# Epsilon-Closed Starts, Move-Closure Transitions and Prefix Invariants

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s0, m0, f0, e10, e20, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s0, [d0, m0, s0], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s0, [d0, m0, s0], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0001/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 2. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s1, m1, f1, e11, e21, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1], [f1, null, e11], [e11, null, e21]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s1, [d1, m1, s1], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s1, [d1, m1, s1], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0002/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 3. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s2, m2, f2, e12, e22, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s2, [d2, m2, s2], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s2, [d2, m2, s2], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0003/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 4. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s3, m3, f3, e13, e23, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s3, [d3, m3, s3], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s3, [d3, m3, s3], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0004/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 5. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s4, m4, f4, e14, e24, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s4, [d4, m4, s4], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s4, [d4, m4, s4], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0005/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 6. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s5, m5, f5, e15, e25, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s5, [d5, m5, s5], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s5, [d5, m5, s5], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0006/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 7. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s6, m6, f6, e16, e26, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6], [f6, null, e16], [e16, null, e26]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s6, [d6, m6, s6], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s6, [d6, m6, s6], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0007/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 8. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s7, [d7, m7, s7], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s7, [d7, m7, s7], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0008/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 9. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s8, m8, f8, e18, e28, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s8, [d8, m8, s8], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s8, [d8, m8, s8], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0009/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 10. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s9, m9, f9, e19, e29, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9], [f9, null, e19], [e19, null, e29]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s9, [d9, m9, s9], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s9, [d9, m9, s9], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0010/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 11. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s10, m10, f10, e110, e210, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10], [f10, null, e110], [e110, null, e210]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s10, [d10, m10, s10], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s10, [d10, m10, s10], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0011/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 12. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s11, m11, f11, e111, e211, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s11, [d11, m11, s11], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s11, [d11, m11, s11], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0012/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 13. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s12, m12, f12, e112, e212, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s12, [d12, m12, s12], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s12, [d12, m12, s12], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0013/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 14. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s13, m13, f13, e113, e213, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13], [f13, null, e113], [e113, null, e213]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s13, [d13, m13, s13], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s13, [d13, m13, s13], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0014/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 15. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s14, m14, f14, e114, e214, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14], [f14, null, e114], [e114, null, e214]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s14, [d14, m14, s14], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s14, [d14, m14, s14], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0015/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 16. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s15, [d15, m15, s15], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s15, [d15, m15, s15], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0016/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 17. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s16, m16, f16, e116, e216, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16], [f16, null, e116], [e116, null, e216]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s16, [d16, m16, s16], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s16, [d16, m16, s16], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0017/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 18. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s17, m17, f17, e117, e217, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17], [f17, null, e117], [e117, null, e217]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s17, [d17, m17, s17], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s17, [d17, m17, s17], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0018/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 19. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s18, m18, f18, e118, e218, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s18, [d18, m18, s18], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s18, [d18, m18, s18], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0019/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 20. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s19, m19, f19, e119, e219, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s19, [d19, m19, s19], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s19, [d19, m19, s19], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0020/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 21. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s20, m20, f20, e120, e220, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s20, [d20, m20, s20], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s20, [d20, m20, s20], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0021/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 22. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s21, [d21, m21, s21], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s21, [d21, m21, s21], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0022/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 23. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s22, m22, f22, e122, e222, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22], [f22, null, e122], [e122, null, e222]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s22, [d22, m22, s22], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s22, [d22, m22, s22], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0023/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 24. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s23, m23, f23, e123, e223, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23], [f23, null, e123], [e123, null, e223]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s23, [d23, m23, s23], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s23, [d23, m23, s23], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0024/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 25. The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Use complete epsilon-NFA states=[s24, m24, f24, e124, e224, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24], [f24, null, e124], [e124, null, e224]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

View answer and explanation(s24, [d24, m24, s24], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s24, [d24, m24, s24], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0025/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-epsilon-start · candidate





