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Description

The Brzozowski Algebraic Method is a technique used in automata theory, specifically for constructing the minimal deterministic finite automaton (DFA) from a given nondeterministic finite automaton (NFA). It was introduced by Janusz Brzozowski in 1962 and involves the use of algebraic transformations to simplify and optimize automata.

As the workings of the algorithm are clearly described in the linked article they will not be repeated here.

Task

Implement the Brzozowski Algebraic Method to convert an NFA representation into a regular expression. The implementation should:

1. Take as input:

```
  - A 2D array 'a' representing the transition matrix of the NFA
  - An array 'b' representing the final states

```

2. Include functionality to:

```
  - Handle basic regular expression operations (Empty, Epsilon, Character, Union, Concatenation, Star)
  - Print regular expressions in a readable format
  - Simplify regular expressions using algebraic rules

```

3. Apply the Brzozowski algorithm to:

```
  - Process the transition matrix in reverse order
  - Update transitions using Star and Concatenation operations
  - Combine paths using Union operations
  - Generate a final regular expression representing the language accepted by the NFA

```

4. Include simplification rules for:

```
  - Eliminating redundant unions (e + e → e)
  - Handling identity elements (e·1 → e, 1·e → e)
  - Handling zero elements (e·0 → 0, 0·e → 0)
  - Simplifying star operations (0* → 1, 1* → 1)

```

The implementation should be able to process the given example input and produce both the raw and simplified regular expressions as output.
assistant
**Translation of**: Python

```
# 20241120 Raku programming solution

role RE1 { has $.c }
role RE2 { has ( $.e, $.f ) }

class Empty           { method Str { '0' } }
class Epsilon         { method Str { '1' } }
class Car    does RE1 { method Str { $.c } } # Caractère ?
class Star   does RE1 { method Str { "($.c)*" } }
class Union  does RE2 { method Str { "$.e+$.f" } }
class Concat does RE2 { method Str { "($.e)($.f)" } }

my ($empty, $epsilon) = (Empty, Epsilon)>>.new;

sub simple-re($e is rw) {
   sub do-simple($e) {
      given $e {
         when Union {
            my ($ee,$ef) = ($e.e, $e.f).map: { do-simple $_ };
            if    $ee ~~ $ef   { return $ee }
            elsif $ee ~~ Union { do-simple(Union.new(e => $ee.e, f => Union.new(e => $ee.f, f => $ef))) }
            elsif $ee ~~ Empty { return $ef }
            elsif $ef ~~ Empty { return $ee }
            else  { Union.new: e => $ee, f => $ef }
         }
         when Concat {
            my ($ee,$ef) = ($e.e, $e.f).map: { do-simple $_ };
            if    $ee ~~ Epsilon { return $ef }
            elsif $ef ~~ Epsilon {  return $ee }
            elsif $ee ~~ Empty || $ef ~~ Empty { return Empty }
            elsif $ee ~~ Concat { do-simple(Concat.new(e => $ee.e, f => Concat.new(e => $ee.f, f => $ef))) }
            else { Concat.new: e => $ee, f => $ef }
         }
         when Star {
            given do-simple($e.c) {
               $_ ~~ Empty | Epsilon ?? return Epsilon !! Star.new: c => $_
            }      
         }
         default { $e }
      }
   }
   my $prev = '';
   until $e === $prev or $e.Str eq $prev { ($prev, $e) = $e, do-simple($e) }
   return $e;
}

sub brzozowski(@a, @b) {
   for @a.elems - 1 ... 0 -> \n {
      @b[n] = Concat.new(e => Star.new(c => @a[n;n]), f => @b[n]);
      for ^n -> \j {
         @a[n;j] = Concat.new(e => Star.new(c => @a[n;n]), f => @a[n;j]);
      }
      for ^n -> \i {
         @b[i] = Union.new(e => @b[i], f => Concat.new(e => @a[i;n], f => @b[n]));
         for ^n -> \j {
            @a[i;j] = Union.new(e => @a[i;j], f => Concat.new(e => @a[i;n], f => @a[n;j]));
         }
      }
      for ^n -> \i { @a[i;n] = $empty }
   }
   return @b[0]
}

my @a = [
    [$empty, Car.new(c => 'a'), Car.new(c => 'b')],
    [Car.new(c => 'b'), $empty, Car.new(c => 'a')],
    [Car.new(c => 'a'), Car.new(c => 'b'), $empty],
];
my @b = [$epsilon, $empty, $empty];

say ( my $re = brzozowski(@a, @b) ).Str;
say simple-re($re).Str;

```

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