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// a class for deterministic finite automata,
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// the type of states is kept polymorphic
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case class Automaton[A](start: A, states: Set[A], delta: Map[(A, Char), A], fins: Set[A]) {
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// the transition function lifted to list of characters
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def deltas(q: A, cs: List[Char]) : Either[A, String] =
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if (states.contains(q)) cs match {
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case Nil => Left(q)
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case c::cs =>
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if (delta.isDefinedAt(q, c)) deltas(delta(q, c), cs)
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else Right(q + " does not have a transition for " + c)
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}
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else Right(q + " is not a state of the automaton")
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// wether a string is accepted by the automaton
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def accepts(s: String) = deltas(start, s.toList) match {
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case Left(q) => fins.contains(q)
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case _ => false
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}
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}
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// translating a regular expression into a finite
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// automaton
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abstract class Rexp
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case object NULL extends Rexp
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case object EMPTY extends Rexp
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case class CHAR(c: Char) extends Rexp
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case class ALT(r1: Rexp, r2: Rexp) extends Rexp
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case class SEQ(r1: Rexp, r2: Rexp) extends Rexp
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case class STAR(r: Rexp) extends Rexp
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implicit def string2rexp(s : String) = {
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def chars2rexp (cs: List[Char]) : Rexp = cs match {
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case Nil => EMPTY
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case c::Nil => CHAR(c)
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case c::cs => SEQ(CHAR(c), chars2rexp(cs))
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}
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chars2rexp(s.toList)
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}
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def nullable (r: Rexp) : Boolean = r match {
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case NULL => false
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case EMPTY => true
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case CHAR(_) => false
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case ALT(r1, r2) => nullable(r1) || nullable(r2)
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case SEQ(r1, r2) => nullable(r1) && nullable(r2)
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case STAR(_) => true
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}
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def der (r: Rexp, c: Char) : Rexp = r match {
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case NULL => NULL
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case EMPTY => NULL
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case CHAR(d) => if (c == d) EMPTY else NULL
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case ALT(r1, r2) => ALT(der(r1, c), der(r2, c))
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case SEQ(r1, r2) => if (nullable(r1)) ALT(SEQ(der(r1, c), r2), der(r2, c))
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else SEQ(der(r1, c), r2)
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case STAR(r) => SEQ(der(r, c), STAR(r))
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}
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// Here we construct an automaton whose
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// states are regular expressions
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type State = Rexp
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type States = Set[State]
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type Transition = Map[(State, Char), State]
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35
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// we use as an alphabet all lowercase letters
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val alphabet = "abcdefghijklmnopqrstuvwxyz".toSet
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def goto(q: State, c: Char, qs: States, delta: Transition) : (States, Transition) = {
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val q_der : State = der(q, c)
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if (qs.contains(q_der)) (qs, delta + ((q, c) -> q))
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else explore(qs + q_der, delta + ((q, c) -> q_der), q_der)
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}
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def explore (qs: States, delta: Transition, q: State) : (States, Transition) =
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alphabet.foldRight[(States, Transition)] (qs, delta) ((c, qsd) => goto(q, c, qsd._1, qsd._2))
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def mk_automaton (r: Rexp) : Automaton[Rexp] = {
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val (qs, delta) = explore(Set(r), Map(), r);
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val fins = for (q <- qs if nullable(q)) yield q;
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Automaton[Rexp](r, qs, delta, fins)
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}
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val A = mk_automaton(ALT("ab","ac"))
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println(A.accepts("bd"))
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println(A.accepts("ab"))
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println(A.accepts("ac"))
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