// 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(a,b)=>nullable(a)||nullable(b)
  case SEQ(a,b) => nullable(a) && nullable(b)
  case STAR(_) => true
}


/*val rex = "1~0.%|11"

assert(der('1',rex) == SEQ(ONE,SEQ(CHAR(~),SEQ(CHAR(0),SEQ(CHAR(.),SEQ(CHAR(%),SEQ(CHAR(|),SEQ(CHAR(1),CHAR(1)))))))))

assert(der('1',der('1',rex)) ==
        ALT(SEQ(ZERO,SEQ(CHAR(~),SEQ(CHAR(0),SEQ(CHAR(.),SEQ(CHAR(%),SEQ(CHAR(|),
        SEQ(CHAR(1),CHAR(1)))))))),SEQ(ZERO,SEQ(CHAR(0),SEQ(CHAR(.),SEQ(CHAR(%),
        SEQ(CHAR(|),SEQ(CHAR(1),CHAR(1))))))))*/

// (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.

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 ALT(a,b) => der(c,a)|der(c,b)
  case SEQ(a,b) => if(nullable(a)) {(der(c,a)~b)|der(c,b)}
                   else der(c,a)~b
  case STAR(a) => der(c,a)~STAR(a)
}

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


//ALT(SEQ(ZERO,ZERO),ZERO)
//ALT(ALT(ZERO,ZERO),ALT(ZERO,ZERO))

// * == |
// + == ~
// (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 SEQ(ZERO,_) => ZERO
  case SEQ(_,ZERO) => ZERO
  case SEQ(ONE,a) => simp(a)
  case SEQ(a,ONE) => simp(a)
  case ALT(ZERO,a) => simp(a)
  case ALT(a,ZERO) => simp(a)
  case ALT(a,b) => if(a == b) simp(a) else r
  case _ => r
}*/

def simp(r: Rexp) : Rexp = r match{
  case SEQ(a,b) =>{ val sa = simp(a)
                    val sb = simp(b)
                    if(sa == ZERO || sb == ZERO) ZERO
                    else if(sa == ONE) sb
                    else if(sb == ONE) sa
                    else SEQ(sa,sb)
                    }
  case ALT(a,b) =>{ val sa = simp(a)
                    val sb = simp(b)
                    if(sa == ONE || sb == ONE) ONE
                    else if(sa == ZERO) sb
                    else if(sb == ZERO) sa
                    else if(sa == sb) sa
                    else ALT(sa,sb)
                    }
  //case STAR(STAR(a)) => simp(STAR(a))
  //case STAR(a) => STAR(simp(a))
  case _ => r
  /*
  case SEQ(ZERO,_) => ZERO
  case SEQ(_,ZERO) => ZERO
  case SEQ(ONE,a) => simp(a)
  case SEQ(a,ONE) => simp(a)
  case SEQ(a,b) => SEQ(simp(a),simp(b))
  //case ALT(ZERO,a) => simp(a)
  case ALT(a,ZERO) => simp(a)
  case ALT(ONE,_) => ONE
  case ALT(_,ONE) => ONE
  case ALT(a,b) => {val sa = simp(a)
                    if(sa == simp(b)) sa else r
                    }
  case STAR(STAR(a)) => simp(STAR(a))
  case STAR(a) => STAR(simp(a))
  case _ => r*/
}


/*val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))
println("TEST: " + simp(der('a', der('a', EVIL))))
println(simp(ONE))
val r1 = ALT(ZERO,ONE)
val r2 = SEQ(ONE,ZERO)
val r3 = SEQ(r1,SEQ(r2,r1))
println("R1 = " + simp(r1))
println(simp(r2))
println(simp(r3))
*/

// (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 a::z => ders(z,simp(der(a,r)))
}

def matcher(r: Rexp, s: String): Boolean = {
  val derivatives = simp(ders(s.toList,r))
  nullable(derivatives)
}

// (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(a,b) => 1 + size(a) + size(b)
  case ALT(a,b) => 1 + size(a) + size(b)
  case STAR(a) => 1 + size(a)
}

println(der('a', ZERO | ONE))// == (ZERO | ZERO)
println(der('a', (CHAR('a') | ONE) ~ CHAR('a')))// ==ALT((ONE | ZERO) ~ CHAR('a'), ONE)
println(der('a', STAR(CHAR('a'))))// == (ONE ~ STAR(CHAR('a')))
println(der('b', STAR(CHAR('a'))))// == (ZERO ~ STAR(CHAR('a'))))
// some testing data
/*

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


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


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

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

// size with simplification
assert("8 " + size(simp(der('a', der('a', EVIL))))           ==8)// => 8
assert("8 " + size(simp(der('a', der('a', der('a', EVIL))))) ==8) // => 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
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.

*/