R0046-23997 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0047
Original question and givens
Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s21, m21, d21], input symbol=b21. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).
G001 — Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s21, m21, d21], input symbol=b21. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).
Step-by-step derivation
S001 · implies
Before
source operation=subsetdfa-symbol-transition;original snapshot and mathematics retained
After
explicit ordered source output fields=[subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate]
Conditions: none
Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL states=[s21, m21, f21, e121, e221, d21],alphabet=[a21, b21],start=s21;ORIGINAL transition order=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]
After
final set normalized UTF16=[f21];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model
Conditions: none
This model differs from the earlier NFA node:transition triples are unique and finals are sorted.Subsets must be actual carrier subsets;do not import old duplicate/final-order conventions.
Step identifiers and dependencies
Rule: subsetdfa_qualified_NFA_signature_DUPLICATE_edges_REJECTED_here · Uses: G001, S001
S003 · implies
Before
Subset DFA ORIGINAL one-symbol transition:EXACT ORIGINAL supplied S=[d21, m21, s21];symbol=b21;ALL matching ORIGINAL edge rows=[[3, m21, f21]]
After
direct symbol targets=[f21];NO EPS preclosure inserted into S;now close ONLY these targets
Conditions: none
An arbitrary supplied S need not be EPS-closed;preclosing it would change this operation.Apply move(S,a) and then close its targets.Empty targets yield the legal empty dead state.
Step identifiers and dependencies
Rule: subsetdfa_GIVEN_S_symbol_move_BEFORE_closure_NO_preclose · Uses: G001, S002
S004 · implies
Before
Subset DFA ORIGINAL one-symbol transition post-symbol EPS closure:stage=0;seed set=[f21];previous set=[f21]
After
ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, Yes], [5, e121, e221, No]];next set=[e121, f21];stable=No
Conditions: none
Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input.
Step identifiers and dependencies
Rule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S003
S005 · implies
Before
Subset DFA ORIGINAL one-symbol transition post-symbol EPS closure:stage=1;seed set=[f21];previous set=[e121, f21]
After
ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, Yes], [5, e121, e221, Yes]];next set=[e121, e221, f21];stable=No
Conditions: none
Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input.
Step identifiers and dependencies
Rule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S004
S006 · implies
Before
Subset DFA ORIGINAL one-symbol transition post-symbol EPS closure:stage=2;seed set=[f21];previous set=[e121, e221, f21]
After
ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, Yes], [5, e121, e221, Yes]];next set=[e121, e221, f21];stable=Yes
Conditions: none
Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input.
Step identifiers and dependencies
Rule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S005
Result
([d21, m21, s21], b21, [f21], [e121, e221, f21], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.