Question 23286: An Epsilon-NFA Relation May Be Empty, Partial or Multivalued

Original question and givens

Use the complete finite epsilon-NFA: states=[s10, m10, f10, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]. 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=[s10, m10, f10, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]. 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]=[s10, null, m10];state carrier=[s10, m10, f10, d10];alphabet=[a10, b10]

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]=[s10, null, d10];state carrier=[s10, m10, f10, d10];alphabet=[a10, b10]

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]=[m10, a10, m10];state carrier=[s10, m10, f10, d10];alphabet=[a10, b10]

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]=[m10, b10, f10];state carrier=[s10, m10, f10, d10];alphabet=[a10, b10]

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

ORIGINAL states=[s10, m10, f10, d10],alphabet=[a10, b10],start=s10,final_states=[f10];complete relation=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]

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=[s10, m10, f10, d10];ORIGINAL alphabet order=[a10, b10];complete relation=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]

ALL [state,symbol,ORIGINAL-order targets] rows=[[s10, a10, []], [s10, b10, []], [m10, a10, [m10]], [m10, b10, [f10]], [f10, a10, []], [f10, b10, []], [d10, a10, []], [d10, b10, []]];EMPTY target rows=[[s10, a10, []], [s10, b10, []], [f10, a10, []], [f10, b10, []], [d10, a10, []], [d10, b10, []]];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

([s10, m10, f10, d10], [a10, b10], s10, [f10], [[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]], [[s10, a10, []], [s10, b10, []], [f10, a10, []], [f10, b10, []], [d10, a10, []], [d10, b10, []]], 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.