Question 23291: An Epsilon-NFA Relation May Be Empty, Partial or Multivalued

Original question and givens

Use the complete finite epsilon-NFA: states=[s15, m15, f15, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]. 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=[s15, m15, f15, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]. 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]=[s15, null, m15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15]

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]=[s15, null, d15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15]

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]=[m15, a15, m15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15]

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]=[m15, b15, f15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15]

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

ORIGINAL states=[s15, m15, f15, d15],alphabet=[a15, b15],start=s15,final_states=[f15];complete relation=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]

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=[s15, m15, f15, d15];ORIGINAL alphabet order=[a15, b15];complete relation=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]

ALL [state,symbol,ORIGINAL-order targets] rows=[[s15, a15, []], [s15, b15, []], [m15, a15, [m15]], [m15, b15, [f15]], [f15, a15, []], [f15, b15, []], [d15, a15, []], [d15, b15, []]];EMPTY target rows=[[s15, a15, []], [s15, b15, []], [f15, a15, []], [f15, b15, []], [d15, a15, []], [d15, b15, []]];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

([s15, m15, f15, d15], [a15, b15], s15, [f15], [[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]], [[s15, a15, []], [s15, b15, []], [f15, a15, []], [f15, b15, []], [d15, a15, []], [d15, b15, []]], 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.