1 // uses bitcode sequences and annotated regular 

     2 // expressions

     3 //

     4 // for basic regular expressions and RECD

     5 //

     6 // uses a non-standard extraction method for generating

     7 // tokens (this is tail-recursive) 

     8 //

     9 // can match 60 copies of the fib-program (size 10500)

    10 // in about 20 secs

    11 

    12 

    13 import scala.language.implicitConversions    

    14 import scala.language.reflectiveCalls

    15 import scala.annotation.tailrec   

    16 

    17 // standard regular expressions

    18 abstract class Rexp 

    19 case object ZERO extends Rexp

    20 case object ONE extends Rexp

    21 case class CHAR(c: Char) extends Rexp

    22 case class ALT(r1: Rexp, r2: Rexp) extends Rexp 

    23 case class SEQ(r1: Rexp, r2: Rexp) extends Rexp 

    24 case class STAR(r: Rexp) extends Rexp 

    25 case class RECD(x: String, r: Rexp) extends Rexp

    26 

    27 abstract class Bit

    28 case object Z extends Bit

    29 case object S extends Bit

    30 

    31 type Bits = List[Bit]

    32 

    33 // annotated regular expressions

    34 abstract class ARexp 

    35 case object AZERO extends ARexp

    36 case class AONE(bs: Bits) extends ARexp

    37 case class ACHAR(bs: Bits, c: Char) extends ARexp

    38 case class AALTS(bs: Bits, rs: List[ARexp]) extends ARexp 

    39 case class ASEQ(bs: Bits, r1: ARexp, r2: ARexp) extends ARexp 

    40 case class ASTAR(bs: Bits, r: ARexp) extends ARexp 

    41 

    42 // an abbreviation for binary alternatives

    43 def AALT(bs: Bits, r1: ARexp, r2: ARexp) = AALTS(bs, List(r1, r2))

    44 

    45 abstract class Val

    46 case object Empty extends Val

    47 case class Chr(c: Char) extends Val

    48 case class Sequ(v1: Val, v2: Val) extends Val

    49 case class Left(v: Val) extends Val

    50 case class Right(v: Val) extends Val

    51 case class Stars(vs: List[Val]) extends Val

    52 case class Recd(x: String, v: Val) extends Val

    53    

    54 // some convenience for typing in regular expressions

    55 def charlist2rexp(s: List[Char]): Rexp = s match {

    56   case Nil => ONE

    57   case c::Nil => CHAR(c)

    58   case c::s => SEQ(CHAR(c), charlist2rexp(s))

    59 }

    60 implicit def string2rexp(s: String) : Rexp = charlist2rexp(s.toList)

    61 

    62 implicit def RexpOps(r: Rexp) = new {

    63   def | (s: Rexp) = ALT(r, s)

    64   def % = STAR(r)

    65   def ~ (s: Rexp) = SEQ(r, s)

    66 }

    67 

    68 implicit def stringOps(s: String) = new {

    69   def | (r: Rexp) = ALT(s, r)

    70   def | (r: String) = ALT(s, r)

    71   def % = STAR(s)

    72   def ~ (r: Rexp) = SEQ(s, r)

    73   def ~ (r: String) = SEQ(s, r)

    74   def $ (r: Rexp) = RECD(s, r)

    75 }

    76 

    77 def size(r: Rexp) : Int = r match {

    78   case ZERO => 1

    79   case ONE => 1

    80   case CHAR(_) => 1

    81   case ALT(r1, r2) => 1 + size(r1) + size(r2)

    82   case SEQ(r1, r2) => 1 + size(r1) + size(r2)

    83   case STAR(r) => 1 + size(r)

    84   case RECD(_, r) => 1 + size(r)

    85 }

    86 

    87 

    88 

    89 // Bitcoded + Annotation

    90 //=======================

    91 

    92 //erase function: extracts the Rexp from ARexp

    93 def erase(r:ARexp): Rexp = r match{

    94   case AZERO => ZERO

    95   case AONE(_) => ONE

    96   case ACHAR(bs, c) => CHAR(c)

    97   case AALTS(bs, Nil) => ZERO

    98   case AALTS(bs, r::Nil) => erase(r)

    99   case AALTS(bs, r::rs) => ALT(erase(r), erase(AALTS(bs, rs)))

   100   case ASEQ(bs, r1, r2) => SEQ (erase(r1), erase(r2))

   101   case ASTAR(cs, r)=> STAR(erase(r))

   102 }

   103 

   104 def fuse(bs: Bits, r: ARexp) : ARexp = r match {

   105   case AZERO => AZERO

   106   case AONE(cs) => AONE(bs ++ cs)

   107   case ACHAR(cs, c) => ACHAR(bs ++ cs, c)

   108   case AALTS(cs, rs) => AALTS(bs ++ cs, rs)

   109   case ASEQ(cs, r1, r2) => ASEQ(bs ++ cs, r1, r2)

   110   case ASTAR(cs, r) => ASTAR(bs ++ cs, r)

   111 }

   112 

   113 def internalise(r: Rexp) : ARexp = r match {

   114   case ZERO => AZERO

   115   case ONE => AONE(Nil)

