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Question 16237: A DFA Needs a Complete Total Deterministic Transition Table

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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R0031-16237 · q-20261014-advance-20261101-r07-finite-automaton-0012

Original question and givens

Use a complete deterministic finite automaton with states=Fin(3)={0,…,2}, alphabet=Fin(2)={0,…,1}, total row-state/column-symbol transition=[[2, 0], [1, 0], [0, 0]], unique start=2, and increasing accepting subset=[2]. Every transition output lies in this same state set. There are no epsilon transitions or nondeterministic branches. Read a finite word in symbol order; the empty word leaves the automaton at its start state. Give (state_count,alphabet_count,start,accepting,transition_cell_count,qualified?,complete_total_deterministic_table).

  • G001 — Use a complete deterministic finite automaton with states=Fin(3)={0,…,2}, alphabet=Fin(2)={0,…,1}, total row-state/column-symbol transition=[[2, 0], [1, 0], [0, 0]], unique start=2, and increasing accepting subset=[2]. Every transition output lies in this same state set. There are no epsilon transitions or nondeterministic branches. Read a finite word in symbol order; the empty word leaves the automaton at its start state. Give (state_count,alphabet_count,start,accepting,transition_cell_count,qualified?,complete_total_deterministic_table).

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 · calculation

Before

DFA total state-symbol cell count := mul(3, 2)

After

DFA total state-symbol cell count = 6

Conditions: none

Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001

S002 · implies

Before

same complete states Fin(3);alphabet Fin(2);row-state column-symbol table=[[2, 0], [1, 0], [0, 0]]

After

single original start=2;canonical accepting subset=[2];required transition cells=6

Conditions: none

Fix states,alphabet,one start state and accepting subset.Rows index states and columns index symbols;every output must remain in the same state set.Check the whole table,not just cells visited by this word.

Step identifiers and dependencies

Rule: dfa_complete_typed_carriers_start_and_accepting_subset · Uses: G001, S001

S003 · implies

Before

original transition cell state=0;symbol=0;complete original row=[2, 0]

After

single same-carrier next state=2

Conditions: none

Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.

Step identifiers and dependencies

Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S002

S004 · implies

Before

original transition cell state=0;symbol=1;complete original row=[2, 0]

After

single same-carrier next state=0

Conditions: none

Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.

Step identifiers and dependencies

Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S003

S005 · implies

Before

original transition cell state=1;symbol=0;complete original row=[1, 0]

After

single same-carrier next state=1

Conditions: none

Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.

Step identifiers and dependencies

Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S004

S006 · implies

Before

original transition cell state=1;symbol=1;complete original row=[1, 0]

After

single same-carrier next state=0

Conditions: none

Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.

Step identifiers and dependencies

Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S005

S007 · implies

Before

original transition cell state=2;symbol=0;complete original row=[0, 0]

After

single same-carrier next state=0

Conditions: none

Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.

Step identifiers and dependencies

Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S006

S008 · implies

Before

original transition cell state=2;symbol=1;complete original row=[0, 0]

After

single same-carrier next state=0

Conditions: none

Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.

Step identifiers and dependencies

Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S007

Reverse checks

  • 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 — Recompute DFA total state-symbol cell count with exact rational arithmetic, not rounded decimals.

Result

(3, 2, 2, [2], 6, Yes, complete_total_deterministic_table)

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: 26f5640d4b80d211c781d3fb18c6f15afb48a25655f786dcd95da55b34576ae1
  • candidate record SHA-256: a66e592b2bf08b4b7430870176828a8367df783abac21a9257b2c1bb62faa856
  • semantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a
  • question-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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