// Another automaton construction
//================================

import $file.dfa, dfa._ 

// regular expressions
abstract class Rexp
case object ZERO extends Rexp                    // matches nothing
case object ONE extends Rexp                     // matches the empty string
case class CHAR(c: Char) extends Rexp            // matches a character c
case class ALT(r1: Rexp, r2: Rexp) extends Rexp  // alternative
case class SEQ(r1: Rexp, r2: Rexp) extends Rexp  // sequence
case class STAR(r: Rexp) extends Rexp            // star

// the nullable function: tests whether the regular 
// expression can recognise the empty string
def nullable (r: Rexp) : Boolean = r match {
  case ZERO => false
  case ONE => true
  case CHAR(_) => false
  case ALT(r1, r2) => nullable(r1) || nullable(r2)
  case SEQ(r1, r2) => nullable(r1) && nullable(r2)
  case STAR(_) => true
}

// the derivative of a regular expression w.r.t. a character
def der (c: Char, r: Rexp) : Rexp = r match {
  case ZERO => ZERO
  case ONE => ZERO
  case CHAR(d) => if (c == d) ONE else ZERO
  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))
  case SEQ(r1, r2) => 
    if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))
    else SEQ(der(c, r1), r2)
  case STAR(r1) => SEQ(der(c, r1), STAR(r1))
}


def flaw(r: Rexp) : DFA[Rexp, Char] = {
  DFA(r, 
      { case (r, c) => der(c, r) }, 
      nullable(_))
}

val r = STAR(CHAR('a'))
val pseudo = flaw(r)
println(pseudo.accepts("".toList))    // true
println(pseudo.accepts("a".toList))   // true
println(pseudo.accepts("aa".toList))  // true
println(pseudo.accepts("bb".toList))  // false