// Main Part 3 about Regular Expression Matching//==============================================object M3 {abstract class Rexpcase object ZERO extends Rexpcase object ONE extends Rexpcase class CHAR(c: Char) extends Rexpcase class ALTs(rs: List[Rexp]) extends Rexp // alternatives case class SEQs(rs: List[Rexp]) extends Rexp // sequencescase class STAR(r: Rexp) extends Rexp // star//the usual binary choice and binary sequence can be defined //in terms of ALTs and SEQsdef ALT(r1: Rexp, r2: Rexp) = ALTs(List(r1, r2))def SEQ(r1: Rexp, r2: Rexp) = SEQs(List(r1, r2))// some convenience for typing regular expressionsdef charlist2rexp(s: List[Char]): Rexp = s match { case Nil => ONE case c::Nil => CHAR(c) case c::s => SEQ(CHAR(c), charlist2rexp(s))}import scala.language.implicitConversionsgiven Conversion[String, Rexp] = (s => charlist2rexp(s.toList))extension (r: Rexp) { def | (s: Rexp) = ALT(r, s) def % = STAR(r) def ~ (s: Rexp) = SEQ(r, s)}// some examples for the conversion and extension:// val areg : Rexp = "a" | "b"// => ALTs(List(CHAR('a'), CHAR('b')))//// val sreg : Rexp = "a" ~ "b"// => SEQs(List(CHAR('a'), CHAR('b'))) //// val star_reg : Rexp = ("a" ~ "b").%// => STAR(SEQs(List(CHAR('a'), CHAR('b')))) // ADD YOUR CODE BELOW//======================// (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 functiondef 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")) // => trueprintln(matcher(("a" ~ "b") ~ "c", "ab")) // => false// the supposedly 'evil' regular expression (a*)* bval EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))println(matcher(EVIL, "a" * 1000 ++ "b")) // => trueprintln(matcher(EVIL, "a" * 1000)) // => false// size without simplificationsprintln(size(der('a', der('a', EVIL)))) // => 36println(size(der('a', der('a', der('a', EVIL))))) // => 83// size with simplificationprintln(simp(der('a', der('a', EVIL)))) println(simp(der('a', der('a', der('a', EVIL)))))println(size(simp(der('a', der('a', EVIL))))) // => 7println(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.*/ }// This template code is subject to copyright // by King's College London, 2022. Do not // make the template code public in any shape // or form, and do not exchange it with other // students under any circumstance.