// Main Part 3 about Regular Expression Matching
//==============================================

object M3 {

abstract class Rexp
case object ZERO extends Rexp
case object ONE extends Rexp
case class CHAR(c: Char) extends Rexp
case class ALTs(rs: List[Rexp]) extends Rexp  // alternatives 
case class SEQs(rs: List[Rexp]) extends Rexp  // sequences
case class STAR(r: Rexp) extends Rexp         // star


//the usual binary choice and binary sequence can be defined 
//in terms of ALTs and SEQs
def ALT(r1: Rexp, r2: Rexp) = ALTs(List(r1, r2))
def SEQ(r1: Rexp, r2: Rexp) = SEQs(List(r1, r2))


// some convenience for typing regular expressions
import scala.language.implicitConversions    
import scala.language.reflectiveCalls 

def charlist2rexp(s: List[Char]): Rexp = s match {
  case Nil => ONE
  case c::Nil => CHAR(c)
  case c::s => SEQ(CHAR(c), charlist2rexp(s))
}
implicit def string2rexp(s: String): Rexp = charlist2rexp(s.toList)

implicit def RexpOps (r: Rexp) = new {
  def | (s: Rexp) = ALT(r, s)
  def % = STAR(r)
  def ~ (s: Rexp) = SEQ(r, s)
}

implicit def stringOps (s: String) = new {
  def | (r: Rexp) = ALT(s, r)
  def | (r: String) = ALT(s, r)
  def % = STAR(s)
  def ~ (r: Rexp) = SEQ(s, r)
  def ~ (r: String) = SEQ(s, r)
}

// examples for the implicits:
// ALT(CHAR('a'), CHAR('b'))
// val areg : Rexp = "a" | "b"

// SEQ(CHAR('a'), CHAR('b')) 
// val sreg : Rexp = "a" ~ "b"


// ADD YOUR CODE BELOW
//======================

// (1)
def nullable (r: Rexp) : Boolean = r match {
  case ZERO     => false
  case ONE      => true
  case CHAR(_)  => false
  case ALTs(rs) => (for(reg <- rs) yield nullable(reg)).exists(_ == true)
  case SEQs(rs) => (for(reg <- rs) yield nullable(reg)).forall(_ == true)
  case STAR(_)  => true
}

/*
nullable(ZERO) == false
nullable(ONE) == true
nullable(CHAR('a')) == false
nullable(ZERO | ONE) == true
nullable(ZERO | CHAR('a')) == false
nullable(ONE ~ ONE) == true
nullable(ONE ~ CHAR('a')) == false
nullable(STAR(ZERO)) == true
nullable(ALTs(List(ONE, CHAR('a'), ZERO))) == true
nullable(SEQs(List(ONE, ALTs(List(ONE, CHAR('a'), ZERO)), STAR(ZERO)))) == true
*/

// (2) 
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 ALTs(rs)       => ALTs(for(reg <- rs) yield der(c, reg))
  case SEQs(Nil)      => ZERO
  case SEQs(r :: rs)  => if(nullable(r)) ALT(SEQs(der(c, r) :: rs), der(c, SEQs(rs))) else SEQs(der(c, r) :: rs)
  case STAR(r)        => SEQ(der(c,r), STAR(r))
}

/*
der('a', ZERO | ONE) == (ZERO | ZERO)
der('a', (CHAR('a') | ONE) ~ CHAR('a')) == ALT((ONE | ZERO) ~ CHAR('a'), SEQs(List(ONE)))
der('a', (CHAR('a') | CHAR('a')) ~ CHAR('a')) == (ONE | ONE) ~ CHAR('a')
der('a', STAR(CHAR('a'))) == (ONE ~ STAR(CHAR('a')))
der('b', STAR(CHAR('a'))) == (ZERO ~ STAR(CHAR('a')))
*/

// (3) 
def denest(rs: List[Rexp]) : List[Rexp] = rs match {
  case Nil                => Nil
  case ZERO :: rest       => denest(rest)
  case ALTs(rgs) :: rest  => rgs ::: denest(rest)
  case r :: rest          => r :: denest(rest)
}

