Question 23292: An Epsilon-NFA Relation May Be Empty, Partial or Multivalued

Original question and givens

Use the complete finite epsilon-NFA: states=[s16, m16, f16, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]. 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=[s16, m16, f16, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]. 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]=[s16, null, m16];state carrier=[s16, m16, f16, d16];alphabet=[a16, b16]

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]=[s16, null, d16];state carrier=[s16, m16, f16, d16];alphabet=[a16, b16]

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]=[m16, a16, m16];state carrier=[s16, m16, f16, d16];alphabet=[a16, b16]

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]=[m16, b16, f16];state carrier=[s16, m16, f16, d16];alphabet=[a16, b16]

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

ORIGINAL states=[s16, m16, f16, d16],alphabet=[a16, b16],start=s16,final_states=[f16];complete relation=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]

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=[s16, m16, f16, d16];ORIGINAL alphabet order=[a16, b16];complete relation=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]

ALL [state,symbol,ORIGINAL-order targets] rows=[[s16, a16, []], [s16, b16, []], [m16, a16, [m16]], [m16, b16, [f16]], [f16, a16, []], [f16, b16, []], [d16, a16, []], [d16, b16, []]];EMPTY target rows=[[s16, a16, []], [s16, b16, []], [f16, a16, []], [f16, b16, []], [d16, a16, []], [d16, b16, []]];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

([s16, m16, f16, d16], [a16, b16], s16, [f16], [[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]], [[s16, a16, []], [s16, b16, []], [f16, a16, []], [f16, b16, []], [d16, a16, []], [d16, b16, []]], 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.