Question 23289: An Epsilon-NFA Relation May Be Empty, Partial or Multivalued

Original question and givens

Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. 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=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. 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]=[s13, null, m13];state carrier=[s13, m13, f13, d13];alphabet=[a13, b13]

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]=[s13, null, d13];state carrier=[s13, m13, f13, d13];alphabet=[a13, b13]

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]=[m13, a13, m13];state carrier=[s13, m13, f13, d13];alphabet=[a13, b13]

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]=[m13, b13, f13];state carrier=[s13, m13, f13, d13];alphabet=[a13, b13]

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

ORIGINAL states=[s13, m13, f13, d13],alphabet=[a13, b13],start=s13,final_states=[f13];complete relation=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]

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=[s13, m13, f13, d13];ORIGINAL alphabet order=[a13, b13];complete relation=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]

ALL [state,symbol,ORIGINAL-order targets] rows=[[s13, a13, []], [s13, b13, []], [m13, a13, [m13]], [m13, b13, [f13]], [f13, a13, []], [f13, b13, []], [d13, a13, []], [d13, b13, []]];EMPTY target rows=[[s13, a13, []], [s13, b13, []], [f13, a13, []], [f13, b13, []], [d13, a13, []], [d13, b13, []]];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

([s13, m13, f13, d13], [a13, b13], s13, [f13], [[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]], [[s13, a13, []], [s13, b13, []], [f13, a13, []], [f13, b13, []], [d13, a13, []], [d13, b13, []]], 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.