// A version with simplification of derivatives;+
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// this keeps the regular expressions small, which+
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// is good for the run-time+
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// +
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// call the test cases with X = {1,2}+
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//+
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//   amm re3.sc testX+
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//+
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// or +
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//+
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//   amm re3.sc all+
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+
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+
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abstract class Rexp+
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case object ZERO extends Rexp+
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case object ONE extends Rexp+
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case class CHAR(c: Char) extends Rexp+
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case class ALT(r1: Rexp, r2: Rexp) extends Rexp +
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case class SEQ(r1: Rexp, r2: Rexp) extends Rexp +
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case class STAR(r: Rexp) extends Rexp +
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case class NTIMES(r: Rexp, n: Int) extends Rexp +
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+
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+
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+
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// the nullable function: tests whether the regular +
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// expression can recognise the empty string+
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def nullable (r: Rexp) : Boolean = r match {+
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  case ZERO => false+
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  case ONE => true+
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  case CHAR(_) => false+
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  case ALT(r1, r2) => nullable(r1) || nullable(r2)+
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  case SEQ(r1, r2) => nullable(r1) && nullable(r2)+
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  case STAR(_) => true+
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  case NTIMES(r, i) => if (i == 0) true else nullable(r)+
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}+
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+
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// the derivative of a regular expression w.r.t. a character+
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def der(c: Char, r: Rexp) : Rexp = r match {+
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  case ZERO => ZERO+
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  case ONE => ZERO+
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  case CHAR(d) => if (c == d) ONE else ZERO+
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  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))+
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  case SEQ(r1, r2) => +
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    if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))+
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    else SEQ(der(c, r1), r2)+
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  case STAR(r1) => SEQ(der(c, r1), STAR(r1))+
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  case NTIMES(r, i) => +
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    if (i == 0) ZERO else SEQ(der(c, r), NTIMES(r, i - 1))+
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}+
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+
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def simp(r: Rexp) : Rexp = r match {+
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  case ALT(r1, r2) => (simp(r1), simp(r2)) match {+
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    case (ZERO, r2s) => r2s+
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    case (r1s, ZERO) => r1s+
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    case (r1s, r2s) => if (r1s == r2s) r1s else ALT (r1s, r2s)+
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  }+
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  case SEQ(r1, r2) =>  (simp(r1), simp(r2)) match {+
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    case (ZERO, _) => ZERO+
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    case (_, ZERO) => ZERO+
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    case (ONE, r2s) => r2s+
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    case (r1s, ONE) => r1s+
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    case (r1s, r2s) => SEQ(r1s, r2s)+
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  }+
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  case r => r+
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}+
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+
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+
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// the derivative w.r.t. a string (iterates der)+
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def ders(s: List[Char], r: Rexp) : Rexp = s match {+
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  case Nil => r+
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  case c::s => ders(s, simp(der(c, r)))+
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}+
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+
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+
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// the main matcher function+
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def matcher(r: Rexp, s: String) : Boolean = +
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  nullable(ders(s.toList, r))+
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+
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+
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// one or zero+
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def OPT(r: Rexp) = ALT(r, ONE)+
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+
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+
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// Test Cases+
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+
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// evil regular expressions: (a?){n} a{n}  and (a*)* b+
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def EVIL1(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))+
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val EVIL2 = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))+
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+
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+
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def time_needed[T](i: Int, code: => T) = {+
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  val start = System.nanoTime()+
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  for (j <- 1 to i) code+
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  val end = System.nanoTime()+
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  (end - start)/(i * 1.0e9)+
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}+
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+
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+
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@arg(doc = "Test (a?{n}) (a{n})")+
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@main+
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def test1() = {+
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  println("Test (a?{n}) (a{n})")+
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+
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  for (i <- 0 to 9000 by 1000) {+
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    println(f"$i: ${time_needed(3, matcher(EVIL1(i), "a" * i))}%.5f")+
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  }+
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}  +
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+
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@arg(doc = "Test (a*)* b")+
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@main+
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def test2() = {+
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  println("Test (a*)* b")+
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+
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  for (i <- 0 to 6000000 by 500000) {+
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    println(f"$i: ${time_needed(3, matcher(EVIL2, "a" * i))}%.5f")+