## 26. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s0, m0, f0, e10, e20, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s0, m0, d0], input symbol=a0. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d0, m0, s0], a0, [m0], [m0], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d0, m0, s0], a0, [m0], [m0], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0026/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 27. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s1, m1, f1, e11, e21, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1], [f1, null, e11], [e11, null, e21]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s1, m1, d1], input symbol=b1. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d1, m1, s1], b1, [f1], [e11, e21, f1], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d1, m1, s1], b1, [f1], [e11, e21, f1], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0027/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 28. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s2, m2, f2, e12, e22, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s2, m2, d2], input symbol=a2. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d2, m2, s2], a2, [m2], [m2], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d2, m2, s2], a2, [m2], [m2], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0028/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 29. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s3, m3, f3, e13, e23, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s3, m3, d3], input symbol=b3. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d3, m3, s3], b3, [f3], [e13, e23, f3], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d3, m3, s3], b3, [f3], [e13, e23, f3], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0029/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 30. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s4, m4, f4, e14, e24, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s4, m4, d4], input symbol=a4. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d4, m4, s4], a4, [m4], [m4], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d4, m4, s4], a4, [m4], [m4], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0030/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 31. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s5, m5, f5, e15, e25, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s5, m5, d5], input symbol=b5. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d5, m5, s5], b5, [f5], [e15, e25, f5], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d5, m5, s5], b5, [f5], [e15, e25, f5], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0031/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 32. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s6, m6, f6, e16, e26, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6], [f6, null, e16], [e16, null, e26]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s6, m6, d6], input symbol=a6. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d6, m6, s6], a6, [m6], [m6], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d6, m6, s6], a6, [m6], [m6], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0032/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 33. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s7, m7, d7], input symbol=b7. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d7, m7, s7], b7, [f7], [e17, e27, f7], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d7, m7, s7], b7, [f7], [e17, e27, f7], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0033/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 34. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s8, m8, f8, e18, e28, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s8, m8, d8], input symbol=a8. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d8, m8, s8], a8, [m8], [m8], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d8, m8, s8], a8, [m8], [m8], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0034/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 35. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s9, m9, f9, e19, e29, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9], [f9, null, e19], [e19, null, e29]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s9, m9, d9], input symbol=b9. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d9, m9, s9], b9, [f9], [e19, e29, f9], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d9, m9, s9], b9, [f9], [e19, e29, f9], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0035/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 36. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s10, m10, f10, e110, e210, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10], [f10, null, e110], [e110, null, e210]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s10, m10, d10], input symbol=a10. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d10, m10, s10], a10, [m10], [m10], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d10, m10, s10], a10, [m10], [m10], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0036/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 37. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s11, m11, f11, e111, e211, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s11, m11, d11], input symbol=b11. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d11, m11, s11], b11, [f11], [e111, e211, f11], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d11, m11, s11], b11, [f11], [e111, e211, f11], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0037/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 38. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s12, m12, f12, e112, e212, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s12, m12, d12], input symbol=a12. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d12, m12, s12], a12, [m12], [m12], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d12, m12, s12], a12, [m12], [m12], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0038/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 39. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s13, m13, f13, e113, e213, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13], [f13, null, e113], [e113, null, e213]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s13, m13, d13], input symbol=b13. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d13, m13, s13], b13, [f13], [e113, e213, f13], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d13, m13, s13], b13, [f13], [e113, e213, f13], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0039/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 40. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s14, m14, f14, e114, e214, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14], [f14, null, e114], [e114, null, e214]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s14, m14, d14], input symbol=a14. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d14, m14, s14], a14, [m14], [m14], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d14, m14, s14], a14, [m14], [m14], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0040/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 41. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s15, m15, d15], input symbol=b15. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d15, m15, s15], b15, [f15], [e115, e215, f15], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d15, m15, s15], b15, [f15], [e115, e215, f15], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0041/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 42. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s16, m16, f16, e116, e216, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16], [f16, null, e116], [e116, null, e216]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s16, m16, d16], input symbol=a16. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d16, m16, s16], a16, [m16], [m16], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d16, m16, s16], a16, [m16], [m16], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0042/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 43. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s17, m17, f17, e117, e217, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17], [f17, null, e117], [e117, null, e217]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s17, m17, d17], input symbol=b17. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d17, m17, s17], b17, [f17], [e117, e217, f17], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d17, m17, s17], b17, [f17], [e117, e217, f17], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0043/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 44. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s18, m18, f18, e118, e218, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s18, m18, d18], input symbol=a18. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d18, m18, s18], a18, [m18], [m18], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d18, m18, s18], a18, [m18], [m18], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0044/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 45. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s19, m19, f19, e119, e219, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s19, m19, d19], input symbol=b19. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d19, m19, s19], b19, [f19], [e119, e219, f19], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d19, m19, s19], b19, [f19], [e119, e219, f19], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0045/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 46. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s20, m20, f20, e120, e220, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s20, m20, d20], input symbol=a20. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d20, m20, s20], a20, [m20], [m20], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d20, m20, s20], a20, [m20], [m20], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0046/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 47. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s21, m21, d21], input symbol=b21. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d21, m21, s21], b21, [f21], [e121, e221, f21], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d21, m21, s21], b21, [f21], [e121, e221, f21], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0047/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 48. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s22, m22, f22, e122, e222, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22], [f22, null, e122], [e122, null, e222]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s22, m22, d22], input symbol=a22. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d22, m22, s22], a22, [m22], [m22], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d22, m22, s22], a22, [m22], [m22], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0048/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 49. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s23, m23, f23, e123, e223, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23], [f23, null, e123], [e123, null, e223]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s23, m23, d23], input symbol=b23. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d23, m23, s23], b23, [f23], [e123, e223, f23], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d23, m23, s23], b23, [f23], [e123, e223, f23], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0049/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 50. A DFA Subset Transition Applies Symbol Move before Epsilon Closure

