diff -r 3b8e3a156200 -r 20a57552d722 exps/bit.scala
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/exps/bit.scala	Wed Mar 13 10:36:29 2019 +0000
@@ -0,0 +1,829 @@
+
+import scala.language.implicitConversions    
+import scala.language.reflectiveCalls
+import scala.annotation.tailrec   
+import scala.util.Try
+
+// for escaping strings
+def escape(raw: String) : String = {
+  import scala.reflect.runtime.universe._
+  Literal(Constant(raw)).toString
+}
+
+def esc2(r: (String, String)) = (escape(r._1), escape(r._2))
+
+def distinctBy[B, C](xs: List[B], f: B => C, acc: List[C] = Nil): List[B] = xs match {
+  case Nil => Nil
+  case (x::xs) => {
+    val res = f(x)
+    if (acc.contains(res)) distinctBy(xs, f, acc)  
+    else x::distinctBy(xs, f, res::acc)
+  }
+} 
+
+abstract class Bit
+case object Z extends Bit
+case object S extends Bit
+case class C(c: Char) extends Bit
+
+type Bits = List[Bit]
+
+// usual regular expressions
+abstract class Rexp 
+case object ZERO extends Rexp
+case object ONE extends Rexp
+case class PRED(f: Char => Boolean, s: String = "_") extends Rexp 
+case class ALTS(rs: List[Rexp]) extends Rexp 
+case class SEQ(r1: Rexp, r2: Rexp) extends Rexp 
+case class STAR(r: Rexp) extends Rexp 
+case class RECD(x: String, r: Rexp) extends Rexp
+
+
+// abbreviations
+def CHAR(c: Char) = PRED(_ == c, c.toString)
+def ALT(r1: Rexp, r2: Rexp) = ALTS(List(r1, r2))
+def PLUS(r: Rexp) = SEQ(r, STAR(r))
+val ANYCHAR = PRED(_ => true, ".")
+
+// annotated regular expressions
+abstract class ARexp 
+case object AZERO extends ARexp
+case class AONE(bs: Bits) extends ARexp
+case class APRED(bs: Bits, f: Char => Boolean, s: String = "_") extends ARexp
+case class AALTS(bs: Bits, rs: List[ARexp]) extends ARexp 
+case class ASEQ(bs: Bits, r1: ARexp, r2: ARexp) extends ARexp 
+case class ASTAR(bs: Bits, r: ARexp) extends ARexp 
+
+// abbreviations
+def AALT(bs: Bits, r1: ARexp, r2: ARexp) = AALTS(bs, List(r1, r2))
+
+// values
+abstract class Val
+case object Empty extends Val
+case class Chr(c: Char) extends Val
+case class Sequ(v1: Val, v2: Val) extends Val
+case class Left(v: Val) extends Val
+case class Right(v: Val) extends Val
+case class Stars(vs: List[Val]) extends Val
+case class Rec(x: String, v: Val) extends Val
+
+
+   
+// some convenience for typing in regular expressions
+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)
+  def $ (r: Rexp) = RECD(s, r)
+}
+
+
+// string of a regular expressions - for testing purposes
+def string(r: Rexp): String = r match {
+  case ZERO => "0"
+  case ONE => "1"
+  case PRED(_, s) => s
+  case ALTS(rs) => rs.map(string).mkString("[", "|", "]")
+  case SEQ(r1, r2) => s"(${string(r1)} ~ ${string(r2)})"
+  case STAR(r) => s"{${string(r)}}*"
+  case RECD(x, r) => s"(${x}! ${string(r)})"
+}
+
+// string of an annotated regular expressions - for testing purposes
+
+def astring(a: ARexp): String = a match {
+  case AZERO => "0"
+  case AONE(_) => "1"
+  case APRED(_, _, s) => s
+  case AALTS(_, rs) => rs.map(astring).mkString("[", "|", "]")
