// epsilon NFAs...immediately translated into NFAs

import $file.dfa, dfa._ 

import $file.nfa, nfa._ 

// a fixpoint construction

import scala.annotation.tailrec

@tailrec

def fixpT[S](f: S => S, x: S): S = {

  val fx = f(x)

  if (fx == x) x else fixpT(f, fx) 

}

// translates eNFAs directly into NFAs 

def eNFA[A, C](starts: Set[A],                     // the starting states

               delta: (A, Option[C]) :=> Set[A],   // the epsilon-transitions

               fins: A => Boolean) : NFA[A, C] = { // the final states 

  // epsilon transitions

  def enext(q: A) : Set[A] = 

    applyOrElse(delta, (q, None))

  def enexts(qs: Set[A]) : Set[A] = 

    qs | qs.flatMap(enext(_))     // | is the set-union in Scala

  // epsilon closure

  def ecl(qs: Set[A]) : Set[A] = 

    fixpT(enexts, qs)

  // "normal" transitions

  def next(q: A, c: C) : Set[A] = 

    applyOrElse(delta, (q, Some(c)))

  def nexts(qs: Set[A], c: C) : Set[A] = 

    ecl(ecl(qs).flatMap(next(_, c)))

  // result NFA

  NFA(ecl(starts), 

      { case (q, c) => nexts(Set(q), c) }, 

      q => ecl(Set(q)) exists fins)

}

// eNFA examples

val enfa_trans1 : (State, Option[Char]) :=> Set[State] =

  { case (Q0, Some('a')) => Set(Q0)

    case (Q0, None) => Set(Q1, Q2)

    case (Q1, Some('a')) => Set(Q1)

    case (Q2, Some('b')) => Set(Q2) 

  }

val nfa1 = 

  eNFA(Set[State](Q0), enfa_trans1, Set[State](Q2))

// more examples

case object R1 extends State

case object R2 extends State

case object R3 extends State

val enfa_trans2 : (State, Option[Char]) :=> Set[State] =

  { case (R1, Some('b')) => Set(R3)

    case (R1, None) => Set(R2)

    case (R2, Some('a')) => Set(R1, R3) 

  }

val nfa2 = 

  eNFA(Set[State](R1), enfa_trans1, Set[State](R3))