// Main Part 3 about Regular Expression Matching
//==============================================
object M3 {
// Regular Expressions
abstract class Rexp
case object ZERO extends Rexp
case object ONE extends Rexp
case class CHAR(c: Char) extends Rexp
case class ALTs(rs: List[Rexp]) extends Rexp // alternatives
case class SEQs(rs: List[Rexp]) extends Rexp // sequences
case class STAR(r: Rexp) extends Rexp // star
//the usual binary choice and binary sequence can be defined
//in terms of ALTs and SEQs
def ALT(r1: Rexp, r2: Rexp) = ALTs(List(r1, r2))
def SEQ(r1: Rexp, r2: Rexp) = SEQs(List(r1, r2))
// some convenience for typing in regular expressions
import scala.language.implicitConversions
import scala.language.reflectiveCalls
def charlist2rexp(s: List[Char]): Rexp = s match {
case Nil => ONE
case c::Nil => CHAR(c)
case c::s => SEQ(CHAR(c), charlist2rexp(s))
}
implicit def string2rexp(s: String): Rexp = charlist2rexp(s.toList)
implicit def RexpOps (r: Rexp) = new {
def | (s: Rexp) = ALT(r, s)
def % = STAR(r)
def ~ (s: Rexp) = SEQ(r, s)
}
implicit def stringOps (s: String) = new {
def | (r: Rexp) = ALT(s, r)
def | (r: String) = ALT(s, r)
def % = STAR(s)
def ~ (r: Rexp) = SEQ(s, r)
def ~ (r: String) = SEQ(s, r)
}
// (1)
def nullable (r: Rexp) : Boolean = r match {
case ZERO => false
case ONE => true
case CHAR(_) => false
case ALTs(rs) => rs.exists(nullable)
case SEQs(rs) => rs.forall(nullable)
case STAR(_) => true
}
// (2)
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 ALTs(rs) => ALTs(rs.map(der(c, _)))
case SEQs(Nil) => ZERO
case SEQs(r1::rs) =>
if (nullable(r1)) ALT(SEQs(der(c, r1)::rs), der(c, SEQs(rs)))
else SEQs(der(c, r1):: rs)
case STAR(r1) => SEQ(der(c, r1), STAR(r1))
}
// (3)
def denest(rs: List[Rexp]) : List[Rexp] = rs match {
case Nil => Nil
case ZERO::tl => denest(tl)
case ALTs(rs1)::rs2 => rs1 ::: denest(rs2)
case r::rs => r :: denest(rs)
}
// (4)
def flts(rs: List[Rexp], acc: List[Rexp] = Nil) : List[Rexp] = rs match {
case Nil => acc
case ZERO::rs => ZERO::Nil
case ONE::rs => flts(rs, acc)
case SEQs(rs1)::rs => flts(rs, acc ::: rs1)
case r::rs => flts(rs, acc :+ r)
}
// (5)
def ALTs_smart(rs: List[Rexp]) : Rexp = rs match {
case Nil => ZERO
case r::Nil => r
case rs => ALTs(rs)
}
def SEQs_smart(rs: List[Rexp]) : Rexp = rs match {
case Nil => ONE
case ZERO::nil => ZERO
case r::Nil => r
case rs => SEQs(rs)
}
// (6)
def simp(r: Rexp) : Rexp = r match {
case ALTs(rs) =>
ALTs_smart(denest(rs.map(simp)).distinct)
case SEQs(rs) =>
SEQs_smart(flts(rs.map(simp)))
case r => r
}
//println("Simp tests")
//println(simp(ALT(ONE | CHAR('a'), CHAR('a') | ONE)))
//println(simp(((CHAR('a') | ZERO) ~ ONE) |
// (((ONE | CHAR('b')) | CHAR('c')) ~ (CHAR('d') ~ ZERO))))
// (7)
def ders (s: List[Char], r: Rexp) : Rexp = s match {
case Nil => r
case c::s => ders(s, simp(der(c, r)))
}
// main matcher function
def matcher(r: Rexp, s: String) = nullable(ders(s.toList, r))
// (8)
def size(r: Rexp): Int = r match {
case ZERO => 1
case ONE => 1
case CHAR(_) => 1
case ALTs(rs) => 1 + rs.map(size).sum
case SEQs(rs) => 1 + rs.map(size).sum
case STAR(r1) => 1 + size(r1)
}
// some testing data
/*
println(matcher(("a" ~ "b") ~ "c", "abc")) // => true
println(matcher(("a" ~ "b") ~ "c", "ab")) // => false
// the supposedly 'evil' regular expression (a*)* b
val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))
println(matcher(EVIL, "a" * 1000 ++ "b")) // => true
println(matcher(EVIL, "a" * 1000)) // => false
// size without simplifications
println(size(der('a', der('a', EVIL)))) // => 36
println(size(der('a', der('a', der('a', EVIL))))) // => 83
// size with simplification
println(simp(der('a', der('a', EVIL))))
println(simp(der('a', der('a', der('a', EVIL)))))
println(size(simp(der('a', der('a', EVIL))))) // => 7
println(size(simp(der('a', der('a', der('a', EVIL)))))) // => 7
// Python needs around 30 seconds for matching 28 a's with EVIL.
// Java 9 and later increase this to an "astonishing" 40000 a's in
// around 30 seconds.
//
// Lets see how long it takes to match strings with
// 5 Million a's...it should be in the range of a
// few seconds.
def time_needed[T](i: Int, code: => T) = {
val start = System.nanoTime()
for (j <- 1 to i) code
val end = System.nanoTime()
"%.5f".format((end - start)/(i * 1.0e9))
}
for (i <- 0 to 5000000 by 500000) {
println(s"$i ${time_needed(2, matcher(EVIL, "a" * i))} secs.")
}
// another "power" test case
println(simp(Iterator.iterate(ONE:Rexp)(r => SEQ(r, ONE | ONE)).drop(100).next()) == ONE)
// the Iterator produces the rexp
//
// SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE)
//
// where SEQ is nested 100 times.
*/
}