R0045-23284 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0009
Original question and givens
Use the complete finite epsilon-NFA: states=[s8, m8, f8, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]. 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=[s8, m8, f8, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]. 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).
Step-by-step derivation
S001 · implies
Before
source operation=nfa-qualification;original snapshot and mathematics retained
After
explicit ordered source output fields=[states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention]
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]=[s8, null, m8];state carrier=[s8, m8, f8, d8];alphabet=[a8, b8]
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]=[s8, null, d8];state carrier=[s8, m8, f8, d8];alphabet=[a8, b8]
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]=[m8, a8, m8];state carrier=[s8, m8, f8, d8];alphabet=[a8, b8]
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]=[m8, b8, f8];state carrier=[s8, m8, f8, d8];alphabet=[a8, b8]
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=[s8, m8, f8, d8],alphabet=[a8, b8],start=s8,final_states=[f8];complete relation=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]
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
ORIGINAL state order=[s8, m8, f8, d8];ORIGINAL alphabet order=[a8, b8];complete relation=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]
After
ALL [state,symbol,ORIGINAL-order targets] rows=[[s8, a8, []], [s8, b8, []], [m8, a8, [m8]], [m8, b8, [f8]], [f8, a8, []], [f8, b8, []], [d8, a8, []], [d8, b8, []]];EMPTY target rows=[[s8, a8, []], [s8, b8, []], [f8, a8, []], [f8, b8, []], [d8, a8, []], [d8, b8, []]];no missing edge repaired
Conditions: none
List all targets in original state/symbol order,retaining transition order and duplicates.Empty targets are legal;an empty alphabet has no such pairs,and an empty final set does not become accepting.
Step identifiers and dependencies
Rule: nfa_ORIGINAL_state_symbol_relation_targets_not_totality_assumption · Uses: G001, S006
Result
([s8, m8, f8, d8], [a8, b8], s8, [f8], [[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]], [[s8, a8, []], [s8, b8, []], [f8, a8, []], [f8, b8, []], [d8, a8, []], [d8, b8, []]], 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.