# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1539674892 -3600
# Node ID 1a521448d211a59366e4a958dcebc5b0adb0d76e
# Parent  6e5e3adc9eb1d67a20786a5fab9d208d350531c8
updated

diff -r 6e5e3adc9eb1 -r 1a521448d211 progs/token.scala
--- a/progs/token.scala	Tue Oct 16 00:42:10 2018 +0100
+++ b/progs/token.scala	Tue Oct 16 08:28:12 2018 +0100
@@ -5,32 +5,20 @@
 case object ZERO extends Rexp
 case object ONE extends Rexp
 case class CHAR(c: Char) extends Rexp
-case class ALTS(rs: List[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
 
-// ALT is now an abbreviation
-def ALT(r1: Rexp, r2: Rexp) = ALTS(List(r1, r2))
-
-// proj is now used instead for Left and Right
 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 Proj(n: Int, v: 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(s: String, v: Val) extends Val
- 
-// for manipulating projections  
-def IncProj(m: Int, v: Val) = v match {
-  case Proj(n, v) => Proj(n + m, v)
-}
-
-def DecProj(m: Int, v: Val) = v match {
-  case Proj(n, v) => Proj(n - m, v)
-}
-
-
+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
@@ -51,40 +39,19 @@
   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 expression - for testing purposes 
-def string(r: Rexp) : String = r match {
-  case ZERO => "0"
-  case ONE => "1"
-  case CHAR(c) => c.toString 
-  case ALTS(rs) => rs.map(string).mkString("[", "|", "]")
-  case SEQ(CHAR(c), CHAR(d)) => s"${c}${d}"
-  case SEQ(ONE, CHAR(c)) => s"1${c}"
-  case SEQ(r1, r2) => s"(${string(r1)} ~ ${string(r2)})"
-  case STAR(r) => s"(${string(r)})*"
-}
-
-// size of a regular expression - for testing purposes 
-def size(r: Rexp) : Int = r match {
-  case ZERO => 1
-  case ONE => 1
-  case CHAR(_) => 1
-  case ALTS(rs) => 1 + rs.map(size).sum
-  case SEQ(r1, r2) => 1 + size(r1) + size(r2)
-  case STAR(r) => 1 + size(r)
-}
-
-
 // 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 CHAR(_) => false
-  case ALTS(rs) => rs.exists(nullable)
+  case ALT(r1, r2) => nullable(r1) || nullable(r2)
   case SEQ(r1, r2) => nullable(r1) && nullable(r2)
   case STAR(_) => true
+  case RECD(_, r1) => nullable(r1)
 }
 
 // derivative of a regular expression w.r.t. a character
@@ -92,11 +59,12 @@
   case ZERO => ZERO
   case ONE => ZERO
   case CHAR(d) => if (c == d) ONE else ZERO
-  case ALTS(rs) => ALTS(rs.map(der(c, _)))
+  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)
@@ -105,87 +73,72 @@
   case c::s => ders(s, der(c, r))
 }
 
-val test : Rexp= STAR("a" | "aa")
-size(test)
-size(der('a', test))
-size(der('a', der('a', test)))
-
-size(ders("aaaaaa".toList, test)) 
-string(ders("aaaaaa".toList, test))
-
-
 // extracts a string from value
 def flatten(v: Val) : String = v match {
   case Empty => ""
   case Chr(c) => c.toString
-  case Proj(_, v) => flatten(v)
-  case Sequ(v1, v2) => flatten(v1) ++ flatten(v2)
+  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 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))
 }
 
 
-// mkeps
-def mkeps(r: Rexp) : Val = r match {
-  case ONE => Empty
-  case ALTS(r1::rs) => 
-    if (nullable(r1)) Proj(0, mkeps(r1)) 
-    else IncProj(1, mkeps(ALTS(rs)))
-  case SEQ(r1, r2) => Sequ(mkeps(r1), mkeps(r2))
-  case STAR(r) => Stars(Nil)
-}
-
-// injection 
 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), Proj(0, Sequ(v1, v2))) => Sequ(inj(r1, c, v1), v2)
-  case (SEQ(r1, r2), Proj(1, v2)) => Sequ(mkeps(r1), inj(r2, c, v2))
-  case (ALTS(rs), Proj(n, v1)) => Proj(n, inj(rs(n), c, v1))
+  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), Empty) => Chr(c) 
+  case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))
 }
 
 // main lexing function (produces a value)
-//  - does  not simplify
 def lex(r: Rexp, s: List[Char]) : Val = s match {
-  case Nil => {
-    //println(s"Size of the last regex: ${size(r)}")
-    if (nullable(r)) mkeps(r) else throw new Exception("Not matched")
-  }
+  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)
 
-lexing("a" | ZERO, "a")
-lexing(ZERO | "a", "a")
 
 lexing(("ab" | "a") ~ ("b" | ONE), "ab")
 
 
-// removing duplicate regular expressions
-val unit = (ZERO, F_ERROR(_))
-
-def dups2(xs: List[(Rexp, Val => Val)], 
-	 acc: List[(Rexp, Val => Val)] = Nil) : List[(Rexp, Val => Val)] = xs match {
-  case Nil => acc
-  case (x, y)::xs => 
-	     if (acc.map(_._1).contains(x)) dups2(xs, acc :+ unit)
-	     else dups2(xs, acc :+ (x, y))
-}
-
-def dups(xs: List[(Rexp, Val => Val)]) : List[(Rexp, Val => Val)] = {
-  val out = dups2(xs)
-  //if (out != xs) {
-  //  println()
-  //  println(s"Input ${string(ALTS(xs.map(_._1)))}")
-  //  println(s"Ouput ${string(ALTS(out.map(_._1)))}")
-  //}
-  out
-}
-
 
 // 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))
 }
@@ -193,54 +146,22 @@
   (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_ERROR(v: Val): Val = throw new Exception("error")
-def F_PRINT(v: Val): Val = {
-  println(s"value is ${v}")
-  throw new Exception("caught error")
+def F_RECD(f: Val => Val) = (v:Val) => v match {
+  case Rec(x, v) => Rec(x, f(v))
 }
-
-def flats(rs: List[Rexp], seen: Set[Rexp]) : (List[Rexp], Val => Val) = {
-  //println(s"I am flats: ${string(ALTS(rs))}")
-  //println(s"The size of seen is ${seen.size}, ${seen.map(string)}")
-  rs match {
-  case Nil => (Nil, F_ERROR)
-  case r::rs1 if seen.contains(simp(r)._1) => {
-    //println(s" I remove ${string(r)}")
-    val (rs2, f) = flats(rs1, seen)
-    (rs2, (v:Val) => IncProj(1, f(v)))
-  }
-  case ZERO::rs1 => {
-    val (rs2, f) = flats(rs1, seen)
-    (rs2, (v:Val) => IncProj(1, f(v)))
-  }
-  case ALTS(rs0)::rs1 => {
-    val (rss, f1) = flats(rs0, seen)
-    val (rs2, f2) = flats(rs1, rss.toSet ++ seen)
-    (rss:::rs2, (v:Val) => v match {
-      case Proj(n, vn) => 
-     	if (n < rss.length) Proj(0, f1(Proj(n, vn)))
-  	else IncProj(1, f2(Proj(n - rss.length, vn)))
-    })
-  }
-  case r1::rs2 => {
-    val (r1s, f1) = simp(r1)
-    val (rs3, f2) = flats(rs2, seen + r1s)
-    (r1s::rs3, (v:Val) => v match {
-      case Proj(0, vn) => Proj(0, f1(vn))
-      case Proj(n, vn) => IncProj(1, f2(Proj(n - 1, vn)))
-    })
-  }
-}}
+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(rs) => {
-    val (rs_s, fs_s) = flats(rs, Set())
-    rs_s match {
-      case Nil => (ZERO, F_ERROR) 
-      case r::Nil => (r, (v:Val) => fs_s(Proj(0, v)))
-      case rs_sd => (ALTS(rs_sd), fs_s)
+  case ALT(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 (ALT (r1s, r2s), F_ALT(f1s, f2s)) 
     }
   }
   case SEQ(r1, r2) => {
@@ -251,157 +172,116 @@
       case (_, ZERO) => (ZERO, F_ERROR)
       case (ONE, _) => (r2s, F_SEQ_Empty1(f1s, f2s))
       case (_, ONE) => (r1s, F_SEQ_Empty2(f1s, f2s))
-      case (ALTS(rs), r2s) => (ALTS(rs.map(SEQ(_, r2s))), 
-			       (v:Val) => v match {
-				 case Proj(n, Sequ(v1, v2)) => Sequ(f1s(Proj(n, v1)), f2s(v2))
-			       }
-			      )
       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 => {
-    //println(s"Size of the last regex: ${size(r)}")
-    //println(s"${string(r)}")
-    if (nullable(r)) mkeps(r) else throw new Exception("Not matched")
-  }
+  case Nil => if (nullable(r)) mkeps(r) else throw new Exception("Not matched")
   case c::cs => {
-    val rd = der(c, r)     
-    val (r_simp, f_simp) = simp(rd)
-    println(s"BEFORE ${string(rd)}")
-    println(s"AFTER  ${string(r_simp)}")
-    val rec = lex_simp(r_simp, cs)
-    inj(r, c, f_simp(rec))
+    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)
 
-
-lexing_simp("ab" | "aa", "ab")
-lexing_simp("ab" | "aa", "aa")
-
-lexing(STAR("a" | "aa"), "aaaaa")
-lexing_simp(STAR("a" | "aa"), "aaaaa")
-
-lexing(STAR("a" | "aa"), "aaaaaaaaaaa")
-lexing_simp(STAR("a" | "aa"), "aaaaaaaaaaa")
+lexing_simp(("a" | "ab") ~ ("b" | ""), "ab")
 
-lexing_simp(STAR("a" | "aa"), "a" * 2)
-lexing_simp(STAR("a" | "aa"), "a" * 3)
-lexing_simp(STAR("a" | "aa"), "a" * 4)
-
+// Lexing Rules for a Small While Language
 
-lexing_simp(STAR("a" | "aa"), "a" * 20)
-lexing_simp(STAR("a" | "aa"), "a" * 2000)
-
-lexing(ALTS(List("aa", "aa", "aa", "ab", "ab")), "ab")
-lexing_simp(ALTS(List("aa", "aa", "aa", "ab", "ab")), "ab")
+def PLUS(r: Rexp) = r ~ r.%
 
-lexing(ALTS(List(("aa" | "ab" | "aa"), "aa", "ab", "ab")), "ab")
-lexing_simp(ALTS(List(("aa" | "ab" | "aa"), "aa", "ab", "ab")), "ab")
-
-lexing(ALTS(List(ZERO, ZERO, ONE, "aa", ZERO, "aa", "aa")), "aa")
-lexing_simp(ALTS(List(ZERO, ZERO, ONE, "aa", ZERO, "aa", "aa")), "aa")
-
-lexing_simp(ONE | ZERO, "")
-lexing_simp(ZERO | ONE, "")
-
-lexing("a" | ZERO, "a")
-lexing_simp("a" | ZERO, "a")
-lexing(ZERO | "a", "a")
-lexing_simp(ZERO | "a", "a")
-
-lexing(ALTS(List(ZERO, ZERO, ONE, "a", ZERO, "a")), "a")
-lexing_simp(ALTS(List(ZERO, ZERO, ONE, "a", ZERO, "a")), "a")
-lexing(ALTS(List("a")), "a")
-lexing_simp(ALTS(List("a")), "a")
-
-lexing_simp(("a" | ZERO) | ZERO, "a")
-lexing_simp("a" | (ZERO | ZERO), "a")
-lexing_simp(ZERO | ("a" | ZERO), "a")
+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.% ~ "\""
 
