diff -r ccfce105e36b -r 1e1b0fe66107 progs/token.scala
--- a/progs/token.scala	Fri Oct 09 14:48:06 2015 +0100
+++ b/progs/token.scala	Fri Oct 16 08:42:21 2015 +0100
@@ -1,25 +1,33 @@
-import scala.language.implicitConversions
+import scala.language.implicitConversions    
 import scala.language.reflectiveCalls
-import scala.util._
-import scala.annotation.tailrec
+import scala.annotation.tailrec   
 
-sealed abstract class Rexp
-
+abstract class Rexp 
 case object NULL extends Rexp
 case object EMPTY extends Rexp
 case class CHAR(c: Char) extends Rexp
-case class ALT(r1: Rexp, r2: Rexp) extends Rexp
-case class SEQ(r1: Rexp, r2: Rexp) extends Rexp
-case class STAR(r: Rexp) extends Rexp
+case class ALT(r1: Rexp, r2: 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
 
-def charlist2rexp(s : List[Char]) : Rexp = s match {
+abstract class Val
+case object Void 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 => EMPTY
   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)
@@ -32,34 +40,11 @@
   def % = STAR(s)
   def ~ (r: Rexp) = SEQ(s, r)
   def ~ (r: String) = SEQ(s, r)
+  def $ (r: Rexp) = RECD(s, r)
 }
 
-def Range(s : List[Char]) : Rexp = s match {
-  case Nil => NULL
-  case c::Nil => CHAR(c)
-  case c::s => ALT(CHAR(c), Range(s))
-}
-def RANGE(s: String) = Range(s.toList)
-
-def PLUS(r: Rexp) = SEQ(r, STAR(r))
-
-val SYM = RANGE("ABCDEFGHIJKLMNOPQRSTUVXYZabcdefghijklmnopqrstuvwxyz_")
-val DIGIT = RANGE("0123456789")
-val ID = SYM ~ (SYM | DIGIT).% 
-val NUM = PLUS(DIGIT)
-val KEYWORD : Rexp = "skip" | "while" | "do" | "if" | "then" | "else" | "read" | "write" 
-val SEMI: Rexp = ";"
-val OP: Rexp = ":=" | "=" | "-" | "+" | "*" | "!=" | "<" | ">"
-val WHITESPACE = PLUS(RANGE(" \n"))
-val RPAREN: Rexp = ")"
-val LPAREN: Rexp = "("
-val BEGIN: Rexp = "{"
-val END: Rexp = "}"
-
-//regular expressions ranked by position in the list
-val regs: List[Rexp] = 
-  List(KEYWORD, ID, OP, NUM, SEMI, LPAREN, RPAREN, BEGIN, END, WHITESPACE)
-
+// nullable function: tests whether the regular 
+// expression can recognise the empty string
 def nullable (r: Rexp) : Boolean = r match {
   case NULL => false
   case EMPTY => true
@@ -67,79 +52,224 @@
   case ALT(r1, r2) => nullable(r1) || nullable(r2)
   case SEQ(r1, r2) => nullable(r1) && nullable(r2)
   case STAR(_) => true
+  case RECD(_, r1) => nullable(r1)
 }
 
-def zeroable (r: Rexp) : Boolean = r match {
-  case NULL => true
-  case EMPTY => false
-  case CHAR(_) => false
-  case ALT(r1, r2) => zeroable(r1) && zeroable(r2)
-  case SEQ(r1, r2) => zeroable(r1) || zeroable(r2)
-  case STAR(_) => false
-}
-
+// derivative of a regular expression w.r.t. a character
 def der (c: Char, r: Rexp) : Rexp = r match {
   case NULL => NULL
-  case EMPTY => NULL  
+  case EMPTY => NULL
   case CHAR(d) => if (c == d) EMPTY else NULL
   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(r) => SEQ(der(c, r), STAR(r))
+  case RECD(_, r1) => der(c, r1)
+}
+
+// derivative w.r.t. a string (iterates der)
+def ders (s: List[Char], r: Rexp) : Rexp = s match {
+  case Nil => r
+  case c::s => ders(s, der(c, r))
+}
+
+// extracts a string from value
+def flatten(v: Val) : String = v match {
+  case Void => ""
+  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 Void => 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 EMPTY => Void
+  case ALT(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))
 }
 