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  }+
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}+
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+
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// size of a regular expressions - for testing purposes +
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def size(r: Rexp) : Int = r match {+
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  case ZERO => 1+
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  case ONE => 1+
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  case CHAR(_) => 1+
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  case ALT(r1, r2) => 1 + size(r1) + size(r2)+
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  case SEQ(r1, r2) => 1 + size(r1) + size(r2)+
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  case STAR(r) => 1 + size(r)+
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  case NTIMES(r, _) => 1 + size(r)+
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}+
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+
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+
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// now the size of the derivatives grows +
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// much, much slower+
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+
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size(ders("".toList, EVIL2))      // 5+
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size(ders("a".toList, EVIL2))     // 8+
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size(ders("aa".toList, EVIL2))    // 8+
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size(ders("aaa".toList, EVIL2))   // 8+
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size(ders("aaaa".toList, EVIL2))  // 8+
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size(ders("aaaaa".toList, EVIL2)) // 8+
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+
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+
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@arg(doc = "All tests.")+
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@main+
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def all() = { test1(); test2() } +
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+
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+
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+
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+
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+
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// PS:+
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//+
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// If you want to dig deeper into the topic, you can have +
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// a look at these examples which still explode. They+
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// need an even more aggressive simplification.+
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+
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// test: (a + aa)*+
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val EVIL3 = STAR(ALT(CHAR('a'), SEQ(CHAR('a'), CHAR('a'))))+
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+
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+
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@arg(doc = "Test EVIL3")+
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@main+
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def test3() = {+
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  println("Test (a + aa)*")+
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+
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  for (i <- 0 to 30 by 5) {+
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    println(f"$i: ${time_needed(1, matcher(EVIL3, "a" * i))}%.5f")+
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  }+
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}+
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+
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+
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// test: (1 + a + aa)*+
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val EVIL4 = STAR(ALT(ONE, ALT(CHAR('a'), SEQ(CHAR('a'), CHAR('a')))))+
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+
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@arg(doc = "Test EVIL4")+
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@main+
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def test4() = {+
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  println("Test (1 + a + aa)*")+
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+
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  for (i <- 0 to 30 by 5) {+
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    println(f"$i: ${time_needed(1, matcher(EVIL4, "a" * i))}%.5f")+
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  }+
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}+
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+
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@arg(doc = "Tests that show not all is hunky-dory, but a solution leads too far afield.")+
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@main+
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def fail() = { test3(); test4() } +
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+
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+
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// runs with amm2 and amm3+
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+
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def pp(r: Rexp): String = r match {+
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  case SEQ(CHAR(a1), SEQ(r1, r2)) => s"${a1}${pp(r1)}${pp(r2)}"+
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  case SEQ(ONE, SEQ(r1, r2)) => s"1${pp(r1)}${pp(r2)}"+
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  case SEQ(ZERO, SEQ(r1, r2)) => s"0${pp(r1)}${pp(r2)}"+
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  case SEQ(CHAR(a1), CHAR(a2)) => s"${a1}${a2}"+
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  case SEQ(ONE, CHAR(a2)) => s"1${a2}"+
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  case SEQ(ZERO, CHAR(a2)) => s"0${a2}" +
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  case ZERO => "0"+
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  case ONE => "1"+
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  case CHAR(a) => a.toString+
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  case ALT(r1, r2) => s"(${pp(r1)} + ${pp(r2)})"+
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  case SEQ(r1, r2) => s"(${pp(r1)} o ${pp(r2)})"+
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  case STAR(r1) => s"(${pp(r1)})*"+
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}+
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+
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+
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val REG = STAR(ALT(CHAR('a'), SEQ(CHAR('a'), CHAR('a'))))+
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+
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print(pp(ders("".toList, REG)))+
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print(pp(ders("a".toList, REG)))+
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print(pp(ders("aa".toList, REG)))+
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print(pp(ders("aaa".toList, REG)))+
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+
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size(ders("".toList, REG))        // 6+
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size(ders("a".toList, REG))       // 12+
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size(ders("aa".toList, REG))      // 27+
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size(ders("aaa".toList, REG))     // 55+
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size(ders("aaaa".toList, REG))    // 8+
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size(ders("aaaaa".toList, REG))   // 169+
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size(ders("aaaaaa".toList, REG))  // 283+
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size(ders(("a" * 7).toList, REG)) // 468+
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size(ders(("a" * 8).toList, REG)) // 767+
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size(ders(("a" * 9).toList, REG)) // 1251+
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size(ders(("a" * 10).toList, REG))// 2034+
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size(ders(("a" * 11).toList, REG))// 3301+
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+
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for (i <- (0 to 40)) {+
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  println(s"$i:" + size(ders(("a" * i).toList, REG)))+
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}+
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