// Main Part 3 about Regular Expression Matching

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

object M3 {

// Regular Expressions

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 in 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) 

def nullable (r: Rexp) : Boolean = r match {

  case ZERO => false

  case ONE => true

  case CHAR(_) => false

  case ALTs(rs) => rs.exists(nullable)

  case SEQs(rs) => rs.forall(nullable)

  case STAR(_) => 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(rs.map(der(c, _)))

  case SEQs(Nil) => ZERO

  case SEQs(r1::rs) => 

    if (nullable(r1)) ALT(SEQs(der(c, r1)::rs), der(c, SEQs(rs)))

    else SEQs(der(c, r1):: rs)

  case STAR(r1) => SEQ(der(c, r1), STAR(r1))

}

// (3) 

def denest(rs: List[Rexp]) : List[Rexp] = rs match {

  case Nil => Nil

  case ZERO::tl => denest(tl)

  case ALTs(rs1)::rs2 => rs1 ::: denest(rs2)  

  case r::rs => r :: denest(rs) 

}

// (4)

def flts(rs: List[Rexp], acc: List[Rexp] = Nil) : List[Rexp] = rs match {

  case Nil => acc

  case ZERO::rs => ZERO::Nil

  case ONE::rs => flts(rs, acc)

  case SEQs(rs1)::rs => flts(rs, acc ::: rs1)

  case r::rs => flts(rs, acc :+ r) 

}

// (5)

def ALTs_smart(rs: List[Rexp]) : Rexp = rs match {

  case Nil => ZERO

  case r::Nil => r  

  case rs => ALTs(rs)

}

def SEQs_smart(rs: List[Rexp]) : Rexp = rs match {

  case Nil => ONE

  case r::Nil => r

  case rs => SEQs(rs) 

}

// (6) 

def simp(r: Rexp) : Rexp = r match {

  case ALTs(rs) => 

    ALTs_smart(denest(rs.map(simp)).distinct)

  case SEQs(rs) => 

    SEQs_smart(flts(rs.map(simp)))

  case r => r

}

//println("Simp tests")

//println(simp(ALT(ONE | CHAR('a'), CHAR('a') | ONE)))

//println(simp(((CHAR('a') | ZERO) ~ ONE) | 

//              (((ONE | CHAR('b')) | CHAR('c')) ~ (CHAR('d') ~ ZERO))))

// (7) 

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

  case Nil => r

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

}

// main matcher function

def matcher(r: Rexp, s: String) = nullable(ders(s.toList, r))

// (8) 

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

  case ZERO => 1

  case ONE => 1

  case CHAR(_) => 1

  case ALTs(rs) => 1 + rs.map(size).sum

  case SEQs(rs) => 1 + rs.map(size).sum

  case STAR(r1) => 1 + size(r1)

}

// some testing data

/*

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

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

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

val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))

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

println(matcher(EVIL, "a" * 1000))          // => false

// size without simplifications

println(size(der('a', der('a', EVIL))))             // => 36

println(size(der('a', der('a', der('a', EVIL)))))   // => 83

// size with simplification

println(simp(der('a', der('a', EVIL))))          

println(simp(der('a', der('a', der('a', EVIL)))))

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

println(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

// around 30 seconds.

//

// Lets see how long it takes to match strings with 

// 5 Million a's...it should be in the range of a 

// few 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 

println(simp(Iterator.iterate(ONE:Rexp)(r => SEQ(r, ONE | ONE)).drop(100).next()) == ONE)

// the Iterator produces the rexp

//

//      SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE)

//

//    where SEQ is nested 100 times.

*/ 

assert(simp(ZERO | ONE) == ONE)

assert(simp(STAR(ZERO | ONE)) == STAR(ZERO | ONE))

assert(simp(ONE ~ (ONE ~ (ONE ~ CHAR('a')))) == CHAR('a'))

assert(simp(((ONE ~ ONE) ~ ONE) ~ CHAR('a')) == CHAR('a'))

assert(simp(((ONE | ONE) ~ ONE) ~ CHAR('a')) == CHAR('a'))

assert(simp(ONE ~ (ONE ~ (ONE ~ ZERO))) == ZERO)

assert(simp(ALT(ONE ~ (ONE ~ (ONE ~ ZERO)), CHAR('a'))) == CHAR('a'))

assert(simp(CHAR('a') | CHAR('a')) == CHAR('a'))

assert(simp(CHAR('a') ~ CHAR('a')) == CHAR('a') ~ CHAR('a'))

assert(simp(ONE | CHAR('a')) == (ONE | CHAR('a')))

assert(simp(ALT((CHAR('a') | ZERO) ~ ONE,

                  ((ONE | CHAR('b')) | CHAR('c')) ~ (CHAR('d') ~ ZERO))) == CHAR('a'))

assert(simp((ZERO | ((ZERO | ZERO) | (ZERO | ZERO))) ~ ((ONE | ZERO) | ONE ) ~ (CHAR('a'))) == ZERO)

assert(simp(ALT(ONE | ONE, ONE | ONE)) == ONE)

assert(simp(ALT(ZERO | CHAR('a'), CHAR('a') | ZERO)) == CHAR('a'))

assert(simp(ALT(ONE | CHAR('a'), CHAR('a') | ONE)) == ALT(ONE, CHAR('a')))

assert(simp(ALTs(Nil)) == ZERO)

assert(simp(SEQs(List(CHAR('a')))) == CHAR('a'))

}