How to get rid of braces in boolean expressions with AND and OR

Question

Let`s say I have some simple boolean expression

(A || B) && (C || D)

The goal is, to get rid of braces in this expression, for example

(A || B) && (C || D) => A && C || A && D || B && C || B && D

We know that && evaluates before || from left to right. To achieve this I have created the following algebraic data type:

sealed trait Predicate
case class Or(left: Predicate, right: Predicate) extends Predicate
case class And(left: Predicate, right: Predicate) extends Predicate
case object True extends Predicate
case object False extends Predicate

Let`s assume that we already have some kind of string parser, which converts string to Predicate, it builds some kind of Abstract Syntactic Tree. For example for expression (true || true) && (true || true) we will have the following tree: And(Or(True, True), Or(True, True)). Here we get into account the braces. We need to get Or(Or(And(A, C), And(A, D)), Or(And(B, C), And(B,D))). I stuck with the following solution:

def extractOr(pred: Predicate): Predicate = pred match {
    case And(Or(l, r), Or(ll, rr)) => Or(Or(And(l, ll), And(l, rr)), Or(And(r, ll), And(r, rr)))
    case And(Or(l, r), p) => Or(And(l, p), And(r, p))
    case And(p, Or(l, r)) => Or(And(p, l), And(p, r))
    case p => p
}

def popOrPredicateUp(pred: Predicate): Predicate = pred match {
    case And(l, r) => extractOr(And(popOrPredicateUp(l), popOrPredicateUp(r)))
    case Or(l, r) => Or(popOrPredicateUp(l), popOrPredicateUp(r))
    case p => p
}

But it works incorrect for example for this case: And(False, Or(And(Or(True, True), False), True))

UPD: As @coredump pointed out, I need to get DNF(sum of products)

Answer 1

Finally, with great help @coredump(he pointed out the right direction) and Haskell package hatt(particulary that code). I came to the following solution:

sealed trait Predicate
case class Or(left: Predicate, right: Predicate) extends Predicate
case class And(left: Predicate, right: Predicate) extends Predicate
case class Not(pred: Predicate) extends Predicate
case object True extends Predicate
case object False extends Predicate

object PredicateOps{

  def toNNF(pred: Predicate): Predicate = pred match {
    case a @ (True | False) => a
    case a @ Not( True | False) => a
    case Not(Not(p)) => p
    case And(l, r) => And(toNNF(l), toNNF(r))
    case Not(And(l, r)) => toNNF( Or(Not(l), Not(r)))
    case Or(l, r) => Or(toNNF(l), toNNF(r))
    case Not(Or(l,r)) => toNNF( And(Not(l), Not(r)))
  }

  def dist(predL: Predicate, predR: Predicate): Predicate = (predL, predR) match {
    case (Or(l, r), p) => Or(dist(l, p), dist(r, p))
    case (p, Or(l, r)) => Or(dist(p, l), dist(p, r))
    case (l, r) => And(l, r)
  }

  def toDNF(pred: Predicate): Predicate = pred match {
    case And(l, r) => dist(toDNF(l), toDNF(r))
    case Or(l, r) =>  Or(toDNF(l), toDNF(r))
    case p => p
  }

}

Here is how it works:

val expr = And(False, Or(And(Or(True, True), False), True))
val dnf = (PredicateOps.toNNF _ andThen PredicateOps.toDNF _).apply(expr)
println(dnf)

And the output Or(Or(And(False,And(True,False)),And(False,And(True,False))),And(False,True)) which is correct DNF.