R0046-23991 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0041
Original question and givens
Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s15, m15, d15], input symbol=b15. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).
G001 — Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s15, m15, d15], input symbol=b15. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).
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=[s15, m15, f15, e115, e215, d15],alphabet=[a15, b15],start=s15;ORIGINAL transition order=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]
After
final set normalized UTF16=[f15];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=[d15, m15, s15];symbol=b15;ALL matching ORIGINAL edge rows=[[3, m15, f15]]
After
direct symbol targets=[f15];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=[f15];previous set=[f15]
After
ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, No], [1, s15, d15, No], [4, f15, e115, Yes], [5, e115, e215, No]];next set=[e115, f15];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=[f15];previous set=[e115, f15]
After
ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, No], [1, s15, d15, No], [4, f15, e115, Yes], [5, e115, e215, Yes]];next set=[e115, e215, f15];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=[f15];previous set=[e115, e215, f15]
After
ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, No], [1, s15, d15, No], [4, f15, e115, Yes], [5, e115, e215, Yes]];next set=[e115, e215, f15];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
([d15, m15, s15], b15, [f15], [e115, e215, f15], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.