import RexpRelated._

import RexpRelated.Rexp._

import Spiral._

import scala.collection.mutable.ArrayBuffer

abstract class BRexp

case object BZERO extends BRexp

case object BONE extends BRexp

case class BCHAR(c: Char) extends BRexp

case class BALTS(b: Bit, rs: List[BRexp]) extends BRexp 

case class BSEQ(r1: BRexp, r2: BRexp) extends BRexp 

case class BSTAR(r: BRexp) extends BRexp 

object BRexp{

  def brternalise(r: Rexp) : BRexp = r match {//remember the initial shadowed bit should be set to Z as there 

  //are no enclosing STAR or SEQ

    case ZERO => BZERO

    case ONE => BONE

    case CHAR(c) => BCHAR(c)

    case ALTS(rs) => BALTS(Z,  rs.map(brternalise))

    case SEQ(r1, r2) => BSEQ( brternalise(r1), brternalise(r2) )

    case STAR(r) => BSTAR(brternalise(r))

    case RECD(x, r) => brternalise(r)

  }

  def brnullable (r: BRexp) : Boolean = r match {

    case BZERO => false

    case BONE => true

    case BCHAR(_) => false

    case BALTS(bs, rs) => rs.exists(brnullable)

    case BSEQ(r1, r2) => brnullable(r1) && brnullable(r2)

    case BSTAR(_) => true

  }

  //this function tells bspill when converting a brexp into a list of rexps

  //the conversion : (a+b)*c -> {a*c, b*c} can only happen when a+b is generated from scratch (e.g. when deriving a seq)

  //or is from the original regex but have been "touched" (i.e. have been derived)

  //Why make this distinction? Look at the following example:

  //r = (c+cb)+c(a+b)

  // r\c = (1+b)+(a+b)

  //after simplification

  // (r\c)simp= 1+b+a

  //we lost the structure that tells us 1+b should be grouped together and a grouped as itself

  //however in our brexp simplification, 

  //(r\c)br_simp = 1+b+(a+b)

  //we do not allow the bracket to be opened as it is from the original expression and have not been touched

  def brder(c: Char, r: BRexp) : BRexp = r match {

    case BZERO => BZERO

    case BONE => BZERO

    case BCHAR(d) => if (c == d) BONE else BZERO

    case BALTS(bs, rs) => BALTS(S, rs.map(brder(c, _)))//After derivative: Splittable in the sense of PD

    case BSEQ(r1, r2) => 

      if (brnullable(r1)) BALTS(S, List(BSEQ(brder(c, r1), r2), brder(c, r2) ) )//in r1\c~r2 r2's structure is maintained together with its splittablility bit

      else BSEQ(brder(c, r1), r2)

    case BSTAR(r) => BSEQ(brder(c, r), BSTAR(r))

  }

  def bflat(rs: List[BRexp]): List[BRexp] = {

    rs match {

      case Nil => Nil//TODO: Look at this! bs should have been exhausted one bit earlier.

      case BZERO :: rs1 => bflat(rs1)

      case BALTS(S, rs1) :: rs2 => rs1 ::: bflat(rs2)

      case BALTS(Z, rs1) :: rs2 => BALTS(Z, rs1) :: bflat(rs2)

      case r1 :: rs2 => r1 :: bflat(rs2)

    }

  }

  def stflat(rs: List[BRexp]): List[BRexp] = {

    //println("bs: " + bs + "rs: "+ rs + "length of bs, rs" + bs.length + ", " + rs.length)

    //assert(bs.length == rs.length - 1 || bs.length == rs.length)//bs == Nil, rs == Nil  May add early termination later.

    rs match {

      case Nil => Nil//TODO: Look at this! bs should have been exhausted one bit earlier.

      case BZERO :: rs1 => bflat(rs1)

      case BALTS(_, rs1) :: rs2 => rs1 ::: bflat(rs2)

      case r1 :: rs2 => r1 :: bflat(rs2)

    }

  }

  def berase(r:BRexp): Rexp = r match{

    case BZERO => ZERO

    case BONE => ONE

    case BCHAR(f) => CHAR(f)

    case BALTS(bs, rs) => ALTS(rs.map(berase(_)))

    case BSEQ(r1, r2) => SEQ (berase(r1), berase(r2))

    case BSTAR(r)=> STAR(berase(r))

  }

  def br_simp(r: BRexp): BRexp = r match {

    case BSEQ(r1, r2) => (br_simp(r1), br_simp(r2)) match {

        case (BZERO, _) => BZERO

        case (_, BZERO) => BZERO

        case (BONE, r2s) => r2s 

        case (r1s, r2s) => BSEQ(r1s, r2s)