/*
denest(List(ONE, ZERO, ALTs(List(ONE, CHAR('a'))))) == List(ONE, ONE, CHAR('a'))
denest(List(ONE ~ ONE, ZERO, ZERO | ONE)) == List(ONE ~ ONE, ZERO, ONE)
*/

// (4)
def flts(rs: List[Rexp], acc: List[Rexp] = Nil) : List[Rexp] = rs match {
  case Nil                => acc
  case ZERO :: rest       => List(ZERO)
  case ONE :: rest        => flts(rest, acc)
  case SEQs(rgs) :: rest  => flts(rest, acc ::: rgs)
  case r :: rest          => flts(rest, acc ::: List(r)) 
}

/*
flts(List(CHAR('a'), ZERO, ONE), Nil) == List(ZERO)
flts(List(CHAR('a'), ONE, ONE, CHAR('b')), Nil) == List(CHAR('a'), CHAR('b'))
flts(List(ONE ~ CHAR('a'), CHAR('b') ~ ONE), Nil) == List(ONE, CHAR('a'), CHAR('b'), ONE)
*/

// (5)
def ALTs_smart(rs: List[Rexp]) : Rexp = rs match {
  case Nil      => ZERO
  case List(r)  => r
  case _        => ALTs(rs)
}

def SEQs_smart(rs: List[Rexp]) : Rexp = rs match {
  case Nil      => ONE
  case List(r)  => r
  case _        => SEQs(rs)
}

/*
SEQs_smart(Nil) == ONE
SEQs_smart(List(ZERO)) == ZERO
SEQs_smart(List(CHAR('a'))) == CHAR('a')
SEQs_smart(List(ONE ~ ONE)) == ONE ~ ONE
SEQs_smart(List(ONE, ONE)) == SEQs(List(ONE, ONE))
ALTs_smart(Nil) == ZERO
ALTs_smart(List(ONE ~ ONE)) == ONE ~ ONE
ALTs_smart(List(ZERO, ZERO)) == ALTs(List(ZERO, ZERO))
*/

// (6)
def simp(r: Rexp) : Rexp = r match {
  case ALTs(rs) => ALTs_smart(denest(for(reg <- rs) yield simp(reg)).distinct)
  case SEQs(rs) => SEQs_smart(flts(for(reg <- rs) yield simp(reg)))
  case _        => r
}

/*
simp(ZERO | ONE) == ONE
simp(STAR(ZERO | ONE)) == STAR(ZERO | ONE)
simp(ONE ~ (ONE ~ (ONE ~ CHAR('a')))) == CHAR('a')
simp(((ONE ~ ONE) ~ ONE) ~ CHAR('a')) == CHAR('a')
simp(((ONE | ONE) ~ ONE) ~ CHAR('a')) == CHAR('a')
simp(ONE ~ (ONE ~ (ONE ~ ZERO))) == ZERO
simp(ALT(ONE ~ (ONE ~ (ONE ~ ZERO)), CHAR('a'))) == CHAR('a')
simp(CHAR('a') | CHAR('a')) == CHAR('a')
simp(CHAR('a') ~ CHAR('a')) == CHAR('a') ~ CHAR('a')
simp(ONE | CHAR('a')) == (ONE | CHAR('a'))
simp(ALT((CHAR('a') | ZERO) ~ ONE,((ONE | CHAR('b')) | CHAR('c')) ~ (CHAR('d') ~ ZERO))) == CHAR('a')
simp((ZERO | ((ZERO | ZERO) | (ZERO | ZERO))) ~ ((ONE | ZERO) | ONE ) ~ (CHAR('a'))) == ZERO
simp(ALT(ONE | ONE, ONE | ONE)) == ONE
simp(ALT(ZERO | CHAR('a'), CHAR('a') | ZERO)) == CHAR('a')
simp(ALT(ONE | CHAR('a'), CHAR('a') | ONE)) == ALT(ONE, CHAR('a'))
simp(ALTs(Nil)) == ZERO
simp(SEQs(List(CHAR('a')))) == CHAR('a')
*/

