# Question 23279: An Epsilon-NFA Relation May Be Empty, Partial or Multivalued

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

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).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-nondeterministic-finite-automaton](https://ocme.evemisslab.com/en/mko/mko-nondeterministic-finite-automaton/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

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 dependenciesRule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S001





S003 · impliesBefore

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 dependenciesRule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S002





S004 · impliesBefore

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 dependenciesRule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S003





S005 · impliesBefore

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 dependenciesRule: nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · Uses: G001, S004





S006 · impliesBefore

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 dependenciesRule: nfa_qualified_definition_EMPTY_alphabet_finals_edges_allowed · Uses: G001, S005





S007 · impliesBefore

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 dependenciesRule: nfa_ORIGINAL_state_symbol_relation_targets_not_totality_assumption · Uses: G001, S006









## Reverse checks


- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## 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.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 45b0f34c341053cf9262c9436cdd210fc1f2cfe0d9b647f4f9c2ffb66c90cf16
- candidate record SHA-256: fe5ba2d7f4961ffc1436613fe46755c86bda7c970245456eb1fd397dc5e2d633
- semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
- question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0004.json)