+  case ASEQ(_, r1, r2) => s"(${astring(r1)} ~ ${astring(r2)})"
+  case ASTAR(_, r) => s"{${astring(r)}}*"
+}
+ 
+
+//--------------------------------------------------------------
+// START OF NON-BITCODE PART
+//
+
+// nullable function: tests whether the regular 
+// expression can recognise the empty string
+def nullable(r: Rexp) : Boolean = r match {
+  case ZERO => false
+  case ONE => true
+  case PRED(_, _) => false
+  case ALTS(rs) => rs.exists(nullable)
+  case SEQ(r1, r2) => nullable(r1) && nullable(r2)
+  case STAR(_) => true
+  case RECD(_, r) => nullable(r)
+}
+
+// derivative of a regular expression w.r.t. a character
+def der(c: Char, r: Rexp) : Rexp = r match {
+  case ZERO => ZERO
+  case ONE => ZERO
+  case PRED(f, _) => if (f(c)) ONE else ZERO
+  case ALTS(List(r1, r2)) => ALTS(List(der(c, r1), der(c, r2)))
+  case SEQ(r1, r2) => 
+    if (nullable(r1)) ALTS(List(SEQ(der(c, r1), r2), der(c, r2)))
+    else SEQ(der(c, r1), r2)
+  case STAR(r) => SEQ(der(c, r), STAR(r))
+  case RECD(_, r1) => der(c, r1)
+}
+
+
+def flatten(v: Val) : String = v match {
+  case Empty => ""
+  case Chr(c) => c.toString
+  case Left(v) => flatten(v)
+  case Right(v) => flatten(v)
+  case Sequ(v1, v2) => flatten(v1) + flatten(v2)
+  case Stars(vs) => vs.map(flatten).mkString
+  case Rec(_, v) => flatten(v)
+}
+
+// extracts an environment from a value
+def env(v: Val) : List[(String, String)] = v match {
+  case Empty => Nil
+  case Chr(c) => Nil
+  case Left(v) => env(v)
+  case Right(v) => env(v)
+  case Sequ(v1, v2) => env(v1) ::: env(v2)
+  case Stars(vs) => vs.flatMap(env)
+  case Rec(x, v) => (x, flatten(v))::env(v)
+}
+
+
+// injection part
+def mkeps(r: Rexp) : Val = r match {
+  case ONE => Empty
+  case ALTS(List(r1, r2)) => 
+    if (nullable(r1)) Left(mkeps(r1)) else Right(mkeps(r2))
+  case SEQ(r1, r2) => Sequ(mkeps(r1), mkeps(r2))
+  case STAR(r) => Stars(Nil)
+  case RECD(x, r) => Rec(x, mkeps(r))
+}
+
+def inj(r: Rexp, c: Char, v: Val) : Val = (r, v) match {
+  case (STAR(r), Sequ(v1, Stars(vs))) => Stars(inj(r, c, v1)::vs)
+  case (SEQ(r1, r2), Sequ(v1, v2)) => Sequ(inj(r1, c, v1), v2)
+  case (SEQ(r1, r2), Left(Sequ(v1, v2))) => Sequ(inj(r1, c, v1), v2)
+  case (SEQ(r1, r2), Right(v2)) => Sequ(mkeps(r1), inj(r2, c, v2))
+  case (ALTS(List(r1, r2)), Left(v1)) => Left(inj(r1, c, v1))
+  case (ALTS(List(r1, r2)), Right(v2)) => Right(inj(r2, c, v2))
+  case (PRED(_, _), Empty) => Chr(c) 
+  case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))
+}
+
+// lexing without simplification
+def lex(r: Rexp, s: List[Char]) : Val = s match {
+  case Nil => if (nullable(r)) mkeps(r) else throw new Exception("Not matched")
+  case c::cs => inj(r, c, lex(der(c, r), cs))
+}
+
+def lexing(r: Rexp, s: String) : Val = lex(r, s.toList)
+
+//println(lexing(("ab" | "ab") ~ ("b" | ONE), "ab"))
+
+// some "rectification" functions for simplification
+def F_ID(v: Val): Val = v
+def F_RIGHT(f: Val => Val) = (v:Val) => Right(f(v))
+def F_LEFT(f: Val => Val) = (v:Val) => Left(f(v))
+def F_ALT(f1: Val => Val, f2: Val => Val) = (v:Val) => v match {