// (7)
def ders (s: List[Char], r: Rexp) : Rexp = s match {
  case Nil      => r
  case c :: cs  => ders(cs, simp(der(c, r)))
}
def matcher(r: Rexp, s: String): Boolean = {
  val derivatives = ders(s.toList, r)
  nullable(derivatives)
}

/*
val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))
ders("aaaaa".toList, EVIL) == SEQs(List(STAR(CHAR('a')), STAR(STAR(CHAR('a'))), CHAR('b')))
ders(List('b'), EVIL) == ONE
ders("bb".toList, EVIL) == ZERO
matcher(EVIL, "a" * 5 ++ "b") == true
matcher(EVIL, "b") == true
matcher(EVIL, "bb") == false
matcher("abc", "abc") == true
matcher(("ab" | "a") ~ (ONE | "bc"), "abc") == true
matcher(ONE, "") == true
matcher(ZERO, "") == false
matcher(ONE | CHAR('a'), "") == true
matcher(ONE | CHAR('a'), "a") == true
*/

// (8) 
def size(r: Rexp): Int = r match {
  case ZERO     => 1
  case ONE      => 1
  case CHAR(_)  => 1
  case ALTs(rs) => 1 + (for(reg <- rs) yield size(reg)).sum
  case SEQs(rs) => 1 + (for(reg <- rs) yield size(reg)).sum
  case STAR(r)  => 1 + size(r)
}

/*
val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))
size(der('a', der('a', EVIL))) == 36
size(der('a', der('a', der('a', EVIL)))) == 83
size(ders("aaaaaa".toList, EVIL)) == 7
size(ders(("a" * 50).toList, EVIL)) == 7
*/


// Some testing data
//===================
/*

simp(ALT(ONE | CHAR('a'), CHAR('a') | ONE))   // => ALTs(List(ONE, CHAR(a)))
simp(((CHAR('a') | ZERO) ~ ONE) | (((ONE | CHAR('b')) | CHAR('c')) ~ (CHAR('d') ~ ZERO)))   // => CHAR(a)

matcher(("a" ~ "b") ~ "c", "ab")   // => false
matcher(("a" ~ "b") ~ "c", "abc")  // => true


// the supposedly 'evil' regular expression (a*)* b
val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))

matcher(EVIL, "a" * 1000)          // => false
matcher(EVIL, "a" * 1000 ++ "b")   // => true


// size without simplifications
size(der('a', der('a', EVIL)))             // => 36
size(der('a', der('a', der('a', EVIL))))   // => 83

// size with simplification
size(simp(der('a', der('a', EVIL))))           // => 7
size(simp(der('a', der('a', der('a', EVIL))))) // => 7

// Python needs around 30 seconds for matching 28 a's with EVIL. 
// Java 9 and later increase this to an "astonishing" 40000 a's in
// 30 seconds.
//
// Lets see how long it really takes to match strings with 
// 5 Million a's...it should be in the range of a few
// of seconds.

def time_needed[T](i: Int, code: => T) = {
  val start = System.nanoTime()
  for (j <- 1 to i) code
  val end = System.nanoTime()
  "%.5f".format((end - start)/(i * 1.0e9))
}

for (i <- 0 to 5000000 by 500000) {
  println(s"$i ${time_needed(2, matcher(EVIL, "a" * i))} secs.") 
}

// another "power" test case 
simp(Iterator.iterate(ONE:Rexp)(r => SEQ(r, ONE | ONE)).drop(50).next()) == ONE

// the Iterator produces the rexp
//
//      SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE)
//
//    where SEQ is nested 50 times.

*/

}






// This template code is subject to copyright 
// by King's College London, 2022. Do not 
// make the template code public in any shape 
// or form, and do not exchange it with other 
// students under any circumstance.