// 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