+  case Right(v) => Right(f2(v))
+  case Left(v) => Left(f1(v))
+}
+def F_SEQ(f1: Val => Val, f2: Val => Val) = (v:Val) => v match {
+  case Sequ(v1, v2) => Sequ(f1(v1), f2(v2))
+}
+def F_SEQ_Empty1(f1: Val => Val, f2: Val => Val) = 
+  (v:Val) => Sequ(f1(Empty), f2(v))
+def F_SEQ_Empty2(f1: Val => Val, f2: Val => Val) = 
+  (v:Val) => Sequ(f1(v), f2(Empty))
+def F_RECD(f: Val => Val) = (v:Val) => v match {
+  case Rec(x, v) => Rec(x, f(v))
+}
+def F_ERROR(v: Val): Val = throw new Exception("error")
+
+// simplification of regular expressions returning also an
+// rectification function; no simplification under STAR 
+def simp(r: Rexp): (Rexp, Val => Val) = r match {
+  case ALTS(List(r1, r2)) => {
+    val (r1s, f1s) = simp(r1)
+    val (r2s, f2s) = simp(r2)
+    (r1s, r2s) match {
+      case (ZERO, _) => (r2s, F_RIGHT(f2s))
+      case (_, ZERO) => (r1s, F_LEFT(f1s))
+      case _ => if (r1s == r2s) (r1s, F_LEFT(f1s))
+                else (ALTS(List(r1s, r2s)), F_ALT(f1s, f2s)) 
+    }
+  }
+  case SEQ(r1, r2) => {
+    val (r1s, f1s) = simp(r1)
+    val (r2s, f2s) = simp(r2)
+    (r1s, r2s) match {
+      case (ZERO, _) => (ZERO, F_ERROR)
+      //case (_, ZERO) => (ZERO, F_ERROR)
+      case (ONE, _) => (r2s, F_SEQ_Empty1(f1s, f2s))
+      //case (_, ONE) => (r1s, F_SEQ_Empty2(f1s, f2s))
+      case _ => (SEQ(r1s,r2s), F_SEQ(f1s, f2s))
+    }
+  }
+  case RECD(x, r1) => {
+    val (r1s, f1s) = simp(r1)
+    (RECD(x, r1s), F_RECD(f1s))
+  }
+  case r => (r, F_ID)
+}
+
+def ders_simp(s: List[Char], r: Rexp) : Rexp = s match {
+  case Nil => r
+  case c::s => ders_simp(s, simp(der(c, r))._1)
+}
+
+
+def lex_simp(r: Rexp, s: List[Char]) : Val = s match {
+  case Nil => if (nullable(r)) mkeps(r) else throw new Exception("Not matched")
+  case c::cs => {
+    val (r_simp, f_simp) = simp(der(c, r))
+    inj(r, c, f_simp(lex_simp(r_simp, cs)))
+  }
+}
+
+def lexing_simp(r: Rexp, s: String) : Val = lex_simp(r, s.toList)
+
+//println(lexing_simp(("a" | "ab") ~ ("b" | ""), "ab"))
+
+
+def tokenise_simp(r: Rexp, s: String) = 
+  env(lexing_simp(r, s)).map(esc2)
+
+//--------------------------------------------------------------------
+// Partial Derivatives
+
+
+def pder(c: Char, r: Rexp): Set[Rexp] = r match {
+  case ZERO => Set()
+  case ONE => Set()
+  case PRED(f, _) => if (f(c)) Set(ONE) else Set()
+  case ALTS(rs) => rs.toSet.flatMap(pder(c, _))
+  case SEQ(r1, r2) =>
+    (for (pr1 <- pder(c, r1)) yield SEQ(pr1, r2)) ++
+    (if (nullable(r1)) pder(c, r2) else Set())
+  case STAR(r1) =>
+    for (pr1 <- pder(c, r1)) yield SEQ(pr1, STAR(r1))
+  case RECD(_, r1) => pder(c, r1)
+}
+
+def pders(cs: List[Char], r: Rexp): Set[Rexp] = cs match {
+  case Nil => Set(r)
+  case c::cs => pder(c, r).flatMap(pders(cs, _))
+}
+
+def pders_simp(cs: List[Char], r: Rexp): Set[Rexp] = cs match {
+  case Nil => Set(r)
+  case c::cs => pder(c, r).flatMap(pders_simp(cs, _)).map(simp(_)._1)
+}
+
+def psize(rs: Set[Rexp])  = 
+  rs.map(size).sum
+
+//--------------------------------------------------------------------
+// BITCODED PART
+
+
+def fuse(bs: Bits, r: ARexp) : ARexp = r match {
+  case AZERO => AZERO
+  case AONE(cs) => AONE(bs ++ cs)
+  case APRED(cs, f, s) => APRED(bs ++ cs, f, s)
+  case AALTS(cs, rs) => AALTS(bs ++ cs, rs)
+  case ASEQ(cs, r1, r2) => ASEQ(bs ++ cs, r1, r2)
+  case ASTAR(cs, r) => ASTAR(bs ++ cs, r)
+}
+
+// translation into ARexps
+def internalise(r: Rexp) : ARexp = r match {
+  case ZERO => AZERO
+  case ONE => AONE(Nil)
+  case PRED(f, s) => APRED(Nil, f, s)
+  case ALTS(List(r1, r2)) => 
+    AALTS(Nil, List(fuse(List(Z), internalise(r1)), fuse(List(S), internalise(r2))))
+  case ALTS(r1::rs) => {
+     val AALTS(Nil, rs2) = internalise(ALTS(rs))
+     AALTS(Nil, fuse(List(Z), internalise(r1)) :: rs2.map(fuse(List(S), _)))
+  }
+  case SEQ(r1, r2) => ASEQ(Nil, internalise(r1), internalise(r2))
+  case STAR(r) => ASTAR(Nil, internalise(r))
+  case RECD(x, r) => internalise(r)
+}
+
+internalise(("a" | "ab") ~ ("b" | ""))
+
+// decoding of values from bit sequences
+def decode_aux(r: Rexp, bs: Bits) : (Val, Bits) = (r, bs) match {
+  case (ONE, bs) => (Empty, bs)
+  case (PRED(f, _), C(c)::bs) => (Chr(c), bs)
+  case (ALTS(r::Nil), bs) => decode_aux(r, bs)
+  case (ALTS(rs), bs) => bs match {
+    case Z::bs1 => {
+      val (v, bs2) = decode_aux(rs.head, bs1)
+      (Left(v), bs2)
+    }
+    case S::bs1 => {
+      val (v, bs2) = decode_aux(ALTS(rs.tail), bs1)
+      (Right(v), bs2)			
+    }
+  }
+  case (SEQ(r1, r2), bs) => {
+    val (v1, bs1) = decode_aux(r1, bs)
+    val (v2, bs2) = decode_aux(r2, bs1)
+    (Sequ(v1, v2), bs2)
+  }
+  case (STAR(r1), S::bs) => {
+    val (v, bs1) = decode_aux(r1, bs)
+    val (Stars(vs), bs2) = decode_aux(STAR(r1), bs1)
+    (Stars(v::vs), bs2)
+  }
+  case (STAR(_), Z::bs) => (Stars(Nil), bs)
+  case (RECD(x, r1), bs) => {
+    val (v, bs1) = decode_aux(r1, bs)
+    (Rec(x, v), bs1)
+  }
+}
+
+def decode(r: Rexp, bs: Bits) = decode_aux(r, bs) match {
+  case (v, Nil) => v
+  case _ => throw new Exception("Not decodable")
+}
+
+
+//erase function: extracts a Rexp from Arexp
+def erase(r: ARexp) : Rexp = r match{
+  case AZERO => ZERO
+  case AONE(_) => ONE
+  case APRED(bs, f, s) => PRED(f, s)
+  case AALTS(bs, rs) => ALTS(rs.map(erase(_)))
+  case ASEQ(bs, r1, r2) => SEQ (erase(r1), erase(r2))
+  case ASTAR(cs, r)=> STAR(erase(r))
+}
+
+
+// bnullable function: tests whether the aregular 
+// expression can recognise the empty string
+def bnullable (r: ARexp) : Boolean = r match {
+  case AZERO => false
+  case AONE(_) => true
+  case APRED(_,_,_) => false
+  case AALTS(_, rs) => rs.exists(bnullable)
+  case ASEQ(_, r1, r2) => bnullable(r1) && bnullable(r2)
+  case ASTAR(_, _) => true
+}
+
+def bmkeps(r: ARexp) : Bits = r match {
+  case AONE(bs) => bs
+  case AALTS(bs, rs) => {
+    val n = rs.indexWhere(bnullable)
+    bs ++ bmkeps(rs(n))
+  }
+  case ASEQ(bs, r1, r2) => bs ++ bmkeps(r1) ++ bmkeps(r2)
+  case ASTAR(bs, r) => bs ++ List(Z)
+}
+
+// derivative of a regular expression w.r.t. a character
+def bder(c: Char, r: ARexp) : ARexp = r match {
+  case AZERO => AZERO
+  case AONE(_) => AZERO
