R0045-23256 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0031
Original question and givens
Use the complete finite epsilon-NFA: states=[s5, m5, f5, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]. 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=[s5]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
G001 — Use the complete finite epsilon-NFA: states=[s5, m5, f5, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]. 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=[s5]. 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]=[s5, null, m5];state carrier=[s5, m5, f5, d5];alphabet=[a5, b5]
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]=[s5, null, d5];state carrier=[s5, m5, f5, d5];alphabet=[a5, b5]
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]=[m5, a5, m5];state carrier=[s5, m5, f5, d5];alphabet=[a5, b5]
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]=[m5, b5, f5];state carrier=[s5, m5, f5, d5];alphabet=[a5, b5]
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=[s5, m5, f5, d5],alphabet=[a5, b5],start=s5,final_states=[f5];complete relation=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]
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=[s5];complete state carrier=[s5, m5, f5, d5]
After
initialize unique closure=[s5];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=[s5]
After
ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s5, m5, Yes], [1, s5, d5, Yes]];next unique set=[d5, m5, s5];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=[d5, m5, s5]
After
ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s5, m5, Yes], [1, s5, d5, Yes]];next unique set=[d5, m5, s5];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