R0045-23255 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0030
Original question and givens
Use the complete finite epsilon-NFA: states=[s4, m4, f4, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]. 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=[s4]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
G001 — Use the complete finite epsilon-NFA: states=[s4, m4, f4, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]. 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=[s4]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
Step-by-step derivation
S001 · implies
Before
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 dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL transition index=0;[from,input_or_null,to]=[s4, null, m4];state carrier=[s4, m4, f4, d4];alphabet=[a4, b4]
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 dependencies
Rule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S001
S003 · implies
Before
ORIGINAL transition index=1;[from,input_or_null,to]=[s4, null, d4];state carrier=[s4, m4, f4, d4];alphabet=[a4, b4]
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 dependencies
Rule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S002
S004 · implies
Before
ORIGINAL transition index=2;[from,input_or_null,to]=[m4, a4, m4];state carrier=[s4, m4, f4, d4];alphabet=[a4, b4]
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 dependencies
Rule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S003
S005 · implies
Before
ORIGINAL transition index=3;[from,input_or_null,to]=[m4, b4, f4];state carrier=[s4, m4, f4, d4];alphabet=[a4, b4]
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 dependencies
Rule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S004
S006 · implies
Before
ORIGINAL states=[s4, m4, f4, d4],alphabet=[a4, b4],start=s4,final_states=[f4];complete relation=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]
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 dependencies
Rule: nfa_qualified_definition_EMPTY_alphabet_finals_edges_allowed · Uses: G001, S005
S007 · implies
Before
NFA ORIGINAL EPS closure:ORIGINAL seed list=[s4];complete state carrier=[s4, m4, f4, d4]
After
initialize unique closure=[s4];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 dependencies
Rule: nfa_EPS_closure_contains_ORIGINAL_seeds_duplicates_preserved · Uses: G001, S006
S008 · implies
Before
NFA ORIGINAL EPS closure:stage=0;ORIGINAL-relation EPS pass from previous set=[s4]
After
ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s4, m4, Yes], [1, s4, d4, Yes]];next unique set=[d4, m4, s4];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 dependencies
Rule: nfa_EVERY_EPS_saturation_stage_least_reachable_set · Uses: G001, S007
S009 · implies
Before
NFA ORIGINAL EPS closure:stage=1;ORIGINAL-relation EPS pass from previous set=[d4, m4, s4]
After
ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s4, m4, Yes], [1, s4, d4, Yes]];next unique set=[d4, m4, s4];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 dependencies
Rule: nfa_EVERY_EPS_saturation_stage_least_reachable_set · Uses: G001, S008