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// A simple matcher for basic regular expressions
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abstract class Rexp
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case object ZERO extends Rexp // matches nothing
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case object ONE extends Rexp // matches an empty string
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case class CHAR(c: Char) extends Rexp // matches a character c
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case class ALT(r1: Rexp, r2: Rexp) extends Rexp // alternative
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case class SEQ(r1: Rexp, r2: Rexp) extends Rexp // sequence
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case class STAR(r: Rexp) extends Rexp // star
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// nullable function: tests whether a 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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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case ZERO => false
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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}
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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case ZERO => ZERO
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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case ONE => ZERO
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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case STAR(r1) => SEQ(der(c, r1), STAR(r1))
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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, der(c, r))
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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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// examples from the homework
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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val r = STAR(ALT(SEQ(CHAR('a'), CHAR('b')), CHAR('b')))
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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der('a', r)
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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der('b', r)
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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der('c', r)
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val r2 = SEQ(SEQ(CHAR('x'), CHAR('y')), CHAR('z'))
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der('x', r2)
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der('y', der('x', r2))
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der('z', der('y', der('x', r2)))
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// the optional regular expression (one or zero times)
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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def OPT(r: Rexp) = ALT(r, ONE)
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// the n-times regular expression (explicitly expanded)
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def NTIMES(r: Rexp, n: Int) : Rexp = n match {
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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case 0 => ONE
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case 1 => r
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case n => SEQ(r, NTIMES(r, n - 1))
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}
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// Test Cases
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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// the evil regular expression a?{n} a{n}
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def EVIL1(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))
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// the evil regular expression (a*)*b
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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val EVIL2 = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))
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// for measuring time
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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def time_needed[T](i: Int, code: => T) = {
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val start = System.nanoTime()
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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// test: (a?{n}) (a{n})
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println("Test (a?{n}) (a{n})")
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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for (i <- 0 to 20 by 2) {
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println(f"$i: ${time_needed(2, matcher(EVIL1(i), "a" * i))}%.5f")
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}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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// test: (a*)* b
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println("Test (a*)* b")
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for (i <- 0 to 20 by 2) {
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println(f"$i: ${time_needed(2, matcher(EVIL2, "a" * i))}%.5f")
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}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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// the 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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}
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// the expicit expansion in EVIL1(n) increases
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// drastically its size
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size(EVIL1(1)) // 5
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size(EVIL1(3)) // 17
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size(EVIL1(5)) // 29
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size(EVIL1(7)) // 41
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size(EVIL1(20)) // 119
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// given a regular expression and building successive
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// derivatives might result into bigger and bigger
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// regular expressions...here is an example for this:
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// (a+b)* o a o b o (a+b)*
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val BIG_aux = STAR(ALT(CHAR('a'), CHAR('b')))
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val BIG = SEQ(BIG_aux, SEQ(CHAR('a'),SEQ(CHAR('b'), BIG_aux)))
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size(ders("".toList, BIG)) // 13
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size(ders("ab".toList, BIG)) // 51
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size(ders("abab".toList, BIG)) // 112
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size(ders("ababab".toList, BIG)) // 191
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size(ders("abababab".toList, BIG)) // 288
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size(ders("ababababab".toList, BIG)) // 403
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size(ders("abababababab".toList, BIG)) // 536
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size(ders(("ab" * 200).toList, BIG)) // 366808
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for (i <- 0 to 200 by 10) {
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println(f"$i: ${time_needed(2, matcher(BIG, "ab" * i))}%.5f")
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}
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//////////////////////////////////////
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def concat(A: Set[String], B: Set[String]) : Set[String] =
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for (s1 <- A; s2 <- B) yield s1 ++ s2
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val A = Set("foo", "bar")
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val B = Set("a", "b")
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