 
-// calculates derivatives until all of them are zeroable
-@tailrec
-def munch(s: List[Char], 
-          pos: Int, 
-          rs: List[Rexp], 
-          last: Option[Int]): Option[Int] = rs match {
-  case Nil => last
-  case rs if (s.length <= pos) => last
-  case rs => {
-    val ders = rs.map(der(s(pos), _))
-    val rs_nzero = ders.filterNot(zeroable(_))
-    val rs_nulls = ders.filter(nullable(_))
-    val new_last = if (rs_nulls != Nil) Some(pos) else last
-    munch(s, 1 + pos, rs_nzero, new_last)
+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 (ALT(r1, r2), Left(v1)) => Left(inj(r1, c, v1))
+  case (ALT(r1, r2), Right(v2)) => Right(inj(r2, c, v2))
+  case (CHAR(d), Void) => Chr(c) 
+  case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))
+}
+
+// main lexing function (produces a value)
+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)
+
+// 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_Void1(f1: Val => Val, f2: Val => Val) = (v:Val) => Sequ(f1(Void), f2(v))
+def F_SEQ_Void2(f1: Val => Val, f2: Val => Val) = (v:Val) => Sequ(f1(v), f2(Void))
+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 ALT(r1, r2) => {
+    val (r1s, f1s) = simp(r1)
+    val (r2s, f2s) = simp(r2)
+    (r1s, r2s) match {
+      case (NULL, _) => (r2s, F_RIGHT(f2s))
+      case (_, NULL) => (r1s, F_LEFT(f1s))
+      case _ => if (r1s == r2s) (r1s, F_LEFT(f1s))
+                else (ALT (r1s, r2s), F_ALT(f1s, f2s)) 
+    }
+  }
+  case SEQ(r1, r2) => {
+    val (r1s, f1s) = simp(r1)
+    val (r2s, f2s) = simp(r2)
+    (r1s, r2s) match {
+      case (NULL, _) => (NULL, F_ERROR)
+      case (_, NULL) => (NULL, F_ERROR)
+      case (EMPTY, _) => (r2s, F_SEQ_Void1(f1s, f2s))
+      case (_, EMPTY) => (r1s, F_SEQ_Void2(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 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)))
   }
 }
 
-// iterates the munching function and prints 
-// out the component strings
-@tailrec
-def tokenize(s: String, rs: List[Rexp]) : Unit = munch(s.toList, 0, rs, None) match {
-  case None if (s == "") => println("EOF")
-  case None => println(s"Lexing error: $s")
-  case Some(n) => {
-    val (head, tail) = s.splitAt(n + 1)
-    print(s"|${head.replaceAll("\n","RET")}|")
-    tokenize(tail, rs)
-  }
+def lexing_simp(r: Rexp, s: String) : Val = lex_simp(r, s.toList)
+
+lexing_simp(("a" + "ab") | ("b" + ""), "ab")
+
+// Lexing Rules for a Small While Language
+
+def PLUS(r: Rexp) = r ~ r.%
+val SYM = "a" | "b" | "c" | "d" | "e" | "f" | "g" | "h" | "i" | "j" | "k" | "l" | "m" | "n" | "o" | "p" | "q" | "r" | "s" | "t" | "u" | "v" | "w" | "x" | "y" | "z"
+val DIGIT = "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"
+val ID = SYM ~ (SYM | DIGIT).% 
+val NUM = PLUS(DIGIT)
+val KEYWORD : Rexp = "skip" | "while" | "do" | "if" | "then" | "else" | "read" | "write" | "true" | "false"
+val SEMI: Rexp = ";"
+val OP: Rexp = ":=" | "==" | "-" | "+" | "*" | "!=" | "<" | ">" | "<=" | ">=" | "%" | "/"
+val WHITESPACE = PLUS(" " | "\n" | "\t")
+val RPAREN: Rexp = ")"
+val LPAREN: Rexp = "("
+val BEGIN: Rexp = "{"
+val END: Rexp = "}"
+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)).%
+
+//   Testing
+//============
+
+def time[T](code: => T) = {
+  val start = System.nanoTime()
+  val result = code
+  val end = System.nanoTime()
+  println((end - start)/1.0e9)
+  result
+}
+
+val r1 = ("a" | "ab") ~ ("bcd" | "c")
+println(lexing(r1, "abcd"))
+
+val r2 = ("" | "a") ~ ("ab" | "b")
+println(lexing(r2, "ab"))
+
+
+// Two Simple While Tests
+//========================
+println("prog0 test")
+
+val prog0 = """read n"""
+println(env(lexing_simp(WHILE_REGS, prog0)))
+
+println("prog1 test")
+
+val prog1 = """read  n; write (n)"""
+println(env(lexing_simp(WHILE_REGS, prog1)))
+
+
+// Big Test
+//==========
+
+val prog2 = """
+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("Tokens")
+println(env(lexing_simp(WHILE_REGS, prog2)))
+
+for (i <- 1 to 80) {
+  print(i.toString + ":  ")
+  time(lexing_simp(WHILE_REGS, prog2 * i))
 }
 
 
-val test_prog = """
-start := XXX;
-x := start;
-y := start;
-z := start;
-while 0 < x do {
- while 0 < y do {
-  while 0 < z do {
-    z := z - 1
-  };
-  z := start;
-  y := y - 1
- };     
- y := start;
- x := x - 1
-};
-write x;
-write y;
-write z
-"""
-
-tokenize(test_prog, regs)