R0046-23979 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0029
Original question and givens
Use complete epsilon-NFA states=[s3, m3, f3, e13, e23, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s3, m3, d3], input symbol=b3. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).
G001 — Use complete epsilon-NFA states=[s3, m3, f3, e13, e23, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s3, m3, d3], input symbol=b3. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).
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=[s3, m3, f3, e13, e23, d3],alphabet=[a3, b3],start=s3;ORIGINAL transition order=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]
After
final set normalized UTF16=[f3];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=[d3, m3, s3];symbol=b3;ALL matching ORIGINAL edge rows=[[3, m3, f3]]
After
direct symbol targets=[f3];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=[f3];previous set=[f3]
After
ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s3, m3, No], [1, s3, d3, No], [4, f3, e13, Yes], [5, e13, e23, No]];next set=[e13, f3];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=[f3];previous set=[e13, f3]
After
ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s3, m3, No], [1, s3, d3, No], [4, f3, e13, Yes], [5, e13, e23, Yes]];next set=[e13, e23, f3];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=[f3];previous set=[e13, e23, f3]
After
ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s3, m3, No], [1, s3, d3, No], [4, f3, e13, Yes], [5, e13, e23, Yes]];next set=[e13, e23, f3];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
([d3, m3, s3], b3, [f3], [e13, e23, f3], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.