    }

    case BALTS(bs, rs) => {

      //assert(bs.length == rs.length - 1)

      val dist_res = if(bs == S) {//all S

        val rs_simp = rs.map(br_simp)

        val flat_res = bflat(rs_simp)

        distinctBy(flat_res, berase)

      }

      else{//not allowed to simplify (all Z)

        rs

      }

      dist_res match {

        case Nil => {BZERO}

        case s :: Nil => { s}

        case rs => {BALTS(bs, rs)}

      }

    }

    //case BSTAR(r) => BSTAR(r)

    case r => r

  }

  def strong_br_simp(r: BRexp): BRexp = r match {

    case BSEQ(r1, r2) => (strong_br_simp(r1), strong_br_simp(r2)) match {

        case (BZERO, _) => BZERO

        case (_, BZERO) => BZERO

        case (BONE, r2s) => r2s 

        case (r1s, r2s) => BSEQ(r1s, r2s)

    }

    case BALTS(bs, rs) => {

      //assert(bs.length == rs.length - 1)

      val dist_res = {//all S

        val rs_simp = rs.map(strong_br_simp)

        val flat_res = stflat(rs_simp)

        distinctBy(flat_res, berase)

      }

      dist_res match {

        case Nil => {BZERO}

        case s :: Nil => { s}

        case rs => {BALTS(bs, rs)}

      }

    }

    case BSTAR(r) => BSTAR(strong_br_simp(r))

    case r => r

  }

  //we want to bound the size by a function bspill s.t.

  //bspill(ders_simp(r,s)) ⊂  PD(r) 

  //and bspill is size-preserving (up to a constant factor)

  //so we bound |ders_simp(r,s)| by |PD(r)|

  //where we already have a nice bound for |PD(r)|: t^2*n^2 in Antimirov's paper

  //the function bspill converts a brexp into a list of rexps

  //the conversion mainly about this: (r1+r2)*r3*r4*.....rn -> {r1*r3*r4*....rn, r2*r3*r4*.....rn} 

  //but this is not always allowed

  //sometimes, we just leave the expression as it is: 

  //eg1: (a+b)*c -> {(a+b)*c}

  //eg2: r1+r2 -> {r1+r2} instead of {r1, r2}

  //why?

  //if we always return {a, b} when we encounter a+b, the property

  //bspill(ders_simp(r,s)) ⊂  PD(r) 

  //does not always hold

  //for instance 

  //if we call the bspill that always returns {r1, r2} when encountering r1+r2 "bspilli"

  //then bspilli( ders_simp( (c+cb)+c(a+b), c ) ) == bspilli(1+b+a) = {1, b, a}

  //However the letter a does not belong to PD( (c+cb)+c(a+b) )

  //then we can no longer say ders_simp(r,s)'s size is bounded by PD(r) because the former contains something the latter does not have

  //In order to make sure the inclusion holds, we have to find out why new terms appeared in the bspilli set that don't exist in the PD(r) set

  //Why a does not belong to PD((c+cb)+c(a+b))?

  //PD(r1+r2) = PD(r1) union PD(r2) => PD((c+cb)+c(a+b)) = PD(c+cb) union PD(c(a+b))

  //So the only possibility for PD to include a must be in the latter part of the regex, namely, c(a+b)

  //we have lemma that PD(r) = union of pders(s, r) where s is taken over all strings whose length does not exceed depth(r)

  //so PD(r) ⊂ pder(empty_string, r) union pder(c, r) union pder(ca, r) union pder(cb, r) where r = c(a+b)

  //RHS = {1} union pder(c, c(a+b))

  //Observe that pder(c, c(a+b)) == {a+b}

  //a and b are together, from the original regular expression (c+cb)+c(a+b).

  //but by our simplification, we first flattened this a+b into the same level with 1+b, then

  //removed duplicates of b, thereby destroying the structure in a+b and making this term a, instead of a+b

  //But in PD(r) we only have a+b, no a

  //one ad hoc solution might be to try this bspill(ders_simp(r,s)) ⊂  PD(r) union {all letters}

  //But this does not hold either according to experiment.

  //So we need to make sure the structure r1+r2 is not simplified away if it is from the original expression

  def bspill(r: BRexp): List[Rexp] = {

      r match {

          case BALTS(b, rs) => {

            if(b == S)

              rs.flatMap(bspill)

            else

              List(ALTS(rs.map(berase)))

          }

          case BSEQ(r2, r3) => bspill(r2).map( a => if(a == ONE) berase(r3) else SEQ(a, berase(r3)) )

          case BZERO => List()

          case r => List(berase(r))

      }

  }

}