dfa minimization

Step $1$: Remove unreachable states.

Step $2$: Mark the distinguishable pairs of states.

To achieve this task, we first mark all pairs p$,$ q$,$ where p$2$F and qZF as distinguishable. Then, we proceed as follows:

repeat

for all non $-$ marked pairs p$,$ q do

for each letter a do

if the pair $($p$,$ a$),($q$,$ a$)$ is marked

then mark p$,$ q

until no new pairs are marked

Step $3$: Construct the reduced automaton A$.$

implement dfa minimization with python from scratch with no other library simple function and no ambugity

def remove$\_$unreachable$\_$states$($states$,$ alphabet, transitions, initial$\_$state, accepting$\_$states$)$:

reachable$\_$states $=$ set$()$

stack $=[$initial$\_$state$]$

while stack:

current$\_$state $=$ stack.pop$()$

reachable$\_$states.add$($current$\_$state$)$

for letter in alphabet:

next$\_$state $=$ transitions.get$(($current$\_$state, letter$),$ None$)$

if next$\_$state is not None and next$\_$state not in reachable$\_$states:

stack.append$($next$\_$state$)$

unreachable$\_$states $=$ set$($states$)-$ reachable$\_$states

for state in unreachable$\_$states:

states.remove$($state$)$

accepting$\_$states.discard$($state$)$

for letter in alphabet:

transitions.pop$(($state$,$ letter$),$ None$)$

def mark$\_$distinguishable$\_$pairs$($states$,$ alphabet, transitions, accepting$\_$states$)$:

marked$\_$pairs $=$ set$()$

for i$,$ state$1$ in enumerate$($states$)$:

for state$2$ in states$[$i $+1$:$]$:

if $($state$1$ in accepting$\_$states and state$2$ not in accepting$\_$states$)$ or $($state$2$ in accepting$\_$states and state$1$ not in accepting$\_$states$)$:

marked$\_$pairs.add$(($state$1,$ state$2))$

changed $=$ True

while changed:

changed $=$ False

for state$1$ in states:

for state$2$ in states:

if $($state$1,$ state$2)$ not in marked$\_$pairs and $($state$2,$ state$1)$ not in marked$\_$pairs:

for letter in alphabet:

next$\_$state$1=$ transitions.get$(($state$1,$ letter$),$ None$)$

next$\_$state$2=$ transitions.get$(($state$2,$ letter$),$ None$)$

if $($next$\_$state$1,$ next$\_$state$2)$ in marked$\_$pairs or $($next$\_$state$2,$ next$\_$state$1)$ in marked$\_$pairs:

marked$\_$pairs.add$(($state$1,$ state$2))$

changed $=$ True

return marked$\_$pairs

def minimize$\_$dfa$($states$,$ alphabet, transitions, initial$\_$state, accepting$\_$states$)$:

remove$\_$unreachable$\_$states$($states$,$ alphabet, transitions, initial$\_$state, accepting$\_$states$)$

marked$\_$pairs $=$ mark$\_$distinguishable$\_$pairs$($states$,$ alphabet, transitions, accepting$\_$states$)$

while marked$\_$pairs:

state$1,$ state$2=$ marked$\_$pairs.pop$()$

for letter in alphabet:

next$\_$state$1=$ transitions.pop$(($state$1,$ letter$),$ None$)$

next$\_$state$2=$ transitions.pop$(($state$2,$ letter$),$ None$)$

if next$\_$state$1$ is not None and next$\_$state$2$ is not None:

new$\_$state $=$ min$($next$\_$state$1,$ next$\_$state$2),$ max$($next$\_$state$1,$ next$\_$state$2)$

transitions$[($state$1,$ letter$)]=$ new$\_$state

transitions$[($state$2,$ letter$)]=$ new$\_$state

if state$1$ in accepting$\_$states and state$2$ not in accepting$\_$states:

accepting$\_$states.remove$($state$1)$

elif state$2$ in accepting$\_$states and state$1$ not in accepting$\_$states:

accepting$\_$states.remove$($state$2)$

remove$\_$unreachable$\_$states$($states$,$ alphabet, transitions, initial$\_$state, accepting$\_$states$)$

marked$\_$pairs $=$ mark$\_$distinguishable$\_$pairs$($states$,$ alphabet, transitions, accepting$\_$states$)$

return states, alphabet, transitions, initial$\_$state, accepting$\_$states

# Example usage:

states $=\{0,1,2\}$

alphabet $=\{\text{'}$a$\text{'},\text{'}$b$\text{'}\}$

transitions $=\{(0,\text{'}$a$\text{'})$: $1,(0,\text{'}$b$\text{'})$: $2,(1,\text{'}$a$\text{'})$: $1,(1,\text{'}$b$\text{'})$: $2,(2,\text{'}$a$\text{'})$: $0,(2,\text{'}$b$\text{'})$: $1\}$

initial$\_$state $=0$

accepting$\_$states $=\{1\}$

minimized$\_$dfa $=$ minimize$\_$dfa$($states$,$ alphabet, transitions, initial$\_$state, accepting$\_$states$)$

print$($minimized$\_$dfa$)$

line $25,$ in mark$\_$distinguishable$\_$pairs

for state$2$ in states$[$i $+1$:$]$:

TypeError: 'set' object is not subscriptable

fix the error please