefix=Yes;next state=6

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S011

S013 · implies

Before

prefix state=5;represented prefix=[1, 0, 0, 1];next symbol=1;extended word=[1, 0, 0, 1, 1]

After

extension is declared prefix=No;next state=8

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S012

S014 · implies

Before

prefix state=6;represented prefix=[1, 0, 0, 1, 0];next symbol=0;extended word=[1, 0, 0, 1, 0, 0]

After

extension is declared prefix=No;next state=8

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S013

S015 · implies

Before

prefix state=6;represented prefix=[1, 0, 0, 1, 0];next symbol=1;extended word=[1, 0, 0, 1, 0, 1]

After

extension is declared prefix=Yes;next state=7

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S014

S016 · implies

Before

prefix state=7;represented prefix=[1, 0, 0, 1, 0, 1];next symbol=0;extended word=[1, 0, 0, 1, 0, 1, 0]

After

extension is declared prefix=No;next state=8

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S015

S017 · implies

Before

prefix state=7;represented prefix=[1, 0, 0, 1, 0, 1];next symbol=1;extended word=[1, 0, 0, 1, 0, 1, 1]

After

extension is declared prefix=No;next state=8

Conditions: none

Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching.

Step identifiers and dependencies

Rule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S016

S018 · implies

Before

dead state=8;all alphabet symbols Fin(2);full supplied words=[[0], [1, 0, 0, 1, 0, 1]]

After

dead loops row=[8, 8];accepting prefix-state indices=[1, 7];complete table=[[1, 2], [8, 8], [3, 8], [4, 8], [8, 5], [6, 8], [8, 7], [8, 8], [8, 8]]

Conditions: none

Every symbol loops at dead.Accept exactly states equal to full supplied words,not every proper prefix.If the empty word is supplied,start0 is accepting.

Step identifiers and dependencies

Rule: regular_trie_dead_absorbing_accepts_full_words_only · Uses: G001, S017

S019 · implies

Before

same complete prefix-state table and full word family=[[0], [1, 0, 0, 1, 0, 1]];start state0 represents empty prefix

After

for EVERY finite input prefix,its state is that exact prefix index if present,otherwise dead;prefix extension closure and dead absorption preserve this by induction;accepted iff input is one FULL supplied word

Conditions: none

The empty prefix is the base case.Each transition follows exact extension.If a consumed word is not a supplied-word prefix,no later extension can become one,so dead absorption is valid.Induction over arbitrary word length gives acceptance exactly for full listed words,not an inference from a successful probe.

Step identifiers and dependencies

Rule: regular_trie_all_word_induction_exact_language_not_probe_only · Uses: G001, S018

S020 · implies

Before

finite-language probe:original ordered word=[1, 0, 0, 1, 0, 1];start=0

After

trace before consuming symbols=[0]

Conditions: none

Include the initial state.An empty word performs no transition;do not reorder symbols merely because a final answer might coincide.

Step identifiers and dependencies

Rule: regular_word_initial_state_empty_word_retained · Uses: G001, S019

S021 · implies

Before

finite-language probe:word position=0;symbol=1;current state=0;complete original row=[1, 2]

After

next state=2;ordered prefix trace=[0, 2]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S020

S022 · implies

Before

finite-language probe:word position=1;symbol=0;current state=2;complete original row=[3, 8]

After

next state=3;ordered prefix trace=[0, 2, 3]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S021

S023 · implies

Before

finite-language probe:word position=2;symbol=0;current state=3;complete original row=[4, 8]

After

next state=4;ordered prefix trace=[0, 2, 3, 4]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S022

S024 · implies

Before

finite-language probe:word position=3;symbol=1;current state=4;complete original row=[8, 5]

After

next state=5;ordered prefix trace=[0, 2, 3, 4, 5]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S023

S025 · implies

Before

finite-language probe:word position=4;symbol=0;current state=5;complete original row=[6, 8]

After

next state=6;ordered prefix trace=[0, 2, 3, 4, 5, 6]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S024

S026 · implies

Before

finite-language probe:word position=5;symbol=1;current state=6;complete original row=[8, 7]

After

next state=7;ordered prefix trace=[0, 2, 3, 4, 5, 6, 7]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S025

S027 · implies

Before

original probe=[1, 0, 0, 1, 0, 1];final state=7;accepting indices=[1, 7]

After

probe accepted=Yes;separate all-word construction certificate retained

Conditions: none

The probe is one run.The complete construction and induction above supply the full finite-language certificate.

Step identifiers and dependencies

Rule: regular_probe_result_separate_universal_construction · Uses: G001, S026

Reverse checks

  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([[], [0], [1], [1, 0], [1, 0, 0], [1, 0, 0, 1], [1, 0, 0, 1, 0], [1, 0, 0, 1, 0, 1]], 8, [1, 7], [[1, 2], [8, 8], [3, 8], [4, 8], [8, 5], [6, 8], [8, 7], [8, 8], [8, 8]], [1, 0, 0, 1, 0, 1], [0, 2, 3, 4, 5, 6, 7], Yes, explicit_DFA_recognizes_exact_finite_language)

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: af3918d4c2d11ea34ed86cd67d6ecfd34def5edf66665eb92ca6b971d00ba675
  • candidate record SHA-256: 00e9fd5da5f8a810a90fa6603f562cbbede367cb853a9e63703b252070be4feb
  • semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3
  • question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON