import scala.annotation.tailrec

//some list functions
def tl[A](xs: List[A]) : List[A] = xs match {
  case Nil => Nil
  case x::xs => xs
}

def hd[A](xs: List[A]) : A = xs.head

def nth_of[A](xs: List[A], n: Int) = 
  if (n <= xs.length) Some(xs(n)) else None 


//some map functions
def dget(m: Map[Int, Int], n: Int) = m.getOrElse(n, 0)


//some string functions
def join[A](xs: List[A]) = xs.foldLeft("")(_.toString + _)



sealed abstract class Cell {
  def * (n: Int) : List[Cell] = n match {
    case 0 => Nil
    case n => this :: this * (n - 1)
  }
}
case object Bk extends Cell { override def toString = "0" }
case object Oc extends Cell { override def toString = "1" }

sealed abstract class Action
case object WBk extends Action
case object WOc extends Action
case object R extends Action
case object L extends Action
case object Nop extends Action

type State = Int
type Inst = (Action, State)
type Prog = List[Inst]

//tapes
case class Tape(l: List[Cell], r: List[Cell]) {
  def update(a: Action) = a match {
    case WBk => Tape(l, Bk::tl(r))
    case WOc => Tape(l, Oc::tl(r))
    case L => if (l == Nil) Tape(Nil, Bk::r) else Tape (tl(l), hd(l)::r)
    case R => if (r == Nil) Tape(Bk::l, Nil) else Tape(hd(r)::l, tl(r))
    case Nop => Tape(l, r)
  }

  def read = r match {
    case Nil => Bk
    case x::_ => x
  }

  override def toString = join(l.reverse) + ">" + join(r)
}

// standard tapes
class STape(ns: Int*) extends 
  Tape(Nil, ns.map(n => Oc * (n + 1)).reduceLeft(_ ::: List(Bk) ::: _))
  

// configurations
case class Config(s: State, tp: Tape) {
  def is_final = s == 0

  override def toString = tp.toString
}


// Turing machines
case class TM(p: Prog) {

  def ++ (that: TM) = TM(this.p ::: that.p)

  def shift(n: Int) =
    TM(p.map{case (a, s) => (a, if (s == 0) 0 else s + n)})
  def adjust(n: Int) : TM =
    TM(p.map{case (a, s) => (a, if (s == 0) n else s)})
  def adjust : TM = adjust(p.length / 2 + 1)
  
  def fetch(s: State, a: Cell) = (s, a) match {
    case (0, _) => (Nop, 0)
    case (s, Bk) => nth_of(p, 2 * (s - 1)) match {
      case None => (Nop, 0)
      case Some(i) => i
    }
    case (s, Oc) => nth_of(p, 2 * (s - 1) + 1) match {
      case None => (Nop, 0)
      case Some(i) => i
    }
  } 

  def step(cf: Config) : Config = fetch (cf.s, cf.tp.read) match {
    case (a, s_next) => Config(s_next, cf.tp.update(a))
  }

  def steps(cf: Config, n: Int) : Config = n match {
    case 0 => cf
    case n => steps(step(cf), n - 1)
  } 

  @tailrec
  final def run(cf: Config) : Config = {
    if (cf.is_final) cf else run(step(cf))
  }
 
  def run(tp: Tape) : Tape = run(Config(1, tp)).tp
}


// Turing machine examples
val TMCopy = TM(List((WBk, 5), (R, 2), (R, 3), (R, 2), (WOc, 3), 
                     (L, 4), (L, 4), (L, 5), (R, 11), (R, 6), 
                     (R, 7), (WBk, 6), (R, 7), (R, 8), (WOc, 9), 
                     (R, 8), (L, 10), (L, 9), (L, 10), (L, 5), 
                     (L, 0), (R, 12), (WOc, 13), (L, 14), (R, 12), 
                     (R, 12), (L, 15), (WBk, 14), (R, 0), (L, 15)))

def TMFindnth(n: Int) : TM = n match {
  case 0 => TM(Nil)
  case n => TMFindnth(n - 1) ++ TM(List((WOc, 2 * n - 1), (R, 2 * n), (R, 2 * n + 1), (R, 2 * n)))
}

def TMMopup(n: Int) = {
  def TMMopup1(n: Int) : TM = n match {
    case 0 => TM(Nil)
    case n => TMMopup1(n - 1) ++ TM(List((R, 2 * n + 1), (WBk, 2 * n), (R, 2 * n - 1), (WOc, 2 * n)))
  }

  val TMMopup2 = TM(List((R, 2), (R, 1), (L, 5), (WBk, 3), (R, 4), (WBk, 3),
                         (R, 2), (WBk, 3), (L, 5), (L, 6), (R, 0), (L, 6)))

  TMMopup1(n) ++ TMMopup2.shift(2 * n)
}

println("TMCopy:    " + (TMCopy.run(new STape(3))))
println("TMfindnth: " + (TMFindnth(3).run(new STape(1,2,3,4,5))))
println("TMMopup: "   + (TMMopup(3).run(new STape(1,2,3,4,5))))


// Abacus machines
sealed abstract class AInst 
case class Inc(n: Int) extends AInst
case class Dec(n: Int, l: Int) extends AInst
case class Goto(l: Int) extends AInst


type AProg = List[AInst]
type Regs = Map[Int, Int]

case class AConfig(s: Int, regs: Regs)  

case class Abacus(p: AProg) {

  def ++ (that: Abacus) = Abacus(this.p ::: that.p)

  def shift(offset: Int, jump: Int) = Abacus(p.map(_ match {
    case Inc(n) => Inc(n)
    case Dec(n, l) => if (l == jump) Dec(n, l) else Dec(n, l + offset)
    case Goto(l) => if (l == jump) Goto(l) else Goto(l + offset)
  }))
      
  def adjust(old_jump: Int, jump: Int) = Abacus(p.map(_ match {
    case Inc(n) => Inc(n)
    case Dec(n, l) => if (l == old_jump) Dec(n, jump) else Dec(n, l)
    case Goto(l) => if (l == old_jump) Goto(jump) else Goto(l)
  }))

  def step(cf: AConfig) : AConfig = (nth_of(p, cf.s), cf.s) match {
    case (None, _) => cf
    case (Some(Inc(n)), s) => AConfig(s + 1, cf.regs + (n -> (dget(cf.regs, n) + 1)))
    case (Some(Dec(n, l)), s) => {
      if (dget(cf.regs, n) == 0) AConfig(l, cf.regs)
      else AConfig(s + 1, cf.regs + (n -> (dget(cf.regs, n) - 1)))
    }
    case (Some(Goto(l)), _) => AConfig(l, cf.regs)
  }  

  def steps(cf: AConfig, n: Int) : AConfig = n match {
    case 0 => cf
    case n => steps(step(cf), n - 1)
  } 

  def steps(regs: Regs, n: Int) : AConfig = steps(AConfig(0, regs), n)

  def run(cf: AConfig) : AConfig = {
    if (cf.s >= p.length || cf.s < 0) cf else run(step(cf))
  }
 
  def run(regs: Regs): Regs = run(AConfig(0, regs)).regs
}

// Abacus examples
def Copy(in: Int, out: Int, jump: Int) = 
  Abacus(List(Dec(in, jump), Inc(out), Goto(0))) 

def Plus(m: Int, n: Int, tmp: Int, jump: Int) =
  Abacus(List(Dec(m, 4), Inc(n), Inc(tmp), Goto(0), Dec(tmp, jump), Inc(m), Goto(4))) 

def Mult(in1: Int, in2: Int, out: Int, tmp: Int, jump: Int) = 
  Abacus(List(Dec(in1, jump))) ++ Plus(in2, out, tmp, -1).shift(1, -1).adjust(-1, 0)

def Expo(in1: Int, in2: Int, out: Int, tmp1: Int, tmp2: Int, jump: Int) = {
  Abacus(List(Inc(out), Dec(in1, jump))) ++ 
  Mult(out, in2, tmp2, tmp1, -1).shift(2, -1).adjust(-1, 10) ++
  Copy(tmp2, out, -1).shift(10, -1). adjust(-1, 1)
}

println("Copy: 3     " + (Copy(0, 1, -1).run(Map(0 -> 3, 1 -> 0))))
println("Plus: 3 + 4 " + (Plus(0, 1, 2, -1).run(Map(0 -> 3, 1 -> 4, 2 -> 0))))
println("Mult: 3 * 5 " + (Mult(0, 1, 2, 3, -1).run(Map(0 -> 3, 1 -> 5, 2 -> 0, 3 -> 0))))
println("Expo: 3 ^ 4 " + (Expo(0, 1, 2, 3, 4, -1).run(Map(0 -> 4, 1 -> 3, 2 -> 0, 3 -> 0, 4 -> 0))))



// Abacus to TM translation
def layout(p: AProg) = p.map(_ match {
  case Inc(n) => 2 * n + 9 
  case Dec(n, _) => 2 * n + 16 
  case Goto(n) => 1
})

def start(p: AProg, n: Int) = layout(p).take(n).sum + 1

def compile_Inc(s: Int, n: Int) = {
  val TMInc = TM(List((WOc, 1), (R, 2), (WOc, 3), (R, 2), (WOc, 3), (R, 4), 
                      (L, 7), (WBk, 5), (R, 6), (WBk, 5), (WOc, 3), (R, 6),
                      (L, 8), (L, 7), (R, 9), (L, 7), (R, 10), (WBk, 9)))
  TMFindnth(n).shift(s - 1) ++ TMInc.shift(2 * n).shift(s - 1)
}

def compile_Dec(s: Int, n: Int, e: Int) = {
  val TMDec = TM(List((WOc, 1), (R, 2), (L, 14), (R, 3), (L, 4), (R, 3),
                      (R, 5), (WBk, 4), (R, 6), (WBk, 5), (L, 7), (L, 8),
                      (L, 11), (WBk, 7), (WOc, 8), (R, 9), (L, 10), (R, 9),
                      (R, 5), (WBk, 10), (L, 12), (L, 11), (R, 13), (L, 11),
                      (R, 17), (WBk, 13), (L, 15), (L, 14), (R, 16), (L, 14),
                      (R, 0), (WBk, 16)))
  TMFindnth(n).shift(s - 1) ++ TMDec.shift(2 * n).shift(s - 1).adjust(e)
}

def compile_Goto(s: Int) = {
  val TMGoto = TM(List((Nop, 1), (Nop, 1)))
  TMGoto.shift(s - 1)
}

def compile(p: AProg, s: Int, i: AInst) = i match {
  case Inc(n) => compile_Inc(s, n)
  case Dec(n, e) => compile_Dec(s, n, start(p, e))
  case Goto(e) => compile_Goto(start(p, e))
}

def tms(p: AProg) = {
  val ss = (0 until p.length).map (start(p,_))
  (ss zip p).map{case (n, i) => compile(p, n, i)}
}

def toTM(p: AProg) = tms(p).reduceLeft(_ ++ _)

//examples
println("Copy 3:     " + toTM(Copy(0, 1, Int.MaxValue).p).run(new STape(3,0,0)))
println("Plus 3 + 4: " + toTM(Plus(0, 1, 2, Int.MaxValue).p).run(new STape(3,4,0,0)))
println("Mult 3 * 5: " + toTM(Mult(0, 1, 2, 3, Int.MaxValue).p).run(new STape(3,5,0,0)))
println("Expo 3 ^ 4: " + toTM(Expo(0, 1, 2, 3, 4, 10000).p).run(new STape(3,4,0,0,0)))


//Recursive Functions
abstract class Rec {
  def eval(ns: List[Int]) : Int
}
case object Z extends Rec {
  override def eval(ns: List[Int]) = ns match {
    case n::Nil => 0
    case _ => throw new IllegalArgumentException("Z: args")
  }
} 
case object S extends Rec {
  override def eval(ns: List[Int]) = ns match {
    case n::Nil => n + 1
    case _ => throw new IllegalArgumentException("S: args")
  }
} 
case class Id(n: Int, m: Int) extends Rec {
  override def eval(ns: List[Int]) = 
    if (ns.length == n && m < n) ns(m)
    else throw new IllegalArgumentException("Id: args")
}
case class Cn(n: Int, f: Rec, gs: List[Rec]) extends Rec {
  override def eval(ns: List[Int]) = 
    if (ns.length == n) f.eval(gs.map(_.eval(ns)))
    else throw new IllegalArgumentException("Cn: args")
}
case class Pr(n: Int, f: Rec, g: Rec) extends Rec {
  override def eval(ns: List[Int]) = 
    if (ns.length == n - 1) {
      if (ns.last == 0) f.eval(ns.init)
      else {
        val r = Pr(n, f, g).eval(ns.init ::: List(ns.last - 1))
        g.eval(ns.init ::: List(ns.last - 1, r))
      }
    }
    else throw new IllegalArgumentException("Cn: args")
}
case class Mn(n: Int, f: Rec) extends Rec {
  override def eval(ns: List[Int]) = 0
    
}