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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Subset construction from an NFA to a DFA · qf-mod43-subsetdfa-full-word-trace-invariant · candidate
Canonical question JSONL · English JSONL · Validation report