// Thompson Construction

//=======================

import $file.dfa, dfa._ 

import $file.nfa, nfa._

import $file.enfa, enfa._

// states for Thompson construction

case class TState(i: Int) extends State

object TState {

  var counter = 0

  def apply() : TState = {

    counter += 1;

    new TState(counter)

  }

}

// some types abbreviations

type NFAt = NFA[TState, Char]

type NFAtrans = (TState, Char) :=> Set[TState]

type eNFAtrans = (TState, Option[Char]) :=> Set[TState]

// for composing an eNFA transition with an NFA transition

// | is for set union

implicit def nfaOps(f: eNFAtrans) = new {

  def +++(g: NFAtrans) : eNFAtrans = 

  { case (q, None) =>  applyOrElse(f, (q, None)) 

    case (q, Some(c)) => applyOrElse(f, (q, Some(c))) | applyOrElse(g, (q, c))  }

}

// NFA that does not accept any string

def NFA_ZERO(): NFAt = {

  val Q = TState()

  NFA(Set(Q), { case _ => Set() }, Set())

}

// NFA that accepts the empty string

def NFA_ONE() : NFAt = {

  val Q = TState()

  NFA(Set(Q), { case _ => Set() }, Set(Q))

}

// NFA that accepts the string "c"

def NFA_CHAR(c: Char) : NFAt = {

  val Q1 = TState()

  val Q2 = TState()

  NFA(Set(Q1), { case (Q1, d) if (c == d) => Set(Q2) }, Set(Q2))

}

// sequence of two NFAs

def NFA_SEQ(enfa1: NFAt, enfa2: NFAt) : NFAt = {

  val new_delta : eNFAtrans = 

    { case (q, None) if enfa1.fins(q) => enfa2.starts }

  eNFA(enfa1.starts, 

       new_delta +++ enfa1.delta +++ enfa2.delta, 

       enfa2.fins)

}

// alternative of two NFAs

def NFA_ALT(enfa1: NFAt, enfa2: NFAt) : NFAt = {

  val new_delta : NFAtrans = { 

    case (q, c) =>  applyOrElse(enfa1.delta, (q, c)) | 

                    applyOrElse(enfa2.delta, (q, c)) }

  val new_fins = (q: TState) => enfa1.fins(q) || enfa2.fins(q)

  NFA(enfa1.starts | enfa2.starts, new_delta, new_fins)

}

// star of a NFA

def NFA_STAR(enfa: NFAt) : NFAt = {

  val Q = TState()

  val new_delta : eNFAtrans = 

    { case (Q, None) => enfa.starts

      case (q, None) if enfa.fins(q) => Set(Q) }

  eNFA(Set(Q), new_delta +++ enfa.delta, Set(Q))

}

// We are now ready to translate regular expressions

// into DFAs (via eNFAs and NFAs, and the subset construction)

// 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

// thompson construction 

def thompson (r: Rexp) : NFAt = r match {

  case ZERO => NFA_ZERO()

  case ONE => NFA_ONE()

  case CHAR(c) => NFA_CHAR(c)  

  case ALT(r1, r2) => NFA_ALT(thompson(r1), thompson(r2))

  case SEQ(r1, r2) => NFA_SEQ(thompson(r1), thompson(r2))

  case STAR(r1) => NFA_STAR(thompson(r1))

}

//optional regular expression (one or zero times)

def OPT(r: Rexp) = ALT(r, ONE)

//n-times regular expression (explicitly expanded)

def NTIMES(r: Rexp, n: Int) : Rexp = n match {

  case 0 => ONE

  case 1 => r

  case n => SEQ(r, NTIMES(r, n - 1))

}

def tmatches_nfa(r: Rexp, s: String) : Boolean =

  thompson(r).accepts(s.toList)

def tmatches_nfa2(r: Rexp, s: String) : Boolean =

  thompson(r).accepts2(s.toList)

// dfas via subset construction

def tmatches_dfa(r: Rexp, s: String) : Boolean =

  subset(thompson(r)).accepts(s.toList)

// Test Cases

//============

// the evil regular expression  a?{n} a{n}

def EVIL1(n: Int) : Rexp = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))

// the evil regular expression (a*)*b

val EVIL2 : Rexp = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))

//for measuring time

def time_needed[T](i: Int, code: => T) = {

  val start = System.nanoTime()

  for (j <- 1 to i) code

  val end = System.nanoTime()

  (end - start)/(i * 1.0e9)

}

// the size of the NFA can be large, 

// thus slowing down the breadth-first search

for (i <- 1 to 13) {

  println(i + ": " + "%.5f".format(time_needed(2, tmatches_nfa(EVIL1(i), "a" * i))))

}

for (i <- 1 to 100 by 5) {

  println(i + " " + "%.5f".format(time_needed(2, tmatches_nfa(EVIL2, "a" * i))))

}

// the backtracking that is needed in depth-first 

// search can be painfully slow

for (i <- 1 to 8) {

  println(i + " " + "%.5f".format(time_needed(2, tmatches_nfa2(EVIL2, "a" * i))))

}

// while my thompson->enfa->subset->partial-function-chain

// is probably not the most effcient way to obtain a fast DFA 

// (the test below should be much faster with a more direct 

// construction), in general the DFAs can be slow because of 

// the state explosion in the subset construction

for (i <- 1 to 13) {

  println(i + ": " + "%.5f".format(time_needed(2, tmatches_dfa(EVIL1(i), "a" * i))))

}

for (i <- 1 to 100 by 5) {

  println(i + " " + "%.5f".format(time_needed(2, tmatches_dfa(EVIL2, "a" * i))))

}