Finite Epsilon Closures, Existential Accepting Paths and Bounded-Search Limits
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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-20261111-r07-nondeterministic-finite-automaton-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. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s0, m0, f0, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s0, m0, f0, d0], [a0, b0], s0, [f0], [[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]], [[s0, a0, []], [s0, b0, []], [f0, a0, []], [f0, b0, []], [d0, a0, []], [d0, b0, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s0, m0, f0, d0], [a0, b0], s0, [f0], [[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]], [[s0, a0, []], [s0, b0, []], [f0, a0, []], [f0, b0, []], [d0, a0, []], [d0, b0, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
2. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s1, m1, f1, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s1, m1, f1, d1], [a1, b1], s1, [f1], [[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]], [[s1, a1, []], [s1, b1, []], [f1, a1, []], [f1, b1, []], [d1, a1, []], [d1, b1, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s1, m1, f1, d1], [a1, b1], s1, [f1], [[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]], [[s1, a1, []], [s1, b1, []], [f1, a1, []], [f1, b1, []], [d1, a1, []], [d1, b1, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
3. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s2, m2, f2, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s2, m2, f2, d2], [a2, b2], s2, [f2], [[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]], [[s2, a2, []], [s2, b2, []], [f2, a2, []], [f2, b2, []], [d2, a2, []], [d2, b2, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s2, m2, f2, d2], [a2, b2], s2, [f2], [[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]], [[s2, a2, []], [s2, b2, []], [f2, a2, []], [f2, b2, []], [d2, a2, []], [d2, b2, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
4. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s3, m3, f3, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s3, m3, f3, d3], [a3, b3], s3, [f3], [[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]], [[s3, a3, []], [s3, b3, []], [f3, a3, []], [f3, b3, []], [d3, a3, []], [d3, b3, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s3, m3, f3, d3], [a3, b3], s3, [f3], [[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]], [[s3, a3, []], [s3, b3, []], [f3, a3, []], [f3, b3, []], [d3, a3, []], [d3, b3, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
5. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s4, m4, f4, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s4, m4, f4, d4], [a4, b4], s4, [f4], [[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]], [[s4, a4, []], [s4, b4, []], [f4, a4, []], [f4, b4, []], [d4, a4, []], [d4, b4, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s4, m4, f4, d4], [a4, b4], s4, [f4], [[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]], [[s4, a4, []], [s4, b4, []], [f4, a4, []], [f4, b4, []], [d4, a4, []], [d4, b4, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
6. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s5, m5, f5, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s5, m5, f5, d5], [a5, b5], s5, [f5], [[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]], [[s5, a5, []], [s5, b5, []], [f5, a5, []], [f5, b5, []], [d5, a5, []], [d5, b5, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s5, m5, f5, d5], [a5, b5], s5, [f5], [[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]], [[s5, a5, []], [s5, b5, []], [f5, a5, []], [f5, b5, []], [d5, a5, []], [d5, b5, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
7. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s6, m6, f6, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s6, m6, f6, d6], [a6, b6], s6, [f6], [[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]], [[s6, a6, []], [s6, b6, []], [f6, a6, []], [f6, b6, []], [d6, a6, []], [d6, b6, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s6, m6, f6, d6], [a6, b6], s6, [f6], [[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]], [[s6, a6, []], [s6, b6, []], [f6, a6, []], [f6, b6, []], [d6, a6, []], [d6, b6, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
8. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s7, m7, f7, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s7, m7, f7, d7], [a7, b7], s7, [f7], [[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]], [[s7, a7, []], [s7, b7, []], [f7, a7, []], [f7, b7, []], [d7, a7, []], [d7, b7, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s7, m7, f7, d7], [a7, b7], s7, [f7], [[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]], [[s7, a7, []], [s7, b7, []], [f7, a7, []], [f7, b7, []], [d7, a7, []], [d7, b7, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
9. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s8, m8, f8, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s8, m8, f8, d8], [a8, b8], s8, [f8], [[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]], [[s8, a8, []], [s8, b8, []], [f8, a8, []], [f8, b8, []], [d8, a8, []], [d8, b8, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s8, m8, f8, d8], [a8, b8], s8, [f8], [[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]], [[s8, a8, []], [s8, b8, []], [f8, a8, []], [f8, b8, []], [d8, a8, []], [d8, b8, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
10. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s9, m9, f9, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s9, m9, f9, d9], [a9, b9], s9, [f9], [[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]], [[s9, a9, []], [s9, b9, []], [f9, a9, []], [f9, b9, []], [d9, a9, []], [d9, b9, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s9, m9, f9, d9], [a9, b9], s9, [f9], [[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]], [[s9, a9, []], [s9, b9, []], [f9, a9, []], [f9, b9, []], [d9, a9, []], [d9, b9, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
11. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s10, m10, f10, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s10, m10, f10, d10], [a10, b10], s10, [f10], [[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]], [[s10, a10, []], [s10, b10, []], [f10, a10, []], [f10, b10, []], [d10, a10, []], [d10, b10, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s10, m10, f10, d10], [a10, b10], s10, [f10], [[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]], [[s10, a10, []], [s10, b10, []], [f10, a10, []], [f10, b10, []], [d10, a10, []], [d10, b10, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
12. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s11, m11, f11, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s11, m11, f11, d11], [a11, b11], s11, [f11], [[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]], [[s11, a11, []], [s11, b11, []], [f11, a11, []], [f11, b11, []], [d11, a11, []], [d11, b11, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s11, m11, f11, d11], [a11, b11], s11, [f11], [[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]], [[s11, a11, []], [s11, b11, []], [f11, a11, []], [f11, b11, []], [d11, a11, []], [d11, b11, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
13. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s12, m12, f12, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s12, m12, f12, d12], [a12, b12], s12, [f12], [[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]], [[s12, a12, []], [s12, b12, []], [f12, a12, []], [f12, b12, []], [d12, a12, []], [d12, b12, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s12, m12, f12, d12], [a12, b12], s12, [f12], [[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]], [[s12, a12, []], [s12, b12, []], [f12, a12, []], [f12, b12, []], [d12, a12, []], [d12, b12, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
14. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s13, m13, f13, d13], [a13, b13], s13, [f13], [[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]], [[s13, a13, []], [s13, b13, []], [f13, a13, []], [f13, b13, []], [d13, a13, []], [d13, b13, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s13, m13, f13, d13], [a13, b13], s13, [f13], [[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]], [[s13, a13, []], [s13, b13, []], [f13, a13, []], [f13, b13, []], [d13, a13, []], [d13, b13, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
15. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s14, m14, f14, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s14, m14, f14, d14], [a14, b14], s14, [f14], [[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]], [[s14, a14, []], [s14, b14, []], [f14, a14, []], [f14, b14, []], [d14, a14, []], [d14, b14, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s14, m14, f14, d14], [a14, b14], s14, [f14], [[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]], [[s14, a14, []], [s14, b14, []], [f14, a14, []], [f14, b14, []], [d14, a14, []], [d14, b14, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
16. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s15, m15, f15, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s15, m15, f15, d15], [a15, b15], s15, [f15], [[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]], [[s15, a15, []], [s15, b15, []], [f15, a15, []], [f15, b15, []], [d15, a15, []], [d15, b15, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s15, m15, f15, d15], [a15, b15], s15, [f15], [[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]], [[s15, a15, []], [s15, b15, []], [f15, a15, []], [f15, b15, []], [d15, a15, []], [d15, b15, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
17. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s16, m16, f16, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s16, m16, f16, d16], [a16, b16], s16, [f16], [[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]], [[s16, a16, []], [s16, b16, []], [f16, a16, []], [f16, b16, []], [d16, a16, []], [d16, b16, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s16, m16, f16, d16], [a16, b16], s16, [f16], [[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]], [[s16, a16, []], [s16, b16, []], [f16, a16, []], [f16, b16, []], [d16, a16, []], [d16, b16, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
18. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s17, m17, f17, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s17, m17, f17, d17], [a17, b17], s17, [f17], [[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]], [[s17, a17, []], [s17, b17, []], [f17, a17, []], [f17, b17, []], [d17, a17, []], [d17, b17, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s17, m17, f17, d17], [a17, b17], s17, [f17], [[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]], [[s17, a17, []], [s17, b17, []], [f17, a17, []], [f17, b17, []], [d17, a17, []], [d17, b17, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
19. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s18, m18, f18, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s18, m18, f18, d18], [a18, b18], s18, [f18], [[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]], [[s18, a18, []], [s18, b18, []], [f18, a18, []], [f18, b18, []], [d18, a18, []], [d18, b18, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s18, m18, f18, d18], [a18, b18], s18, [f18], [[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]], [[s18, a18, []], [s18, b18, []], [f18, a18, []], [f18, b18, []], [d18, a18, []], [d18, b18, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
20. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s19, m19, f19, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s19, m19, f19, d19], [a19, b19], s19, [f19], [[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]], [[s19, a19, []], [s19, b19, []], [f19, a19, []], [f19, b19, []], [d19, a19, []], [d19, b19, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s19, m19, f19, d19], [a19, b19], s19, [f19], [[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]], [[s19, a19, []], [s19, b19, []], [f19, a19, []], [f19, b19, []], [d19, a19, []], [d19, b19, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
21. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s20, m20, f20, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s20, m20, f20, d20], [a20, b20], s20, [f20], [[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]], [[s20, a20, []], [s20, b20, []], [f20, a20, []], [f20, b20, []], [d20, a20, []], [d20, b20, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s20, m20, f20, d20], [a20, b20], s20, [f20], [[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]], [[s20, a20, []], [s20, b20, []], [f20, a20, []], [f20, b20, []], [d20, a20, []], [d20, b20, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
22. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s21, m21, f21, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s21, m21, f21, d21], [a21, b21], s21, [f21], [[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]], [[s21, a21, []], [s21, b21, []], [f21, a21, []], [f21, b21, []], [d21, a21, []], [d21, b21, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s21, m21, f21, d21], [a21, b21], s21, [f21], [[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]], [[s21, a21, []], [s21, b21, []], [f21, a21, []], [f21, b21, []], [d21, a21, []], [d21, b21, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
23. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s22, m22, f22, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s22, m22, f22, d22], [a22, b22], s22, [f22], [[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]], [[s22, a22, []], [s22, b22, []], [f22, a22, []], [f22, b22, []], [d22, a22, []], [d22, b22, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s22, m22, f22, d22], [a22, b22], s22, [f22], [[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]], [[s22, a22, []], [s22, b22, []], [f22, a22, []], [f22, b22, []], [d22, a22, []], [d22, b22, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
24. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s23, m23, f23, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s23, m23, f23, d23], [a23, b23], s23, [f23], [[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]], [[s23, a23, []], [s23, b23, []], [f23, a23, []], [f23, b23, []], [d23, a23, []], [d23, b23, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s23, m23, f23, d23], [a23, b23], s23, [f23], [[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]], [[s23, a23, []], [s23, b23, []], [f23, a23, []], [f23, b23, []], [d23, a23, []], [d23, b23, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
25. An Epsilon-NFA Relation May Be Empty, Partial or Multivalued
Use the complete finite epsilon-NFA: states=[s24, m24, f24, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).
View answer and explanation
([s24, m24, f24, d24], [a24, b24], s24, [f24], [[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]], [[s24, a24, []], [s24, b24, []], [f24, a24, []], [f24, b24, []], [d24, a24, []], [d24, b24, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)
Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s24, m24, f24, d24], [a24, b24], s24, [f24], [[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]], [[s24, a24, []], [s24, b24, []], [f24, a24, []], [f24, b24, []], [d24, a24, []], [d24, b24, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-qualification · candidate
26. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s0, m0, f0, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s0]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s0], [d0, m0, s0], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s0], [d0, m0, s0], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
27. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s1, m1, f1, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s1]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s1], [d1, m1, s1], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s1], [d1, m1, s1], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
28. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s2, m2, f2, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s2]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s2], [d2, m2, s2], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s2], [d2, m2, s2], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
29. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s3, m3, f3, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s3]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s3], [d3, m3, s3], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s3], [d3, m3, s3], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
30. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s4, m4, f4, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s4]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s4], [d4, m4, s4], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s4], [d4, m4, s4], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
31. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s5, m5, f5, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s5]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s5], [d5, m5, s5], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s5], [d5, m5, s5], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
32. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s6, m6, f6, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s6]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s6], [d6, m6, s6], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s6], [d6, m6, s6], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
33. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s7, m7, f7, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s7]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s7], [d7, m7, s7], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s7], [d7, m7, s7], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
34. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s8, m8, f8, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s8]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s8], [d8, m8, s8], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s8], [d8, m8, s8], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
35. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s9, m9, f9, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s9]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s9], [d9, m9, s9], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s9], [d9, m9, s9], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
36. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s10, m10, f10, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s10]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s10], [d10, m10, s10], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s10], [d10, m10, s10], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
37. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s11, m11, f11, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s11]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s11], [d11, m11, s11], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s11], [d11, m11, s11], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
38. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s12, m12, f12, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s12]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s12], [d12, m12, s12], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s12], [d12, m12, s12], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
39. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s13]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s13], [d13, m13, s13], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s13], [d13, m13, s13], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
40. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s14, m14, f14, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s14]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s14], [d14, m14, s14], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s14], [d14, m14, s14], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
41. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s15, m15, f15, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s15]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s15], [d15, m15, s15], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s15], [d15, m15, s15], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
42. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s16, m16, f16, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s16]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s16], [d16, m16, s16], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s16], [d16, m16, s16], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
43. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s17, m17, f17, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s17]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s17], [d17, m17, s17], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s17], [d17, m17, s17], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
44. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s18, m18, f18, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s18]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s18], [d18, m18, s18], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s18], [d18, m18, s18], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
45. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s19, m19, f19, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s19]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s19], [d19, m19, s19], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s19], [d19, m19, s19], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
46. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s20, m20, f20, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s20]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s20], [d20, m20, s20], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s20], [d20, m20, s20], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
47. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s21, m21, f21, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s21]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s21], [d21, m21, s21], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s21], [d21, m21, s21], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
48. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s22, m22, f22, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s22]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s22], [d22, m22, s22], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s22], [d22, m22, s22], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
49. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s23, m23, f23, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s23]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s23], [d23, m23, s23], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s23], [d23, m23, s23], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
50. Epsilon Closure Reaches States without Consuming Input
Use the complete finite epsilon-NFA: states=[s24, m24, f24, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s24]. Give (seed_states, epsilon_closure, closure_certificate, certificate).
View answer and explanation
([s24], [d24, m24, s24], Yes, finite_epsilon_reachability_without_input_consumption)
Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s24], [d24, m24, s24], Yes, finite_epsilon_reachability_without_input_consumption).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-epsilon-closure · candidate
51. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s0, m0, f0, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b0], supplied path=[[s0, 0], [m0, 0], [f0, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([b0], [[s0, 0], [m0, 0], [f0, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b0], [[s0, 0], [m0, 0], [f0, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
52. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s1, m1, f1, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a1, b1], supplied path=[[s1, 0], [m1, 0], [m1, 1], [f1, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a1, b1], [[s1, 0], [m1, 0], [m1, 1], [f1, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a1, b1], [[s1, 0], [m1, 0], [m1, 1], [f1, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
53. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s2, m2, f2, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a2, a2, b2], supplied path=[[s2, 0], [m2, 0], [m2, 1], [m2, 2], [f2, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a2, a2, b2], [[s2, 0], [m2, 0], [m2, 1], [m2, 2], [f2, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a2, a2, b2], [[s2, 0], [m2, 0], [m2, 1], [m2, 2], [f2, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
54. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s3, m3, f3, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a3, a3, a3, b3], supplied path=[[s3, 0], [m3, 0], [m3, 1], [m3, 2], [m3, 3], [f3, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a3, a3, a3, b3], [[s3, 0], [m3, 0], [m3, 1], [m3, 2], [m3, 3], [f3, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a3, a3, a3, b3], [[s3, 0], [m3, 0], [m3, 1], [m3, 2], [m3, 3], [f3, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
55. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s4, m4, f4, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b4], supplied path=[[s4, 0], [m4, 0], [f4, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([b4], [[s4, 0], [m4, 0], [f4, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b4], [[s4, 0], [m4, 0], [f4, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
56. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s5, m5, f5, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a5, b5], supplied path=[[s5, 0], [m5, 0], [m5, 1], [f5, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a5, b5], [[s5, 0], [m5, 0], [m5, 1], [f5, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a5, b5], [[s5, 0], [m5, 0], [m5, 1], [f5, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
57. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s6, m6, f6, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a6, a6, b6], supplied path=[[s6, 0], [m6, 0], [m6, 1], [m6, 2], [f6, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a6, a6, b6], [[s6, 0], [m6, 0], [m6, 1], [m6, 2], [f6, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a6, a6, b6], [[s6, 0], [m6, 0], [m6, 1], [m6, 2], [f6, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
58. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s7, m7, f7, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a7, a7, a7, b7], supplied path=[[s7, 0], [m7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a7, a7, a7, b7], [[s7, 0], [m7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a7, a7, a7, b7], [[s7, 0], [m7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
59. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s8, m8, f8, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b8], supplied path=[[s8, 0], [m8, 0], [f8, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([b8], [[s8, 0], [m8, 0], [f8, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b8], [[s8, 0], [m8, 0], [f8, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
60. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s9, m9, f9, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a9, b9], supplied path=[[s9, 0], [m9, 0], [m9, 1], [f9, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a9, b9], [[s9, 0], [m9, 0], [m9, 1], [f9, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a9, b9], [[s9, 0], [m9, 0], [m9, 1], [f9, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
61. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s10, m10, f10, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a10, a10, b10], supplied path=[[s10, 0], [m10, 0], [m10, 1], [m10, 2], [f10, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a10, a10, b10], [[s10, 0], [m10, 0], [m10, 1], [m10, 2], [f10, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a10, a10, b10], [[s10, 0], [m10, 0], [m10, 1], [m10, 2], [f10, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
62. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s11, m11, f11, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a11, a11, a11, b11], supplied path=[[s11, 0], [m11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a11, a11, a11, b11], [[s11, 0], [m11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a11, a11, a11, b11], [[s11, 0], [m11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
63. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s12, m12, f12, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b12], supplied path=[[s12, 0], [m12, 0], [f12, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([b12], [[s12, 0], [m12, 0], [f12, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b12], [[s12, 0], [m12, 0], [f12, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
64. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a13, b13], supplied path=[[s13, 0], [m13, 0], [m13, 1], [f13, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a13, b13], [[s13, 0], [m13, 0], [m13, 1], [f13, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a13, b13], [[s13, 0], [m13, 0], [m13, 1], [f13, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
65. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s14, m14, f14, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a14, a14, b14], supplied path=[[s14, 0], [m14, 0], [m14, 1], [m14, 2], [f14, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a14, a14, b14], [[s14, 0], [m14, 0], [m14, 1], [m14, 2], [f14, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a14, a14, b14], [[s14, 0], [m14, 0], [m14, 1], [m14, 2], [f14, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
66. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s15, m15, f15, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a15, a15, a15, b15], supplied path=[[s15, 0], [m15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a15, a15, a15, b15], [[s15, 0], [m15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a15, a15, a15, b15], [[s15, 0], [m15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
67. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s16, m16, f16, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b16], supplied path=[[s16, 0], [m16, 0], [f16, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([b16], [[s16, 0], [m16, 0], [f16, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b16], [[s16, 0], [m16, 0], [f16, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
68. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s17, m17, f17, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a17, b17], supplied path=[[s17, 0], [m17, 0], [m17, 1], [f17, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a17, b17], [[s17, 0], [m17, 0], [m17, 1], [f17, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a17, b17], [[s17, 0], [m17, 0], [m17, 1], [f17, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
69. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s18, m18, f18, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a18, a18, b18], supplied path=[[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a18, a18, b18], [[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a18, a18, b18], [[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
70. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s19, m19, f19, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a19, a19, a19, b19], supplied path=[[s19, 0], [m19, 0], [m19, 1], [m19, 2], [m19, 3], [f19, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a19, a19, a19, b19], [[s19, 0], [m19, 0], [m19, 1], [m19, 2], [m19, 3], [f19, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a19, a19, a19, b19], [[s19, 0], [m19, 0], [m19, 1], [m19, 2], [m19, 3], [f19, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
71. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s20, m20, f20, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b20], supplied path=[[s20, 0], [m20, 0], [f20, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([b20], [[s20, 0], [m20, 0], [f20, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b20], [[s20, 0], [m20, 0], [f20, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
72. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s21, m21, f21, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a21, b21], supplied path=[[s21, 0], [m21, 0], [m21, 1], [f21, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a21, b21], [[s21, 0], [m21, 0], [m21, 1], [f21, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a21, b21], [[s21, 0], [m21, 0], [m21, 1], [f21, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
73. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s22, m22, f22, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a22, a22, b22], supplied path=[[s22, 0], [m22, 0], [m22, 1], [m22, 2], [f22, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a22, a22, b22], [[s22, 0], [m22, 0], [m22, 1], [m22, 2], [f22, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a22, a22, b22], [[s22, 0], [m22, 0], [m22, 1], [m22, 2], [f22, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
74. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s23, m23, f23, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a23, a23, a23, b23], supplied path=[[s23, 0], [m23, 0], [m23, 1], [m23, 2], [m23, 3], [f23, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([a23, a23, a23, b23], [[s23, 0], [m23, 0], [m23, 1], [m23, 2], [m23, 3], [f23, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a23, a23, a23, b23], [[s23, 0], [m23, 0], [m23, 1], [m23, 2], [m23, 3], [f23, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
75. One Full-Input Final Path Suffices Despite Dead Branches
Use the complete finite epsilon-NFA: states=[s24, m24, f24, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b24], supplied path=[[s24, 0], [m24, 0], [f24, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).
valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it.
View answer and explanation
([b24], [[s24, 0], [m24, 0], [f24, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)
Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b24], [[s24, 0], [m24, 0], [f24, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-accepting-path-witness · candidate
76. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s0, m0, f0, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b0], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([b0], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b0], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
77. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s1, m1, f1, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a1, b1], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a1, b1], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a1, b1], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
78. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s2, m2, f2, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a2, a2, b2], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a2, a2, b2], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a2, a2, b2], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
79. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s3, m3, f3, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a3, a3, a3, b3], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a3, a3, a3, b3], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a3, a3, a3, b3], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
80. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s4, m4, f4, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b4], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([b4], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b4], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
81. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s5, m5, f5, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a5, b5], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a5, b5], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a5, b5], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
82. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s6, m6, f6, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a6, a6, b6], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a6, a6, b6], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a6, a6, b6], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
83. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s7, m7, f7, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a7, a7, a7, b7], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a7, a7, a7, b7], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a7, a7, a7, b7], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
84. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s8, m8, f8, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b8], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([b8], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b8], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
85. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s9, m9, f9, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a9, b9], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a9, b9], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a9, b9], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
86. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s10, m10, f10, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a10, a10, b10], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a10, a10, b10], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a10, a10, b10], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
87. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s11, m11, f11, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a11, a11, a11, b11], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a11, a11, a11, b11], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a11, a11, a11, b11], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
88. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s12, m12, f12, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b12], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([b12], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b12], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
89. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a13, b13], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a13, b13], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a13, b13], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
90. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s14, m14, f14, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a14, a14, b14], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a14, a14, b14], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a14, a14, b14], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
91. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s15, m15, f15, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a15, a15, a15, b15], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a15, a15, a15, b15], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a15, a15, a15, b15], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
92. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s16, m16, f16, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b16], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([b16], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b16], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
93. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s17, m17, f17, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a17, b17], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a17, b17], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a17, b17], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
94. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s18, m18, f18, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a18, a18, b18], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a18, a18, b18], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a18, a18, b18], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
95. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s19, m19, f19, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a19, a19, a19, b19], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a19, a19, a19, b19], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a19, a19, a19, b19], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
96. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s20, m20, f20, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b20], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([b20], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b20], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
97. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s21, m21, f21, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a21, b21], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a21, b21], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a21, b21], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
98. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s22, m22, f22, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a22, a22, b22], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a22, a22, b22], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a22, a22, b22], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
99. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s23, m23, f23, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a23, a23, a23, b23], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([a23, a23, a23, b23], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a23, a23, a23, b23], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
100. A Depth-Limited No-Hit Is Not an NFA Rejection Proof
Use the complete finite epsilon-NFA: states=[s24, m24, f24, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b24], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).
A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance.
View answer and explanation
([b24], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)
Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b24], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs).
Reviewed step-by-step derivation
Nondeterministic finite automaton · qf-mod42-nfa-bounded-search-boundary · candidate
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