R0045-23219 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0069
Original question and givens
Use the complete finite epsilon-NFA: states=[s18, m18, f18, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]. 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. Input=[a18, a18, b18], supplied path=[[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
G001 — Use the complete finite epsilon-NFA: states=[s18, m18, f18, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]. 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. Input=[a18, a18, b18], supplied path=[[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
Step-by-step derivation
S001 · implies
Before
source operation=nfa-accepting-path-witness;original snapshot and mathematics retained
After
explicit ordered source output fields=[input, path, transition_indices, valid_path, accepted, 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]=[s18, null, m18];state carrier=[s18, m18, f18, d18];alphabet=[a18, b18]
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]=[s18, null, d18];state carrier=[s18, m18, f18, d18];alphabet=[a18, b18]
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]=[m18, a18, m18];state carrier=[s18, m18, f18, d18];alphabet=[a18, b18]
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]=[m18, b18, f18];state carrier=[s18, m18, f18, d18];alphabet=[a18, b18]
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=[s18, m18, f18, d18],alphabet=[a18, b18],start=s18,final_states=[f18];complete relation=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]
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 input=[a18, a18, b18];ALL legal raw path configs=[[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]]
After
first configuration EXACT actual start=[s18, 0];this is not yet validity or acceptance
Conditions: none
First qualify every state/position and the exact initial configuration.A list of legal configurations need not have legal transitions.
Step identifiers and dependencies
Rule: nfa_actual_path_initial_and_structural_configurations_required · Uses: G001, S006
S008 · implies
Before
ORIGINAL path step=0;from=[s18, 0];to=[m18, 0];input=[a18, a18, b18]
After
ALL [ORIGINAL transition index,next_or_null] rows=[[0, [m18, 0]], [1, [d18, 0]], [2, null], [3, null]];FIRST matching index=0
Conditions: none
Take the first actual match in original transition order,including duplicate edges.Null advances0;a symbol advances1 and must match the original input.On failure retain prior partial indices;a later final state cannot repair the step.
Step identifiers and dependencies
Rule: nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · Uses: G001, S007
S009 · implies
Before
ORIGINAL path step=1;from=[m18, 0];to=[m18, 1];input=[a18, a18, b18]
After
ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, [m18, 1]], [3, null]];FIRST matching index=2
Conditions: none
Take the first actual match in original transition order,including duplicate edges.Null advances0;a symbol advances1 and must match the original input.On failure retain prior partial indices;a later final state cannot repair the step.
Step identifiers and dependencies
Rule: nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · Uses: G001, S008
S010 · implies
Before
ORIGINAL path step=2;from=[m18, 1];to=[m18, 2];input=[a18, a18, b18]
After
ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, [m18, 2]], [3, null]];FIRST matching index=2
Conditions: none
Take the first actual match in original transition order,including duplicate edges.Null advances0;a symbol advances1 and must match the original input.On failure retain prior partial indices;a later final state cannot repair the step.
Step identifiers and dependencies
Rule: nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · Uses: G001, S009
S011 · implies
Before
ORIGINAL path step=3;from=[m18, 2];to=[f18, 3];input=[a18, a18, b18]
After
ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, null], [3, [f18, 3]]];FIRST matching index=3
Conditions: none
Take the first actual match in original transition order,including duplicate edges.Null advances0;a symbol advances1 and must match the original input.On failure retain prior partial indices;a later final state cannot repair the step.
Step identifiers and dependencies
Rule: nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · Uses: G001, S010
S012 · implies
Before
ORIGINAL last config=[f18, 3];input length=3;matched prior indices=[0, 2, 2, 3]
After
valid_path=Yes;FULL input consumed=Yes;last final=Yes;accepted=Yes;one successful finite path suffices despite dead branches
Conditions: none
Keep valid separate from accepted:a premature final state does not accept.Only accepted=True gives the finite existential witness for this word,not determinism,totality or DFA equivalence.The tag cannot turn false into true.
Step identifiers and dependencies
Rule: nfa_valid_path_FULL_input_final_existential_acceptance · Uses: G001, S011
Result
([a18, a18, b18], [[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.