R0045-23238 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0088
Original question and givens
Use the complete finite epsilon-NFA: states=[s12, m12, f12, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]. 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=[b12], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
G001 — Use the complete finite epsilon-NFA: states=[s12, m12, f12, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]. 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=[b12], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
Step-by-step derivation
S001 · implies
Before
source operation=nfa-bounded-search-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[input, max_steps, found_accepting_path, rejection_claim, boundary]
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]=[s12, null, m12];state carrier=[s12, m12, f12, d12];alphabet=[a12, b12]
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]=[s12, null, d12];state carrier=[s12, m12, f12, d12];alphabet=[a12, b12]
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]=[m12, a12, m12];state carrier=[s12, m12, f12, d12];alphabet=[a12, b12]
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]=[m12, b12, f12];state carrier=[s12, m12, f12, d12];alphabet=[a12, b12]
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=[s12, m12, f12, d12],alphabet=[a12, b12],start=s12,final_states=[f12];complete relation=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]
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=[b12];actual start configuration=[s12, 0];max_steps=1
After
depth0 accepted=No;one finite FULL-input final path suffices;dead alternatives do not veto success
Conditions: none
The bound counts transitions:an EPS move still costs a step despite consuming no input.An initially final start accepts empty input.One finite full-input final path suffices;not all branches must succeed.
Step identifiers and dependencies
Rule: nfa_bounded_initial_and_finite_path_acceptance_scope · Uses: G001, S006
S008 · implies
Before
ORIGINAL depth=0;frontier in ORIGINAL-state/position display order=[[s12, 0]]
After
ALL [from,ORIGINAL transition index,next_or_null] rows=[[[s12, 0], 0, [m12, 0]], [[s12, 0], 1, [d12, 0]], [[s12, 0], 2, null], [[s12, 0], 3, null]];next distinct configs=[[m12, 0], [d12, 0]];hit FULL-input final=No
Conditions: none
Check every configuration and original relation row,with per-depth deduplication.Input position increases1 or0 according to symbol or null.An EPS cycle permits paths but is not an infinite accepting run.
Step identifiers and dependencies
Rule: nfa_ALL_nondeterministic_bounded_frontier_not_infinite_cycle · Uses: G001, S007
S009 · implies
Before
ORIGINAL bound=1;input=[b12];found_accepting_path=No
After
rejection_claim fixed No;bounded flag alone supplies no GENERAL rejection certificate;any actual completeness argument must be established separately
Conditions: none
The final No withholds a general rejection inference from finite tests,without negating found=True.If a finite model separately proves this bound complete,actual rejection may be decided;do not substitute that different completeness claim for this found flag.
Step identifiers and dependencies
Rule: nfa_bounded_No_rejection_claim_not_found_negation · Uses: G001, S008