// Part 1 about Regular Expression Matching//==========================================abstract class Rexpcase object ZERO extends Rexpcase object ONE extends Rexpcase class CHAR(c: Char) extends Rexpcase class ALT(r1: Rexp, r2: Rexp) extends Rexp case class SEQ(r1: Rexp, r2: Rexp) extends Rexp case class STAR(r: Rexp) extends Rexp // some convenience for typing in regular expressionsimport 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)}// (1a) Complete the function nullable according to// the definition given in the coursework; this // function checks whether a regular expression// can match the empty stringdef 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}// (1b) Complete the function der according to// the definition given in the coursework; this// function calculates the derivative of a // regular expression w.r.t. a characterdef 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))}// (1c) Complete the function der according to// the specification given in the coursework; this// function simplifies a regular expression;// however it does not simplify inside STAR-regular// expressionsdef 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}// (1d) Complete the two functions below; the first // calculates the derivative w.r.t. a string; the second// is the regular expression matcher taking a regular// expression and a string and checks whether the// string matches the regular expressiondef 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): Boolean = nullable(ders(s.toList, r))// (1e) Complete the function below: it searches (from the left to // right) in string s1 all the non-empty substrings that match the // regular expression -- these substrings are assumed to be// the longest substrings matched by the regular expression and// assumed to be non-overlapping. All these substrings in s1 are replaced// by s2.def splits(s: String): List[(String, String)] = (for (i <- (1 to s.length).toList) yield s.splitAt(i)).reversesplits("abcde")splits("")def first(r: Rexp, lst: List[(String, String)]): Option[String] = lst match { case Nil => None case (s1, s2)::xs => if (matcher(r, s1)) Some(s2) else first(r, xs)}"abcd".headdef replace(r: Rexp, s1: String, s2: String): String = first(r, splits(s1)) match { case None if (s1 == "") => "" case None => s1.head.toString ++ replace(r, s1.tail, s2) case Some(s) => s2 ++ replace(r, s, s2) }val s1 = "aabbbaaaaaaabaaaaabbaaaabb"val r: Rexp = "aa".% | "bb"splits(s1)first(r, splits(s1))replace(r, s1, "c")splits("bb")first(r, splits("bb"))replace(r, "abb", "c")// PART 2//========// (2a)import scala.annotation.tailrec@tailrecdef iterT[A](n: Int, f: A => A, x: A): A = if (n == 0) x else iterT(n - 1, f, f(x)) //non-tail recursive iterdef iter[A](n: Int, f: A => A, x: A): A = if (n == 0) x else f(iter(n - 1,f, x)) iter(200000, (x: Int) => x + 1, 0)iterT(200000, (x: Int) => x + 1, 0)iterT(100, (x: Int) => x * 2, 2)iterT(100, (x: BigInt) => x * 2, BigInt(2))iterT(10, (x: String) => x ++ "a", "a")// (2b)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(r1) => 1 + size(r1)}val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))size(iterT(20, (r: Rexp) => der('a', r), EVIL)) // should produce 7340068 size(iterT(20, (r: Rexp) => simp(der('a', r)), EVIL)) // should produce 8// (2c)@tailrecdef fixpT[A](f: A => A, x: A): A = { val fx = f(x) if (fx == x) x else fixpT(f, fx) }fixpT((x:Int) => if (200000 < x) x else x + 1, 0)def ctest(n: Long): Long = if (n == 1) 1 else if (n % 2 == 0) n / 2 else 3 * n + 1fixpT(ctest, 97L)fixpT(ctest, 871L)fixpT(ctest, 77031L)fixpT(ctest, 837799L)def foo(s: String): String = { if (matcher("a", s)) "a" else if (matcher("aa" ~ STAR("aa"), s)) s.take(s.length / 2) else "a" ++ s * 3}fixpT(foo, "a" * 97)fixpT(foo, "a" * 871)