# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1525512660 -3600
# Node ID 352d15782d350bf8f5ff38e2a59e8ce7c285fe0b
# Parent  c6ba1d17aaf3220183c1767e2cbba218a192d475
updated

diff -r c6ba1d17aaf3 -r 352d15782d35 progs/token.scala
--- a/progs/token.scala	Thu Apr 05 02:33:30 2018 +0100
+++ b/progs/token.scala	Sat May 05 10:31:00 2018 +0100
@@ -5,20 +5,32 @@
 case object ZERO extends Rexp
 case object ONE extends Rexp
 case class CHAR(c: Char) extends Rexp
-case class ALT(r1: Rexp, r2: Rexp) 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[A](f: String => A, 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 Left(v: Val) extends Val
-case class Right(v: Val) extends Val
+case class Proj(n: Int, v: Val) extends Val
 case class Stars(vs: List[Val]) extends Val
-case class Rec[A](f: String => A, v: 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)
+}
+
+
 // some convenience for typing in regular expressions
 def charlist2rexp(s : List[Char]): Rexp = s match {
   case Nil => ONE
@@ -39,19 +51,40 @@
   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 ALT(r1, r2) => nullable(r1) || nullable(r2)
+  case ALTS(rs) => rs.exists(nullable)
   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
@@ -59,12 +92,11 @@
   case ZERO => ZERO
   case ONE => ZERO
   case CHAR(d) => if (c == d) ONE else ZERO
-  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))
+  case ALTS(rs) => ALTS(rs.map(der(c, _)))
   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)
@@ -77,66 +109,74 @@
 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 Proj(_, 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[A](v: Val) : List[A] = 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(f, v) => ((f:String => A)(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), 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 (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 (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 => if (nullable(r)) mkeps(r) 
-              else throw new Exception("Not matched")
+  case Nil => {
+    //println(s"Size of the last regex: ${size(r)}")
+    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))
 }
@@ -144,22 +184,54 @@
   (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")
+def F_PRINT(v: Val): Val = {
+  println(s"value is ${v}")
+  throw new Exception("caught error")
 }
-def F_ERROR(v: Val): Val = throw new Exception("error")
+
+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)))
+    })
+  }
+}}
 
 // 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 (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 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 SEQ(r1, r2) => {
@@ -170,134 +242,157 @@
       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 => if (nullable(r)) mkeps(r) else throw new Exception("Not matched")
+  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 c::cs => {
-    val (r_simp, f_simp) = simp(der(c, r))
-    inj(r, c, f_simp(lex_simp(r_simp, 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))
   }
 }
 
 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
+lexing_simp("ab" | "aa", "ab")
+lexing_simp("ab" | "aa", "aa")
 
-def PLUS(r: Rexp) = r ~ r.%
+lexing(STAR("a" | "aa"), "aaaaa")
+lexing_simp(STAR("a" | "aa"), "aaaaa")
 
-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(STAR("a" | "aa"), "aaaaaaaaaaa")
+lexing_simp(STAR("a" | "aa"), "aaaaaaaaaaa")
+
+lexing_simp(STAR("a" | "aa"), "a" * 2)
+lexing_simp(STAR("a" | "aa"), "a" * 3)
+lexing_simp(STAR("a" | "aa"), "a" * 4)
 
 
-val WHILE_REGS = (((s) => Token $ KEYWORD) | 
-                  ("i" $ ID) | 
-                  ("o" $ OP) | 
-                  ("n" $ NUM) | 
-                  ("s" $ SEMI) | 
-                  ("str" $ STRING) |
-                  ("p" $ (LPAREN | RPAREN)) | 
-                  ("b" $ (BEGIN | END)) | 
-                  ("w" $ WHITESPACE)).%
+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")
+
+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")
+
+
+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")
+
+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")
+
+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")
+
+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")
+
+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")
 
 
 
 
-// extracts an environment from a value
-def env(v: Val) : List[Token] = 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(f, v) => (f(flatten(v)))::env(v)
+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))
+  }
 }
 
-//   Testing
-//============
-
-def time[T](code: => T) = {
-  val start = System.nanoTime()
-  val result = code
-  val end = System.nanoTime()
-  println((end - start)/1.0e9)
-  result
+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)
+  }
 }
 
-val r1 = ("a" | "ab") ~ ("bcd" | "c")
-println(lexing(r1, "abcd"))
+import scala.util.Try
 
-val r2 = ("" | "a") ~ ("ab" | "b")
-println(lexing(r2, "ab"))
+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"))
 
 
-// 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)))
-
-
-// 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))
-}
-
-abstract class Token
-case class KeyToken(s: String) extends Token
-case class IdToken(s: String) extends Token
-
-list[(STRING, STRING)]=> List(TOKEN)