 
-lexing_simp("abc", "abc")
-
-lexing_simp("abc" | ONE, "abc")
-
-lexing(("a" | "ab") ~ ("b" | ""), "ab")
-lexing_simp(("a" | "ab") ~ ("b" | ""), "ab")
-lexing_simp(("ba" | "c" | "ab"), "ab")
+val WHILE_REGS = (("k" $ KEYWORD) | 
+                  ("i" $ ID) | 
+                  ("o" $ OP) | 
+                  ("n" $ NUM) | 
+                  ("s" $ SEMI) | 
+                  ("str" $ STRING) |
+                  ("p" $ (LPAREN | RPAREN)) | 
+                  ("b" $ (BEGIN | END)) | 
+                  ("w" $ WHITESPACE)).%
 
-lexing(ALTS(List(ALTS(Nil), "c", "ab")), "ab")
-lexing_simp(ALTS(List(ALTS(Nil), "c", "ab")), "ab")
-
-lexing(ALTS(List(ALTS("ab"::Nil), "c", "ab")), "ab")
-lexing_simp(ALTS(List(ALTS("ab"::Nil), "c", "ab")), "ab")
-
-lexing(ALTS(List(ALTS(List("a","ab")), "c", "ab")), "ab")
-lexing_simp(ALTS(List(ALTS(List("a","ab")), "c", "ab")), "ab")
+//   Testing
+//============
 
-lexing(ALTS(List(ALTS(List("ab","a")), "c", "ab")), "ab")
-lexing_simp(ALTS(List(ALTS(List("ab","a")), "c", "ab")), "ab")
-
-lexing(ALTS(List(ALTS(List("ab","a","a")), "c", "ab")), "ab")
-lexing_simp(ALTS(List(ALTS(List("ab","a","a")), "c", "ab")), "ab")
-
-lexing(ALTS(List(ALTS(List("a","ab","a")), "c", "ab")), "ab")
-lexing_simp(ALTS(List(ALTS(List("a","ab","a")), "c", "ab")), "ab")
+def time[T](code: => T) = {
+  val start = System.nanoTime()
+  val result = code
+  val end = System.nanoTime()
+  println((end - start)/1.0e9)
+  result
+}
 
-lexing(ALTS(List(ALTS(List("b","a","ab")), "c", "ab")), "ab")
-lexing_simp(ALTS(List(ALTS(List("b","a","ab")), "c", "ab")), "ab")
-
-
-lexing_simp(ALTS(List("ba", "c", "ab")), "ab")
+val r1 = ("a" | "ab") ~ ("bcd" | "c")
+println(lexing(r1, "abcd"))
 
-lexing(ALTS(List("a", "ab", "a")), "ab")
-lexing_simp(ALTS(List("a", "ab", "a")), "ab")
-
-lexing(STAR("a" | "aa"), "aa")
-lexing_simp(STAR("a" | "aa"), "aa")
-
+val r2 = ("" | "a") ~ ("ab" | "b")
+println(lexing(r2, "ab"))
 
 
+// Two Simple While Tests
+//========================
+println("prog0 test")
 
-def enum(n: Int, s: String) : Set[Rexp] = n match {
-  case 0 => Set(ZERO, ONE) ++ s.toSet.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))
-  }
+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)))
+
+
+// Bigger 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)))
+println(env(lexing_simp(WHILE_REGS, prog2)).filterNot{_._1 == "w"}.mkString("\n"))
+
+// some more timing tests with
+// i copies of the program
+
+for (i <- 1 to 21 by 10) {
+  print(i.toString + ":  ")
+  time(lexing_simp(WHILE_REGS, prog2 * i))
 }
 
