--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/re1.scala	Thu Jul 30 13:50:54 2020 +0100
@@ -0,0 +1,154 @@
+// A simple matcher for basic 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))
+
+
+// examples from the homework
+
+val r = 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)))
+
+
+// the optional regular expression (one or zero times)
+def OPT(r: Rexp) = ALT(r, ONE)
+
+// the n-times regular expression (explicitly expanded)
+def NTIMES(r: Rexp, n: Int) : Rexp = n match {
+  case 0 => ONE
+  case 1 => r
+  case n => SEQ(r, NTIMES(r, n - 1))
+}
+
+
+// Test Cases
+
+// 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})
+println("Test (a?{n}) (a{n})")
+
+for (i <- 0 to 20 by 2) {
+  println(f"$i: ${time_needed(2, matcher(EVIL1(i), "a" * i))}%.5f")
+}
+
+// test: (a*)* b
+println("Test (a*)* b")
+
+for (i <- 0 to 20 by 2) {
+  println(f"$i: ${time_needed(2, matcher(EVIL2, "a" * i))}%.5f")
+}
+
+
+// the size of a regular expressions - for testing purposes 
+def size(r: Rexp) : Int = r match {
+  case ZERO => 1
+  case ONE => 1
+  case CHAR(_) => 1
+  case ALT(r1, r2) => 1 + size(r1) + size(r2)
+  case SEQ(r1, r2) => 1 + size(r1) + size(r2)
+  case STAR(r) => 1 + size(r)
+}
+
+// the expicit expansion in EVIL1(n) increases
+// drastically its size
+
+size(EVIL1(1))  // 5
+size(EVIL1(3))  // 17
+size(EVIL1(5))  // 29
+size(EVIL1(7))  // 41
+size(EVIL1(20)) // 119
+
+// given a regular expression and building successive
+// derivatives might result into bigger and bigger
+// regular expressions...here is an example for this:
+
+// (a+b)* o a o b o (a+b)*
+val BIG_aux = STAR(ALT(CHAR('a'), CHAR('b')))
+val BIG = SEQ(BIG_aux, SEQ(CHAR('a'),SEQ(CHAR('b'), BIG_aux)))
+
+size(ders("".toList, BIG))              // 13
+size(ders("ab".toList, BIG))            // 51
+size(ders("abab".toList, BIG))          // 112
+size(ders("ababab".toList, BIG))        // 191
+size(ders("abababab".toList, BIG))      // 288
+size(ders("ababababab".toList, BIG))    // 403
+size(ders("abababababab".toList, BIG))  // 536
+
+
+size(ders(("ab" * 200).toList, BIG))    // 366808
+
+for (i <- 0 to 200 by 10) {
+  println(f"$i: ${time_needed(2, matcher(BIG, "ab" * i))}%.5f")
+}
+
+
+
+
+//////////////////////////////////////
+def concat(A: Set[String], B: Set[String]) : Set[String] =
+  for (s1 <- A; s2 <- B) yield s1 ++ s2
+  
+
+val A = Set("foo", "bar")
+val B = Set("a", "b")
+
+