+  case APRED(bs, f, _) => if (f(c)) AONE(bs:::List(C(c))) else AZERO
+  case AALTS(bs, rs) => AALTS(bs, rs.map(bder(c, _)))
+  case ASEQ(bs, r1, r2) => 
+    if (bnullable(r1)) AALT(bs, ASEQ(Nil, bder(c, r1), r2), fuse(bmkeps(r1), bder(c, r2)))
+    else ASEQ(bs, bder(c, r1), r2)
+  case ASTAR(bs, r) => ASEQ(bs, fuse(List(S), bder(c, r)), ASTAR(Nil, r))
+}
+
+
+// derivative w.r.t. a string (iterates bder)
+@tailrec
+def bders (s: List[Char], r: ARexp) : ARexp = s match {
+  case Nil => r
+  case c::s => bders(s, bder(c, r))
+}
+
+def flats(rs: List[ARexp]): List[ARexp] = rs match {
+    case Nil => Nil
+    case AZERO :: rs1 => flats(rs1)
+    case AALTS(bs, rs1) :: rs2 => rs1.map(fuse(bs, _)) ::: flats(rs2)
+    case r1 :: rs2 => r1 :: flats(rs2)
+}
+
+def bsimp(r: ARexp): ARexp = r match {
+  case ASEQ(bs1, r1, r2) => (bsimp(r1), bsimp(r2)) match {
+      case (AZERO, _) => AZERO
+      case (_, AZERO) => AZERO
+      case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s)
+      case (r1s, r2s) => ASEQ(bs1, r1s, r2s)
+  }
+  case AALTS(bs1, rs) => distinctBy(flats(rs.map(bsimp)), erase) match {
+    case Nil => AZERO
+    case r :: Nil => fuse(bs1, r)
+    case rs => AALTS(bs1, rs)  
+  }
+  //case ASTAR(bs1, r1) => ASTAR(bs1, bsimp(r1))
+  case r => r
+}
+
+def bders_simp (s: List[Char], r: ARexp) : ARexp = s match {
+  case Nil => r
+  case c::s => bders_simp(s, bsimp(bder(c, r)))
+}
+
+def blex_simp(r: ARexp, s: List[Char]) : Bits = s match {
+  case Nil => if (bnullable(r)) bmkeps(r)
+	      else throw new Exception("Not matched")
+  case c::cs => blex_simp(bsimp(bder(c, r)), cs)
+}
+
+
+def blexing_simp(r: Rexp, s: String) : Val = 
+ decode(r, blex_simp(internalise(r), s.toList))
+
+
+def btokenise_simp(r: Rexp, s: String) = 
+  env(blexing_simp(r, s)).map(esc2)
+
+
+
+// INCLUDING SIMPLIFICATION UNDER STARS
+
+def bsimp_full(r: ARexp): ARexp = r match {
+  case ASEQ(bs1, r1, r2) => (bsimp_full(r1), bsimp_full(r2)) match {
+      case (AZERO, _) => AZERO
+      case (_, AZERO) => AZERO
+      case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s)
+      case (r1s, r2s) => ASEQ(bs1, r1s, r2s)
+  }
+  case AALTS(bs1, rs) => distinctBy(flats(rs.map(bsimp_full)), erase) match {
+    case Nil => AZERO
+    case r :: Nil => fuse(bs1, r)
+    case rs => AALTS(bs1, rs)  
+  }
+  case ASTAR(bs1, r1) => ASTAR(bs1, bsimp_full(r1))
+  case r => r
+}
+
+def bders_simp_full(s: List[Char], r: ARexp) : ARexp = s match {
+  case Nil => r
+  case c::s => bders_simp_full(s, bsimp_full(bder(c, r)))
+}
+
+def blex_simp_full(r: ARexp, s: List[Char]) : Bits = s match {
+  case Nil => if (bnullable(r)) bmkeps(r)
+	      else throw new Exception("Not matched")
+  case c::cs => blex_simp_full(bsimp_full(bder(c, r)), cs)
+}
+
+
+def blexing_simp_full(r: Rexp, s: String) : Val = 
+ decode(r, blex_simp_full(internalise(r), s.toList))
+
+
+def btokenise_simp_full(r: Rexp, s: String) = env(blexing_simp_full(r, s)).map(esc2)
+
+// bders2 for strings in the ALTS case
+
+def bders2_simp(s: List[Char], a: ARexp) : ARexp = {
+  //println(s"s = ${s.length}   a = ${asize(a)}")
+  //Console.readLine
+  (s, a) match {
+  case (Nil, r) => r
+  case (s, AZERO) => AZERO
+  case (s, AONE(_)) => AZERO
+  case (s, APRED(bs, f, _)) => 
+    if (f(s.head) && s.tail == Nil) AONE(bs:::List(C(s.head))) else AZERO
+  case (s, AALTS(bs, rs)) => bsimp(AALTS(bs, rs.map(bders2_simp(s, _)))) 
+  case (c::s, r) => bders2_simp(s, bsimp(bder(c, r)))
+}}
+
+
+
+def blexing2_simp(r: Rexp, s: String) : Val = {
+  val bder = bders2_simp(s.toList, internalise(r))
+  if (bnullable(bder)) decode(r, bmkeps(bder)) else 
+  throw new Exception("Not matched")
+
+}
+
+def btokenise2_simp(r: Rexp, s: String) = 
+  env(blexing2_simp(r, s)).map(esc2)
+
+
+// Parser for regexes
+
+case class Parser(s: String) {
+  var i = 0
+  
+  def peek() = s(i)
+  def eat(c: Char) = 
+    if (c == s(i)) i = i + 1 else throw new Exception("Expected " + c + " got " + s(i))
+  def next() = { i = i + 1; s(i - 1) }
+  def more() = s.length - i > 0
+
+  def Regex() : Rexp = {
+    val t = Term();
+    if (more() && peek() == '|') {
+      eat ('|') ; 
+      ALT(t, Regex()) 
+    } 
+    else t
+  }
+
+  def Term() : Rexp = {
+    var f : Rexp = 
+      if (more() && peek() != ')' && peek() != '|') Factor() else ONE;
+    while (more() && peek() != ')' && peek() != '|') {
+      f = SEQ(f, Factor()) ;
+    }
+    f
+  }
+
+  def Factor() : Rexp = {
+    var b = Base();
+    while (more() && peek() == '*') {
+      eat('*') ;
+      b = STAR(b) ;
+    }
+    while (more() && peek() == '?') {
+      eat('?') ;
+      b = ALT(b, ONE) ;
+    }
+    while (more() && peek() == '+') {
+      eat('+') ;
+      b = SEQ(b, STAR(b)) ;
+    }
+    b
+  }
+
+  def Base() : Rexp = {
+    peek() match {
+      case '(' => { eat('(') ; val r = Regex(); eat(')') ; r }   // if groups should be groups RECD("",r) }
+      case '.' => { eat('.'); ANYCHAR }
+      case _ => CHAR(next())
+    }
+  }
+}
+
+// two simple examples for the regex parser
+
+println("two simple examples for the regex parser")
+
+println(string(Parser("a|(bc)*").Regex()))
+println(string(Parser("(a|b)*(babab(a|b)*bab|bba(a|b)*bab)(a|b)*").Regex()))
+
+
+
+//System.exit(0)
+
+//   Testing
+//============
+
+def time[T](code: => T) = {
+  val start = System.nanoTime()
+  val result = code
+  val end = System.nanoTime()
+  ((end - start)/1.0e9).toString
+  //result
+}
+
+def timeR[T](code: => T) = {
+  val start = System.nanoTime()
+  for (i <- 1 to 10) code
+  val result = code
+  val end = System.nanoTime()
+  (result, (end - start))
+}
+
+//size: of a Aregx for testing purposes 
+def size(r: Rexp) : Int = r match {
+  case ZERO => 1
+  case ONE => 1
+  case PRED(_,_) => 1
+  case SEQ(r1, r2) => 1 + size(r1) + size(r2)
+  case ALTS(rs) => 1 + rs.map(size).sum
+  case STAR(r) => 1 + size(r)
+  case RECD(_, r) => size(r)
+}
+
+def asize(a: ARexp) = size(erase(a))
+
+
+// Lexing Rules for a Small While Language
+
+//symbols
+val SYM = PRED("abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ".contains(_), "SYM")
+//digits
+val DIGIT = PRED("0123456789".contains(_), "NUM")
+//identifiers
+val ID = SYM ~ (SYM | DIGIT).% 
+//numbers
+val NUM = STAR(DIGIT)
+//keywords
+val KEYWORD : Rexp = "skip" | "while" | "do" | "if" | "then" | "else" | "read" | "write" | "true" | "false"
+//semicolons
+val SEMI: Rexp = ";"
+//operators
+val OP: Rexp = ":=" | "==" | "-" | "+" | "*" | "!=" | "<" | ">" | "<=" | ">=" | "%" | "/"
+//whitespaces
+val WHITESPACE = PLUS(" " | "\n" | "\t")
+//parentheses
+val RPAREN: Rexp = ")"
+val LPAREN: Rexp = "("
+val BEGIN: Rexp = "{"
+val END: Rexp = "}"
+//strings...but probably needs not
+val STRING: Rexp = "\"" ~ SYM.% ~ "\""
+
+
+
+val WHILE_REGS = (("k" $ KEYWORD) | 
+                  ("i" $ ID) | 
+                  ("o" $ OP) | 
+                  ("n" $ NUM) | 
+                  ("s" $ SEMI) | 
+                  ("str" $ STRING) |
+                  ("p" $ (LPAREN | RPAREN)) | 
+                  ("b" $ (BEGIN | END)) | 
+                  ("w" $ WHITESPACE)).%
+
+
+// Some Small Tests
+//==================
+
+println("Small tests")
+
+val q = STAR(STAR("bb" | "ab"))
+val qs = "bbb"
+
+println("Size Bit  " + asize(bders_simp(qs.toList, internalise(q))))
+println("Size Bitf " + asize(bders_simp_full(qs.toList, internalise(q))))
+println("Size Bit2 " + asize(bders2_simp(qs.toList, internalise(q))))
+println("Size Old  " + size(ders_simp(qs.toList, q)))
+println("Size Pder " + psize(pders_simp(qs.toList, q)))
+
+
+val re1 = STAR("a" | "aa")
+println(astring(bders_simp("".toList, internalise(re1))))
+println(astring(bders_simp("a".toList, internalise(re1))))
+println(astring(bders_simp("aa".toList, internalise(re1))))
+println(astring(bders_simp("aaa".toList, internalise(re1))))
+println(astring(bders_simp("aaaaaa".toList, internalise(re1))))
+println(astring(bders_simp("aaaaaaaaa".toList, internalise(re1))))
+println(astring(bders_simp("aaaaaaaaaaaa".toList, internalise(re1))))
+println(astring(bders_simp("aaaaaaaaaaaaaaaaaaaaaaaaa".toList, internalise(re1))))
+println(astring(bders_simp("aaaaaabaaaabbbbbaaaaaaaaaaaaaaa".toList, internalise(re1))))
+
+
+for (i <- 0 to 100 by 5) {
+  //print("Old: " + time(tokenise_simp(re1, "a" * i)))
+  print(" Bit: " + time(btokenise_simp(re1, "a" * i)))
+  print(" Bit full simp: " + time(btokenise_simp_full(re1, "a" * i)))
+  println(" Bit2: " + time(btokenise2_simp(re1, "a" * i)))
+}
+
+Console.readLine
+
+
+// Bigger Tests
+//==============
+
+
+println("Big tests")
+
+val fib_prog = """
+write "Fib";
+read n;
+minus1 := 0;
+minus2 := 1;
+while n > 0 do {
+  temp := minus2;
+  minus2 := minus1 + minus2;
+  minus1 := temp;
+  n := n - 1
+};
+write "Result";
+write minus2
+"""
+
+
+println("fib prog tests :")
+println(tokenise_simp(WHILE_REGS, fib_prog))
+println(btokenise_simp(WHILE_REGS, fib_prog))
+println("equal? " + (tokenise_simp(WHILE_REGS, fib_prog) == btokenise_simp(WHILE_REGS, fib_prog)))
+
+for (i <- 1 to 20) {
+  print("Old: " + time(tokenise_simp(WHILE_REGS, fib_prog * i)))
+  print(" Bit: " + time(btokenise_simp(WHILE_REGS, fib_prog * i)))
+  println(" Bit full simp: " + time(btokenise_simp_full(WHILE_REGS, fib_prog * i)))
+  //println(" Bit2: " + time(btokenise2_simp(WHILE_REGS, fib_prog * i)))
+}
+
+
+println("Original " + size(WHILE_REGS))
+println("Size Bit  " + asize(bders_simp((fib_prog * 1).toList, internalise(WHILE_REGS))))
+println("Size Bitf " + asize(bders_simp_full((fib_prog * 1).toList, internalise(WHILE_REGS))))
+println("Size Bit2 " + asize(bders2_simp((fib_prog * 1).toList, internalise(WHILE_REGS))))
+println("Size Old  " + size(ders_simp((fib_prog * 1).toList, WHILE_REGS)))
+println("Size Pder " + psize(pders_simp((fib_prog * 1).toList, WHILE_REGS)))
+
+System.exit(0)
+
+println("Internal sizes test OK or strange")
+
+def perc(p1: Double, p2: Double) : String =
+  f"${(((p1 - p2) / p2) * 100.0) }%5.0f" + "%"
+
+def ders_test(n: Int, s: List[Char], r: Rexp, a: ARexp) : (Rexp, ARexp) = s match {
+  case Nil => (r, a)
+  case c::s => {
+    // derivative 
+    val (rd1, tr1) = timeR(der(c, r))
+    val (ad1, ta1) = timeR(bder(c, a))
+    val trs1 = f"${tr1}%.5f"
+    val tas1 = f"${ta1}%.5f"
+    if (tr1 < ta1) println(s"Time strange der  (step) ${n} ${perc(ta1, tr1)} sizes  der ${size(rd1)} ${asize(ad1)}")
+    //simplification
+    val (rd, tr) = timeR(simp(rd1)._1)
+    val (ad, ta) = timeR(bsimp(ad1))
+    val trs = f"${tr}%.5f"
+    val tas = f"${ta}%.5f"
+    //full simplification
+    val (adf, taf) = timeR(bsimp_full(ad1))
+    if (tr < ta) println(s"Time strange simp (step) ${n} ${perc(ta, tr)} sizes simp ${size(rd)} ${asize(ad)}")
+    if (n == 1749 || n == 1734) {
+      println{s"Aregex before bder (size: ${asize(a)})\n ${string(erase(a))}"}
+      println{s"Aregex after bder (size: ${asize(ad1)})\n ${string(erase(ad1))}"}
+      println{s"Aregex after bsimp (size: ${asize(ad)})\n ${string(erase(ad))}"}
+      println{s"Aregex after bsimp_full (size: ${asize(adf)})\n ${string(erase(adf))}"}
+    }
+    ders_test(n + 1, s, rd, ad)
+  }
+}
+
+val prg = (fib_prog * 10).toList
+ders_test(0, prg, WHILE_REGS, internalise(WHILE_REGS))
+
+
+//testing the two lexings produce the same value
+//enumerates strings of length n over alphabet cs
+def strs(n: Int, cs: String) : Set[String] = {
+  if (n == 0) Set("")
+  else {
+    val ss = strs(n - 1, cs)
+    ss ++
+    (for (s <- ss; c <- cs.toList) yield c + s)
+  }
+}
+def enum(n: Int, s: String) : Stream[Rexp] = n match {
+  case 0 => ZERO #:: ONE #:: s.toStream.map(CHAR)
+  case n => {  
+    val rs = enum(n - 1, s)
+    rs #:::
+    (for (r1 <- rs; r2 <- rs) yield ALT(r1, r2)) #:::
+    (for (r1 <- rs; r2 <- rs) yield SEQ(r1, r2)) #:::
+    (for (r1 <- rs) yield STAR(r1))
+  }
+}
+
+//tests blexing and lexing
+def tests(ss: Set[String])(r: Rexp) = {
+  //println(s"Testing ${r}")
+  for (s <- ss.par) yield {
+    val res1 = Try(Some(lexing_simp(r, s))).getOrElse(None)
+    val res2 = Try(Some(blexing_simp(r, s))).getOrElse(None)
+    if (res1 != res2) 
+      { println(s"Disagree on ${r} and ${s}")
+	println(s"   ${res1} !=  ${res2}")
+	Some((r, s)) } else None
+  }
+}
+
+
+println("Partial searching: ")
+enum(2, "abc").map(tests(strs(3, "abc"))).toSet
+
+
+
+