Question 23299: An Epsilon-NFA Relation May Be Empty, Partial or Multivalued

Original question and givens

Use the complete finite epsilon-NFA: states=[s23, m23, f23, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).

• G001 — Use the complete finite epsilon-NFA: states=[s23, m23, f23, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).

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-nondeterministic-finite-automaton

Step-by-step derivation

source operation=nfa-qualification;original snapshot and mathematics retained

explicit ordered source output fields=[states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention]

ORIGINAL transition index=0;[from,input_or_null,to]=[s23, null, m23];state carrier=[s23, m23, f23, d23];alphabet=[a23, b23]

state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows

ORIGINAL transition index=1;[from,input_or_null,to]=[s23, null, d23];state carrier=[s23, m23, f23, d23];alphabet=[a23, b23]

state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows

ORIGINAL transition index=2;[from,input_or_null,to]=[m23, a23, m23];state carrier=[s23, m23, f23, d23];alphabet=[a23, b23]

state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows

ORIGINAL transition index=3;[from,input_or_null,to]=[m23, b23, f23];state carrier=[s23, m23, f23, d23];alphabet=[a23, b23]

state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows

ORIGINAL states=[s23, m23, f23, d23],alphabet=[a23, b23],start=s23,final_states=[f23];complete relation=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]

states nonempty with actual start;alphabet/final set/relation MAY be EMPTY;definition qualification does NOT prove determinism,totality,DFA equivalence or a successful run

ORIGINAL state order=[s23, m23, f23, d23];ORIGINAL alphabet order=[a23, b23];complete relation=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]

ALL [state,symbol,ORIGINAL-order targets] rows=[[s23, a23, []], [s23, b23, []], [m23, a23, [m23]], [m23, b23, [f23]], [f23, a23, []], [f23, b23, []], [d23, a23, []], [d23, b23, []]];EMPTY target rows=[[s23, a23, []], [s23, b23, []], [f23, a23, []], [f23, b23, []], [d23, a23, []], [d23, b23, []]];no missing edge repaired

Reverse checks

• 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

([s23, m23, f23, d23], [a23, b23], s23, [f23], [[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]], [[s23, a23, []], [s23, b23, []], [f23, a23, []], [f23, b23, []], [d23, a23, []], [d23, b23, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)

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.