# Question 23266: Epsilon Closure Reaches States without Consuming Input [Return to catalogue page 117](https://ocme.evemisslab.com/en/questions/page/117/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations/#q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0041) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0045-23266 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0041 ## 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. Seed states=[s15]. Give (seed_states, epsilon_closure, closure_certificate, certificate). - 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. Seed states=[s15]. Give (seed_states, epsilon_closure, closure_certificate, certificate). ## 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](https://ocme.evemisslab.com/en/mko/mko-nondeterministic-finite-automaton/) ## Step-by-step derivation S001 · impliesBefore source operation=nfa-epsilon-closure;original snapshot and mathematics retained After explicit ordered source output fields=[seed_states, epsilon_closure, closure_certificate, certificate] Conditions: none Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL transition index=0;[from,input_or_null,to]=[s15, null, m15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15] After state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows Conditions: none This is a relation,not a total single-valued function.Retain duplicate,multivalued and missing edges;null consumes0 and is not an alphabet string. Step identifiers and dependenciesRule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S001 S003 · impliesBefore ORIGINAL transition index=1;[from,input_or_null,to]=[s15, null, d15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15] After state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows Conditions: none This is a relation,not a total single-valued function.Retain duplicate,multivalued and missing edges;null consumes0 and is not an alphabet string. Step identifiers and dependenciesRule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S002 S004 · impliesBefore ORIGINAL transition index=2;[from,input_or_null,to]=[m15, a15, m15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15] After state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows Conditions: none This is a relation,not a total single-valued function.Retain duplicate,multivalued and missing edges;null consumes0 and is not an alphabet string. Step identifiers and dependenciesRule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S003 S005 · impliesBefore ORIGINAL transition index=3;[from,input_or_null,to]=[m15, b15, f15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15] After state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows Conditions: none This is a relation,not a total single-valued function.Retain duplicate,multivalued and missing edges;null consumes0 and is not an alphabet string. Step identifiers and dependenciesRule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S004 S006 · impliesBefore 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]] After 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 Conditions: none An empty final set is legal and recognizes the empty language;an empty alphabet is legal too.A qualified definition supplies no accepting path,determinism or DFA-equivalence proof. Step identifiers and dependenciesRule: nfa_qualified_definition_EMPTY_alphabet_finals_edges_allowed · Uses: G001, S005 S007 · impliesBefore NFA ORIGINAL EPS closure:ORIGINAL seed list=[s15];complete state carrier=[s15, m15, f15, d15] After initialize unique closure=[s15];seeds themselves are reachable by zero EPS steps;RAW seed list not changed Conditions: none Seeds may be empty or repeated.Retain the raw seed list while the closure set deduplicates and includes zero-step reachability.An EPS cycle alone gives no acceptance. Step identifiers and dependenciesRule: nfa_EPS_closure_contains_ORIGINAL_seeds_duplicates_preserved · Uses: G001, S006 S008 · impliesBefore NFA ORIGINAL EPS closure:stage=0;ORIGINAL-relation EPS pass from previous set=[s15] After ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s15, m15, Yes], [1, s15, d15, Yes]];next unique set=[d15, m15, s15];stable=No Conditions: none Each added state has an EPS predecessor in the preceding set and hence a finite seed path.Any EPS-closed superset of the seeds contains every stage.Finiteness gives stabilization at the complete least closure,without consuming input. Step identifiers and dependenciesRule: nfa_EVERY_EPS_saturation_stage_least_reachable_set · Uses: G001, S007 S009 · impliesBefore NFA ORIGINAL EPS closure:stage=1;ORIGINAL-relation EPS pass from previous set=[d15, m15, s15] After ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s15, m15, Yes], [1, s15, d15, Yes]];next unique set=[d15, m15, s15];stable=Yes Conditions: none Each added state has an EPS predecessor in the preceding set and hence a finite seed path.Any EPS-closed superset of the seeds contains every stage.Finiteness gives stabilization at the complete least closure,without consuming input. Step identifiers and dependenciesRule: nfa_EVERY_EPS_saturation_stage_least_reachable_set · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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], [d15, m15, s15], Yes, finite_epsilon_reachability_without_input_consumption) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 6a63015d7960b170073dbdf16c9fd9dcb987df3d70870e28fe15affc58df5ef6 - candidate record SHA-256: 478e81a14f6c29d37bd59ea2fc14c398ff5c244edf43414667bfd2e565e0c6b0 - semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438 - question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0041.json)