-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)
-  }
-}
 
-import scala.util.Try
-
-def tests(n: Int, m: Int, s: String) = {
-  val rs = enum(n, s)
-  val ss = strs(m, s)
-  println(s"cases generated: ${rs.size} regexes and ${ss.size} strings")
-  for (r1 <- rs.par; s1 <- ss.par) yield {
-    val res1 = Try(Some(lexing(r1, s1))).getOrElse(None)
-    val res2 = Try(Some(lexing_simp(r1, s1))).getOrElse(None)
-    if (res1 != res2) println(s"Disagree on ${r1} and ${s1}")
-    if (res1 != res2) Some((r1, s1)) else None
-  }
-}
-
-println("Testing")
-println(tests(2,7,"abc"))
-
-
-
diff -r 6e5e3adc9eb1 -r 1a521448d211 slides/slides04.pdf
Binary file slides/slides04.pdf has changed
diff -r 6e5e3adc9eb1 -r 1a521448d211 slides/slides04.tex
--- a/slides/slides04.tex	Tue Oct 16 00:42:10 2018 +0100
+++ b/slides/slides04.tex	Tue Oct 16 08:28:12 2018 +0100
@@ -60,12 +60,17 @@
 \includegraphics[scale=0.2]{../pics/s2.png}
 \end{textblock}}
 
-\only<4>{
+\only<4->{
 \begin{textblock}{6}(0.6,8.5)
 \includegraphics[scale=0.2]{../pics/s3.png}
 \end{textblock}
 \begin{textblock}{6}(8,8.5)
 \includegraphics[scale=0.2]{../pics/s4.png}
+\end{textblock}}
+
+\only<5>{
+\begin{textblock}{6}(1.5,14.5)  
+  room too hot, 3h lecture
 \end{textblock}}  
 
 % \begin{itemize}
@@ -123,7 +128,7 @@
 \begin{textblock}{6}(1,0.8)
 \begin{bubble}[6.7cm]
 \begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
-\multicolumn{3}{@{}l}{substitute \bl{$\mbox{Q}_1$} into \bl{$\mbox{Q}_0$} + \bl{$\mbox{Q}_2$}:}\\    
+\multicolumn{3}{@{}l}{substitute \bl{$\mbox{Q}_1$} into \bl{$\mbox{Q}_0$} \& \bl{$\mbox{Q}_2$}:}\\    
 \bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\ONE + \mbox{Q}_0\,b + \mbox{Q}_0\,a\,b +  \mbox{Q}_2\,b$}\\
 \bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a + \mbox{Q}_2\,a$}
 \end{tabular}
@@ -238,7 +243,8 @@
 \end{axis}
 \end{tikzpicture}
 
-The punchline is that existing libraries do depth-first search in NFAs.
+The punchline is that many existing libraries do depth-first search
+in NFAs (backtracking).
 
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
@@ -469,7 +475,14 @@
   \end{center}
   
 Today a lexer.  
-  
+
+\only<2>{
+\begin{textblock}{1}(6,9.8)
+\begin{tabular}{c}
+\includegraphics[scale=0.13]{../pics/rosetta.jpg}\\[-2mm]
+\footnotesize lexing $\Rightarrow$ recognising words (Stone of Rosetta)
+\end{tabular}
+\end{textblock}}
 \end{frame}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -767,7 +780,7 @@
   \bl{$inj\,(r_1 \cdot r_2)\,c\,(Seq(v_1,v_2))$} & \bl{$\dn$}  & \bl{$Seq(inj\,r_1\,c\,v_1,v_2)$}\\
   \bl{$inj\,(r_1 \cdot r_2)\,c\,(Left(Seq(v_1,v_2)))$} & \bl{$\dn$}  & \bl{$Seq(inj\,r_1\,c\,v_1,v_2)$}\\
   \bl{$inj\,(r_1 \cdot r_2)\,c\,(Right(v))$} & \bl{$\dn$}  & \bl{$Seq(mkeps(r_1),inj\,r_2\,c\,v)$}\\
-  \bl{$inj\,(r^*)\,c\,(Seq(v,vs))$} & \bl{$\dn$}  & \bl{$Stars\,(inj\,r\,c\,v\,::\,vs)$}\\
+  \bl{$inj\,(r^*)\,c\,(Seq(v,Stars\,vs))$} & \bl{$\dn$}  & \bl{$Stars\,(inj\,r\,c\,v\,::\,vs)$}\\
 \end{tabular}
 \end{center}\bigskip