R0045-23279 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0004
Original question and givens
Use the complete finite epsilon-NFA: states=[s3, m3, f3, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]. 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=[s3, m3, f3, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]. 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]=[s3, null, m3];state carrier=[s3, m3, f3, d3];alphabet=[a3, b3]
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]=[s3, null, d3];state carrier=[s3, m3, f3, d3];alphabet=[a3, b3]
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]=[m3, a3, m3];state carrier=[s3, m3, f3, d3];alphabet=[a3, b3]
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]=[m3, b3, f3];state carrier=[s3, m3, f3, d3];alphabet=[a3, b3]
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=[s3, m3, f3, d3],alphabet=[a3, b3],start=s3,final_states=[f3];complete relation=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]
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=[s3, m3, f3, d3];ORIGINAL alphabet order=[a3, b3];complete relation=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]
After
ALL [state,symbol,ORIGINAL-order targets] rows=[[s3, a3, []], [s3, b3, []], [m3, a3, [m3]], [m3, b3, [f3]], [f3, a3, []], [f3, b3, []], [d3, a3, []], [d3, b3, []]];EMPTY target rows=[[s3, a3, []], [s3, b3, []], [f3, a3, []], [f3, b3, []], [d3, a3, []], [d3, b3, []]];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
([s3, m3, f3, d3], [a3, b3], s3, [f3], [[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]], [[s3, a3, []], [s3, b3, []], [f3, a3, []], [f3, b3, []], [d3, a3, []], [d3, b3, []]], 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.