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