Question 17212: Construct a Complete Prefix-Trie DFA for an Exact Finite Language

Original question and givens

Use alphabet=Fin(2)={0,…,1} and the complete nonempty duplicate-free word family=[[0], [1, 1, 1, 0, 1, 0]]. The family may include the empty word; this constructor does not handle an empty family. Probe=[0]. 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, 1, 1, 0, 1, 0]]. The family may include the empty word; this constructor does not handle an empty family. Probe=[0]. 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.

• mko-regular-language

Step-by-step derivation

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

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

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

extension is declared prefix=Yes;next state=1

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

extension is declared prefix=Yes;next state=2

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

extension is declared prefix=No;next state=8

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

extension is declared prefix=No;next state=8

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

extension is declared prefix=No;next state=8

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

extension is declared prefix=Yes;next state=3

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

extension is declared prefix=No;next state=8

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

extension is declared prefix=Yes;next state=4

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

extension is declared prefix=Yes;next state=5

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

extension is declared prefix=No;next state=8

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

extension is declared prefix=No;next state=8

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

extension is declared prefix=Yes;next state=6

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

extension is declared prefix=Yes;next state=7

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

extension is declared prefix=No;next state=8

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

extension is declared prefix=No;next state=8

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

extension is declared prefix=No;next state=8

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

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

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

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

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

trace before consuming symbols=[0]

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

next state=1;ordered prefix trace=[0, 1]

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

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

Reverse checks

• 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, 1], [1, 1, 1], [1, 1, 1, 0], [1, 1, 1, 0, 1], [1, 1, 1, 0, 1, 0]], 8, [1, 7], [[1, 2], [8, 8], [8, 3], [8, 4], [5, 8], [8, 6], [7, 8], [8, 8], [8, 8]], [0], [0, 1], 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.