// Part 1 about Regular Expression Matching
//==========================================

// Regular Expressions
abstract class Rexp
case object ZERO extends Rexp
case object ONE extends Rexp
case class CHAR(c: Char) extends Rexp
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

 
// 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)
}

// (1) Complete the function nullable according to
// the definition given in the coursework; this 
// function checks whether a regular expression
// can match the empty string and Returns a boolean
// accordingly.

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
}



// (2) Complete the function der according to
// the definition given in the coursework; this
// function calculates the derivative of a 
// regular expression w.r.t. a character.

//TODO: debug
//TODO: understand this more. 
// first test runs
// test 2 fails
// test 3 runs
// test 4 runs
def der (c: Char, r: Rexp) : Rexp = r match {
    //TODO: debug
    case ZERO => ZERO
    case ONE => ZERO
    case CHAR(r1) => if (c == r1) 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))
}


// (3) Complete the simp function according to
// the specification given in the coursework; this
// function simplifies a regular expression from
// the inside out, like you would simplify arithmetic 
// expressions; however it does not simplify inside 
// STAR-regular expressions.

def simp(r: Rexp) : Rexp = r match {
    case STAR(_) => r
    case SEQ(r1, r2) => (simp(r1), simp(r2)) match { // potential failure
        case (_, ZERO) => ZERO
        case (ZERO, _) => ZERO
        case (r1, ONE) => simp(r1)
        case (ONE, r2) => simp(r2)
        case (r1, r2) => SEQ(r1, r2)
    }
    case ALT(r1, r2) => (simp(r1), simp(r2)) match {
        case (r1, ZERO) => simp(r1)
        case (ZERO, r1) => simp(r1)
        case (r1, r2) if r1 == r2 => simp(r1)
        case (r1, r2) => ALT(r1, r2)
    }
    case r => r
}


// (4) Complete the two functions below; the first 
// calculates the derivative w.r.t. a string; the second
// is the regular expression matcher taking a regular
// expression and a string and checks whether the
// string matches the regular expression

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 = {
    nullable(ders(s.toList, r))
}


// (5) Complete the size function for regular
// expressions according to the specification 
// given in the coursework.

def size(r: Rexp): Int = r match {
    case ZERO => 1
    case ONE => 1
    case CHAR(_) => 1
    case SEQ(r1, r2) => 1 + size(r1) + size(r2)
    case ALT(r1, r2) => 1 + size(r1) + size(r2)
    case STAR(r1) => 1 + size(r1)
}


// some testing data

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

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

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

// size without simplifications
//size(der('a', der('a', EVIL)))             // => 28
//size(der('a', der('a', der('a', EVIL))))   // => 58



// size with simplification
//size(simp(der('a', der('a', EVIL))))           // => 8
//size(simp(der('a', der('a', der('a', EVIL))))) // => 8

// 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 couple
// of seconds.

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)
}

//for (i <- 0 to 5000000 by 500000) {
//  println(i + " " + "%.5f".format(time_needed(2, matcher(EVIL, "a" * i))))
//}

// another "power" test case 
println(simp(Iterator.iterate(ONE:Rexp)(r => ALT(r, r)).drop(40).next))

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