// A simple matcher for basic regular expressions

//

// Call the testcases with X = {1,2,3}

//

//   amm re1.sc testX

//

// or 

//

//   amm re1.sc all

//

// regular expressions

abstract class Rexp

case object ZERO extends Rexp                    // matches nothing

case object ONE extends Rexp                     // matches an 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

// nullable function: tests whether a regular 

// expression can recognise the empty string  

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

}

// 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(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))

}

// the derivative w.r.t. a string (iterates der)

def ders(s: List[Char], r: Rexp) : Rexp = s match {

  case Nil => r

  case c::s => ders(s, der(c, r))

}

// the main matcher function

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

  nullable(ders(s.toList, r))

// some examples from the homework

val r = SEQ(CHAR('a'), CHAR('c'))

matcher(r, "ac")

val r1 = STAR(ALT(SEQ(CHAR('a'), CHAR('b')), CHAR('b')))

der('a', r)

der('b', r)

der('c', r)

val r2 = SEQ(SEQ(CHAR('x'), CHAR('y')), CHAR('z'))

der('x', r2)

der('y', der('x', r2))

der('z', der('y', der('x', r2)))

// Test Cases

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

// the optional regular expression (one or zero times)

def OPT(r: Rexp) = ALT(r, ONE)   // r + 1

// the n-times regular expression (explicitly expanded to SEQs)

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

  case 0 => ONE

  case 1 => r

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

}

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

def EVIL1(n: Int) = 

  SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))

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

val EVIL2 = 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)

}

// test: (a?{n}) (a{n})

def test1() = {

  println("Test (a?{n}) (a{n})")

  for (i <- 0 to 20 by 2) {

    println(f"$i: ${time_needed(2, matcher(EVIL1(i), "a" * i))}%.5f")

  }

}

// test: (a*)* b

def test2() = {

  println("Test (a*)* b")

  for (i <- 0 to 20 by 2) {

    println(f"$i: ${time_needed(2, matcher(EVIL2, "a" * i))}%.5f")

  }

}

// the size of a regular expressions - for testing purposes 

def size(r: Rexp) : Int = r match {

  case ZERO => 1

  case ONE => 1

  case CHAR(_) => 1

  case ALT(r1, r2) => 1 + size(r1) + size(r2)

  case SEQ(r1, r2) => 1 + size(r1) + size(r2)

  case STAR(r) => 1 + size(r)

}

// the expicit expansion in EVIL1(n) increases

// drastically its size - (a?){n} a{n}

size(EVIL1(1))  // 5

size(EVIL1(3))  // 17

size(EVIL1(5))  // 29

size(EVIL1(7))  // 41

size(EVIL1(20)) // 119

size(ders(("a" * 20).toList, EVIL1(20))) 

// given a regular expression and building successive

// derivatives might result into bigger and bigger

// regular expressions...here is an example for this:

// (a + aa)*

val BIG = STAR(ALT(CHAR('a'), SEQ(CHAR('a'), CHAR('a'))))

size(ders("".toList, BIG))              // 13

size(ders("aa".toList, BIG))            // 51

size(ders("aaaa".toList, BIG))          // 112

size(ders("aaaaaa".toList, BIG))        // 191

size(ders("aaaaaaaa".toList, BIG))      // 288

size(ders("aaaaaaaaaa".toList, BIG))    // 403

size(ders("aaaaaaaaaaaa".toList, BIG))  // 536

size(ders(("a" * 30).toList, BIG))      // 31010539

def test3() = {

  println("Test (a + aa)*")

  for (i <- 0 to 30 by 5) {

    println(f"$i: ${time_needed(2, matcher(BIG, "a" * i))}%.5f")

  }

}

@main

def all() = { test1(); test2() ; test3() } 

//all()