Use complete epsilon-NFA states=[s24, m24, f24, e124, e224, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24], [f24, null, e124], [e124, null, e224]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s24, m24, d24], input symbol=a24. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).

For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant.

View answer and explanation([d24, m24, s24], a24, [m24], [m24], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d24, m24, s24], a24, [m24], [m24], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0050/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-symbol-transition · candidate





## 51. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s0, m0, f0, e10, e20, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f0, m0]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f0, m0], [f0], [f0], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f0, m0], [f0], [f0], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0051/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 52. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s1, m1, f1, e11, e21, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1], [f1, null, e11], [e11, null, e21]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m1, d1]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d1, m1], [f1], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d1, m1], [f1], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0052/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 53. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s2, m2, f2, e12, e22, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f2, m2]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f2, m2], [f2], [f2], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f2, m2], [f2], [f2], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0053/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 54. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s3, m3, f3, e13, e23, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m3, d3]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d3, m3], [f3], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d3, m3], [f3], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0054/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 55. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s4, m4, f4, e14, e24, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f4, m4]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f4, m4], [f4], [f4], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f4, m4], [f4], [f4], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0055/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 56. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s5, m5, f5, e15, e25, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m5, d5]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d5, m5], [f5], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d5, m5], [f5], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0056/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 57. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s6, m6, f6, e16, e26, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6], [f6, null, e16], [e16, null, e26]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f6, m6]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f6, m6], [f6], [f6], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f6, m6], [f6], [f6], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0057/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 58. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m7, d7]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d7, m7], [f7], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d7, m7], [f7], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0058/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 59. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s8, m8, f8, e18, e28, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f8, m8]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f8, m8], [f8], [f8], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f8, m8], [f8], [f8], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0059/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 60. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s9, m9, f9, e19, e29, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9], [f9, null, e19], [e19, null, e29]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m9, d9]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d9, m9], [f9], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d9, m9], [f9], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0060/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 61. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s10, m10, f10, e110, e210, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10], [f10, null, e110], [e110, null, e210]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f10, m10]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f10, m10], [f10], [f10], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f10, m10], [f10], [f10], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0061/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 62. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s11, m11, f11, e111, e211, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m11, d11]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d11, m11], [f11], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d11, m11], [f11], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0062/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 63. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s12, m12, f12, e112, e212, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f12, m12]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f12, m12], [f12], [f12], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f12, m12], [f12], [f12], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0063/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 64. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s13, m13, f13, e113, e213, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13], [f13, null, e113], [e113, null, e213]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m13, d13]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d13, m13], [f13], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d13, m13], [f13], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0064/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 65. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s14, m14, f14, e114, e214, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14], [f14, null, e114], [e114, null, e214]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f14, m14]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f14, m14], [f14], [f14], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f14, m14], [f14], [f14], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0065/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 66. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m15, d15]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d15, m15], [f15], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d15, m15], [f15], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0066/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 67. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s16, m16, f16, e116, e216, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16], [f16, null, e116], [e116, null, e216]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f16, m16]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f16, m16], [f16], [f16], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f16, m16], [f16], [f16], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0067/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 68. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s17, m17, f17, e117, e217, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17], [f17, null, e117], [e117, null, e217]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m17, d17]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d17, m17], [f17], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d17, m17], [f17], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0068/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 69. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s18, m18, f18, e118, e218, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f18, m18]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f18, m18], [f18], [f18], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f18, m18], [f18], [f18], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0069/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 70. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s19, m19, f19, e119, e219, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m19, d19]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d19, m19], [f19], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d19, m19], [f19], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0070/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 71. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s20, m20, f20, e120, e220, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f20, m20]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f20, m20], [f20], [f20], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f20, m20], [f20], [f20], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0071/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 72. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m21, d21]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d21, m21], [f21], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d21, m21], [f21], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0072/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 73. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s22, m22, f22, e122, e222, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22], [f22, null, e122], [e122, null, e222]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f22, m22]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f22, m22], [f22], [f22], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f22, m22], [f22], [f22], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0073/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 74. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s23, m23, f23, e123, e223, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23], [f23, null, e123], [e123, null, e223]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m23, d23]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([d23, m23], [f23], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d23, m23], [f23], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0074/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 75. A Subset Accepts Exactly When It Intersects the NFA Final Set

Use complete epsilon-NFA states=[s24, m24, f24, e124, e224, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24], [f24, null, e124], [e124, null, e224]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f24, m24]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).

View answer and explanation([f24, m24], [f24], [f24], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)

Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f24, m24], [f24], [f24], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0075/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-accepting-subset · candidate





