I suppose I'd parse the expression into some kind of tree structure of objects along the lines of:

role Node;

class Conjunction with Node {
   has left  => (is => 'ro', does => 'Node');
   has right => (is => 'ro', does => 'Node');
}

class Disjunction with Node {
   has left  => (is => 'ro', does => 'Node');
   has right => (is => 'ro', does => 'Node');
}

class Negation with Node {
   has negated_term => (is => 'ro', does => 'Node');
}

class Primative with Node {
   has label => (is => 'ro', isa => Str);
}

my $fred = Primative->new(label => "Fred");
my $john = Primative->new(label => "John");
my $fred_and_john = Conjunction->new(left => $fred, right => $john);
...;
[download]

Then I'd write a few "visitor" rules that can take a tree and tweak it, for example, finding a candidate for rewriting under De Morgan's laws...

my $demorgans_laws = sub
{
   my $tree = $_[0];
   
   if ($tree->isa('Conjunction')
   and $tree->left->isa('Negation')
   and $tree->right->isa('Negation'))
   {
      return Negation->new(
         node => Disjunction->new(
            left => $tree->left->negated_term,
            right => $tree->right->negated_term,
         ),
      );
   }
   
   if ($tree->isa('Disjunction')
   and $tree->left->isa('Negation')
   and $tree->right->isa('Negation'))
   {
      return Negation->new(
         node => Conjunction->new(
            left => $tree->left->negated_term,
            right => $tree->right->negated_term,
         ),
      );
   }
   
   return $tree;
};
[download]

Then use some kind of hill-climbing algorithm to apply the tweaking rules to the tree, creating new trees; discarding trees that become more complex than the original, and retaining trees that become simpler, or at least as simple.

The problem with hill-climbing is that you can find local maxima. If you're climbing a mountain, sometimes you find a small peak, and you need to climb downwards a bit before you can reach the true summit. This is a hard problem to solve without ending up with a completely undirected search. But luckily, it's a problem that many smart people have thought about, so there's a wealth of ideas on how to solve it if you delve into AI literature.

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