// 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.