// A version which attempts to move whole strings, not+ −
// just characters, under derivatives whenever possible+ −
+ −
abstract class Rexp+ −
case object ZERO extends Rexp+ −
case object ONE extends Rexp+ −
case class CHAR(c: Char) extends Rexp+ −
case class ALT(r1: Rexp, r2: Rexp) extends Rexp + −
case class SEQ(r1: Rexp, r2: Rexp) extends Rexp + −
case class STAR(r: Rexp) extends Rexp + −
case class NTIMES(r: Rexp, n: Int) extends Rexp + −
+ −
// the nullable function: tests whether the 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+ −
case NTIMES(r, i) => if (i == 0) true else nullable(r)+ −
}+ −
+ −
// 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))+ −
case NTIMES(r1, i) => + −
if (i == 0) ZERO else SEQ(der(c, r1), NTIMES(r1, i - 1))+ −
}+ −
+ −
def simp(r: Rexp) : Rexp = r match {+ −
case ALT(r1, r2) => (simp(r1), simp(r2)) match {+ −
case (ZERO, r2s) => r2s+ −
case (r1s, ZERO) => r1s+ −
case (r1s, r2s) => if (r1s == r2s) r1s else ALT (r1s, r2s)+ −
}+ −
case SEQ(r1, r2) => (simp(r1), simp(r2)) match {+ −
case (ZERO, _) => ZERO+ −
case (_, ZERO) => ZERO+ −
case (ONE, r2s) => r2s+ −
case (r1s, ONE) => r1s+ −
case (r1s, r2s) => SEQ(r1s, r2s)+ −
}+ −
case r => r+ −
}+ −
+ −
// an example+ −
val r = SEQ(SEQ(CHAR('x'), CHAR('y')), CHAR('z'))+ −
der('x', r)+ −
der('y', der('x', r))+ −
der('z', der('y', der('x', r)))+ −
simp(der('z', der('y', der('x', r))))+ −
+ −
// *new*+ −
// the derivative w.r.t. a string (iterates der)+ −
def ders2(s: List[Char], r: Rexp) : Rexp = (s, r) match {+ −
case (Nil, r) => r+ −
case (s, ZERO) => ZERO+ −
case (s, ONE) => if (s == Nil) ONE else ZERO+ −
case (s, CHAR(c)) => if (s == List(c)) ONE else + −
if (s == Nil) CHAR(c) else ZERO+ −
case (s, ALT(r1, r2)) => ALT(ders2(s, r1), ders2(s, r2))+ −
case (c::s, r) => ders2(s, simp(der(c, r)))+ −
}+ −
+ −
// the main matcher function+ −
def matcher(r: Rexp, s: String) : Boolean = + −
nullable(ders2(s.toList, r))+ −
+ −
+ −
// one or zero+ −
def OPT(r: Rexp) = ALT(r, ONE)+ −
+ −
+ −
// Test Cases+ −
+ −
def EVIL1(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))+ −
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})+ −
for (i <- 0 to 11000 by 1000) {+ −
println(f"$i: ${time_needed(2, matcher(EVIL1(i), "a" * i))}%.5f")+ −
}+ −
+ −
+ −
// test: (a*)* b+ −
for (i <- 0 to 7000000 by 500000) {+ −
println(f"$i: ${time_needed(2, matcher(EVIL2, "a" * i))}%.5f")+ −
}+ −
+ −
+ −
+ −
// the size of a regular expressions - for testing purposes + −
def size(r: Rexp) : Int = r match {+ −
case ZERO => 1+ −
case ONE => 1+ −
case CHAR(_) => 1+ −
case ALT(r1, r2) => 1 + size(r1) + size(r2)+ −
case SEQ(r1, r2) => 1 + size(r1) + size(r2)+ −
case STAR(r) => 1 + size(r)+ −
case NTIMES(r, _) => 1 + size(r)+ −
}+ −
+ −
+ −
// string of a regular expressions - for testing purposes + −
def string(r: Rexp) : String = r match {+ −
case ZERO => "0"+ −
case ONE => "1"+ −
case CHAR(c) => c.toString + −
case ALT(r1, r2) => s"(${string(r1)} + ${string(r2)})"+ −
case SEQ(CHAR(c), CHAR(d)) => s"${c}${d}"+ −
case SEQ(r1, r2) => s"(${string(r1)} ~ ${string(r2)})"+ −
case STAR(r) => s"(${string(r)})*"+ −
case NTIMES(r, n) => s"(${string(r)}){${n}}"+ −
}+ −
+ −
+ −
// test: ("a" | "aa")*+ −
val EVIL3 = STAR(ALT(CHAR('a'), SEQ(CHAR('a'), CHAR('a'))))+ −
+ −
// test: ("" | "a" | "aa")*+ −
val EVIL4 = STAR(ALT(ONE, ALT(CHAR('a'), SEQ(CHAR('a'), CHAR('a')))))+ −
+ −
val t1 = ders2("a".toList, EVIL3)+ −
val t1s = simp(t1) + −
val t2 = ders2("aa".toList, t1s)+ −
val t2s = simp(t2)+ −
val t3 = ders2("aaa".toList, t2s)+ −
val t3s = simp(t3)+ −
val t4 = ders2("aaaa".toList, t3s)+ −
val t4s = simp(t4)+ −
println(string(t1) + " " + size(t1))+ −
println("s " + string(t1s) + " " + size(t1s))+ −
println(string(t2) + " " + size(t2))+ −
println("s " + string(t2s) + " " + size(t2s))+ −
println(string(t3) + " " + size(t3))+ −
println("s " + string(t3s) + " " + size(t3s))+ −
println(string(t4) + " " + size(t4))+ −
println("s " + string(t4s) + " " + size(t4s))+ −