## 76. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s0, m0, f0, e10, e20, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b0]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([b0], [[d0, m0, s0], [e10, e20, f0]], [[d0, m0, s0], [e10, e20, f0]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b0], [[d0, m0, s0], [e10, e20, f0]], [[d0, m0, s0], [e10, e20, f0]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0076/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 77. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s1, m1, f1, e11, e21, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1], [f1, null, e11], [e11, null, e21]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a1, b1]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a1, b1], [[d1, m1, s1], [m1], [e11, e21, f1]], [[d1, m1, s1], [m1], [e11, e21, f1]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a1, b1], [[d1, m1, s1], [m1], [e11, e21, f1]], [[d1, m1, s1], [m1], [e11, e21, f1]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0077/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 78. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s2, m2, f2, e12, e22, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a2, a2, b2]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a2, a2, b2], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a2, a2, b2], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0078/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 79. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s3, m3, f3, e13, e23, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a3, a3, a3, b3]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a3, a3, a3, b3], [[d3, m3, s3], [m3], [m3], [m3], [e13, e23, f3]], [[d3, m3, s3], [m3], [m3], [m3], [e13, e23, f3]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a3, a3, a3, b3], [[d3, m3, s3], [m3], [m3], [m3], [e13, e23, f3]], [[d3, m3, s3], [m3], [m3], [m3], [e13, e23, f3]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0079/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 80. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s4, m4, f4, e14, e24, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b4]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([b4], [[d4, m4, s4], [e14, e24, f4]], [[d4, m4, s4], [e14, e24, f4]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b4], [[d4, m4, s4], [e14, e24, f4]], [[d4, m4, s4], [e14, e24, f4]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0080/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 81. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s5, m5, f5, e15, e25, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a5, b5]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a5, b5], [[d5, m5, s5], [m5], [e15, e25, f5]], [[d5, m5, s5], [m5], [e15, e25, f5]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a5, b5], [[d5, m5, s5], [m5], [e15, e25, f5]], [[d5, m5, s5], [m5], [e15, e25, f5]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0081/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 82. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s6, m6, f6, e16, e26, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6], [f6, null, e16], [e16, null, e26]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a6, a6, b6]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a6, a6, b6], [[d6, m6, s6], [m6], [m6], [e16, e26, f6]], [[d6, m6, s6], [m6], [m6], [e16, e26, f6]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a6, a6, b6], [[d6, m6, s6], [m6], [m6], [e16, e26, f6]], [[d6, m6, s6], [m6], [m6], [e16, e26, f6]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0082/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 83. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a7, a7, a7, b7]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a7, a7, a7, b7], [[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]], [[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a7, a7, a7, b7], [[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]], [[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0083/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 84. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s8, m8, f8, e18, e28, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b8]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([b8], [[d8, m8, s8], [e18, e28, f8]], [[d8, m8, s8], [e18, e28, f8]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b8], [[d8, m8, s8], [e18, e28, f8]], [[d8, m8, s8], [e18, e28, f8]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0084/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 85. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s9, m9, f9, e19, e29, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9], [f9, null, e19], [e19, null, e29]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a9, b9]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a9, b9], [[d9, m9, s9], [m9], [e19, e29, f9]], [[d9, m9, s9], [m9], [e19, e29, f9]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a9, b9], [[d9, m9, s9], [m9], [e19, e29, f9]], [[d9, m9, s9], [m9], [e19, e29, f9]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0085/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 86. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s10, m10, f10, e110, e210, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10], [f10, null, e110], [e110, null, e210]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a10, a10, b10]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a10, a10, b10], [[d10, m10, s10], [m10], [m10], [e110, e210, f10]], [[d10, m10, s10], [m10], [m10], [e110, e210, f10]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a10, a10, b10], [[d10, m10, s10], [m10], [m10], [e110, e210, f10]], [[d10, m10, s10], [m10], [m10], [e110, e210, f10]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0086/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 87. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s11, m11, f11, e111, e211, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a11, a11, a11, b11]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a11, a11, a11, b11], [[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]], [[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a11, a11, a11, b11], [[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]], [[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0087/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 88. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s12, m12, f12, e112, e212, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b12]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([b12], [[d12, m12, s12], [e112, e212, f12]], [[d12, m12, s12], [e112, e212, f12]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b12], [[d12, m12, s12], [e112, e212, f12]], [[d12, m12, s12], [e112, e212, f12]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0088/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 89. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s13, m13, f13, e113, e213, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13], [f13, null, e113], [e113, null, e213]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a13, b13]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a13, b13], [[d13, m13, s13], [m13], [e113, e213, f13]], [[d13, m13, s13], [m13], [e113, e213, f13]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a13, b13], [[d13, m13, s13], [m13], [e113, e213, f13]], [[d13, m13, s13], [m13], [e113, e213, f13]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0089/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 90. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s14, m14, f14, e114, e214, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14], [f14, null, e114], [e114, null, e214]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a14, a14, b14]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a14, a14, b14], [[d14, m14, s14], [m14], [m14], [e114, e214, f14]], [[d14, m14, s14], [m14], [m14], [e114, e214, f14]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a14, a14, b14], [[d14, m14, s14], [m14], [m14], [e114, e214, f14]], [[d14, m14, s14], [m14], [m14], [e114, e214, f14]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0090/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 91. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a15, a15, a15, b15]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a15, a15, a15, b15], [[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]], [[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a15, a15, a15, b15], [[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]], [[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0091/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 92. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s16, m16, f16, e116, e216, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16], [f16, null, e116], [e116, null, e216]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b16]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([b16], [[d16, m16, s16], [e116, e216, f16]], [[d16, m16, s16], [e116, e216, f16]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b16], [[d16, m16, s16], [e116, e216, f16]], [[d16, m16, s16], [e116, e216, f16]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0092/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 93. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s17, m17, f17, e117, e217, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17], [f17, null, e117], [e117, null, e217]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a17, b17]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a17, b17], [[d17, m17, s17], [m17], [e117, e217, f17]], [[d17, m17, s17], [m17], [e117, e217, f17]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a17, b17], [[d17, m17, s17], [m17], [e117, e217, f17]], [[d17, m17, s17], [m17], [e117, e217, f17]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0093/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 94. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s18, m18, f18, e118, e218, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a18, a18, b18]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a18, a18, b18], [[d18, m18, s18], [m18], [m18], [e118, e218, f18]], [[d18, m18, s18], [m18], [m18], [e118, e218, f18]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a18, a18, b18], [[d18, m18, s18], [m18], [m18], [e118, e218, f18]], [[d18, m18, s18], [m18], [m18], [e118, e218, f18]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0094/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 95. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s19, m19, f19, e119, e219, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a19, a19, a19, b19]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a19, a19, a19, b19], [[d19, m19, s19], [m19], [m19], [m19], [e119, e219, f19]], [[d19, m19, s19], [m19], [m19], [m19], [e119, e219, f19]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a19, a19, a19, b19], [[d19, m19, s19], [m19], [m19], [m19], [e119, e219, f19]], [[d19, m19, s19], [m19], [m19], [m19], [e119, e219, f19]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0095/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 96. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s20, m20, f20, e120, e220, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b20]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([b20], [[d20, m20, s20], [e120, e220, f20]], [[d20, m20, s20], [e120, e220, f20]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b20], [[d20, m20, s20], [e120, e220, f20]], [[d20, m20, s20], [e120, e220, f20]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0096/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 97. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a21, b21]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a21, b21], [[d21, m21, s21], [m21], [e121, e221, f21]], [[d21, m21, s21], [m21], [e121, e221, f21]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a21, b21], [[d21, m21, s21], [m21], [e121, e221, f21]], [[d21, m21, s21], [m21], [e121, e221, f21]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0097/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 98. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s22, m22, f22, e122, e222, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22], [f22, null, e122], [e122, null, e222]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a22, a22, b22]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a22, a22, b22], [[d22, m22, s22], [m22], [m22], [e122, e222, f22]], [[d22, m22, s22], [m22], [m22], [e122, e222, f22]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a22, a22, b22], [[d22, m22, s22], [m22], [m22], [e122, e222, f22]], [[d22, m22, s22], [m22], [m22], [e122, e222, f22]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0098/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 99. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s23, m23, f23, e123, e223, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23], [f23, null, e123], [e123, null, e223]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a23, a23, a23, b23]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([a23, a23, a23, b23], [[d23, m23, s23], [m23], [m23], [m23], [e123, e223, f23]], [[d23, m23, s23], [m23], [m23], [m23], [e123, e223, f23]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a23, a23, a23, b23], [[d23, m23, s23], [m23], [m23], [m23], [e123, e223, f23]], [[d23, m23, s23], [m23], [m23], [m23], [e123, e223, f23]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0099/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate





## 100. Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Use complete epsilon-NFA states=[s24, m24, f24, e124, e224, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24], [f24, null, e124], [e124, null, e224]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b24]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem.

View answer and explanation([b24], [[d24, m24, s24], [e124, e224, f24]], [[d24, m24, s24], [e124, e224, f24]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b24], [[d24, m24, s24], [e124, e224, f24]], [[d24, m24, s24], [e124, e224, f24]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0100/)

Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations/validation-report.json)