   116   case CHAR(c) => ACHAR(Nil, c)

   117   case ALT(r1, r2) => AALT(Nil, fuse(List(Z), internalise(r1)), fuse(List(S), internalise(r2)))

   118   case SEQ(r1, r2) => ASEQ(Nil, internalise(r1), internalise(r2))

   119   case STAR(r) => ASTAR(Nil, internalise(r))

   120   case RECD(_, r) => internalise(r)

   121 }

   122 

   123 // example

   124 // internalise(("a" | "ab") ~ ("b" | ""))

   125 

   126 

   127 // decoding of a value from a bitsequence

   128 // (this is not tail-recursive and therefore a potential bottleneck)

   129 def vdecode_aux(r: Rexp, bs: Bits) : (Val, Bits) = (r, bs) match {

   130   case (ONE, bs) => (Empty, bs)

   131   case (CHAR(c), bs) => (Chr(c), bs)

   132   case (ALT(r1, r2), Z::bs) => {

   133     val (v, bs1) = vdecode_aux(r1, bs)

   134     (Left(v), bs1)

   135   }

   136   case (ALT(r1, r2), S::bs) => {

   137     val (v, bs1) = vdecode_aux(r2, bs)

   138     (Right(v), bs1)

   139   }

   140   case (SEQ(r1, r2), bs) => {

   141     val (v1, bs1) = vdecode_aux(r1, bs)

   142     val (v2, bs2) = vdecode_aux(r2, bs1)

   143     (Sequ(v1, v2), bs2)

   144   }

   145   case (STAR(r1), Z::bs) => {

   146     val (v, bs1) = vdecode_aux(r1, bs)

   147     val (Stars(vs), bs2) = vdecode_aux(STAR(r1), bs1)

   148     (Stars(v::vs), bs2)

   149   }

   150   case (STAR(_), S::bs) => (Stars(Nil), bs)

   151   case (RECD(s, r1), bs) => 

   152     val (v, bs1) = vdecode_aux(r1, bs)

   153     (Recd(s, v), bs1)

   154 }

   155 

   156 def vdecode(r: Rexp, bs: Bits) = vdecode_aux(r, bs) match {

   157   case (v, Nil) => v

   158   case _ => throw new Exception("Not decodable")

   159 }

   160 

   161 // decoding of sequence of string tokens from a bitsequence

   162 // tail-recursive version using an accumulator (alternative for 

   163 // vdecode)

   164 @tailrec

   165 def sdecode_aux(rs: List[Rexp], bs: Bits, acc: List[String]) : List[String] = (rs, bs) match {

   166   case (Nil, _) => acc

   167   case (_, Nil) => acc

   168   case (ONE::rest, bs) => sdecode_aux(rest, bs, acc)

   169   case (CHAR(c)::rest, bs) => 

   170     sdecode_aux(rest, bs, (acc.head ++ c.toString)::acc.tail)

   171   case (ALT(r1, r2)::rest, Z::bs) => sdecode_aux(r1::rest, bs, acc)

   172   case (ALT(r1, r2)::rest, S::bs) => sdecode_aux(r2::rest, bs, acc)

   173   case (SEQ(r1, r2)::rest, bs) => sdecode_aux(r1::r2::rest, bs, acc)

   174   case (STAR(r1)::rest, Z::bs) => sdecode_aux(r1::STAR(r1)::rest, bs, acc)

   175   case (STAR(_)::rest, S::bs) => sdecode_aux(rest, bs, acc)

   176   case (RECD(s, r1)::rest, bs) => 

   177     sdecode_aux(r1::rest, bs, s"$s:"::acc)

   178 }

   179 

   180 def sdecode(r: Rexp, bs: Bits) : List[String] = 

   181   sdecode_aux(List(r), bs, List("")).reverse.tail

   182 

   183 

   184 

   185 // nullable function: tests whether the an (annotated) 

   186 // regular expression can recognise the empty string

   187 def bnullable (r: ARexp) : Boolean = r match {

   188   case AZERO => false

   189   case AONE(_) => true

   190   case ACHAR(_,_) => false

   191   case AALTS(_, rs) => rs.exists(bnullable)

   192   case ASEQ(_, r1, r2) => bnullable(r1) && bnullable(r2)

   193   case ASTAR(_, _) => true

   194 }

   195 

   196 def bmkeps(r: ARexp) : Bits = r match {

   197   case AONE(bs) => bs

   198   case AALTS(bs, r::Nil) => bs ++ bmkeps(r) 

   199   case AALTS(bs, r::rs) => 

   200     if (bnullable(r)) bs ++ bmkeps(r) else bmkeps(AALTS(bs, rs))  

   201   case ASEQ(bs, r1, r2) => bs ++ bmkeps(r1) ++ bmkeps(r2)

   202   case ASTAR(bs, r) => bs ++ List(S)

   203 }

   204 

   205 // derivative of a regular expression w.r.t. a character

   206 def bder(c: Char, r: ARexp) : ARexp = r match {

   207   case AZERO => AZERO

   208   case AONE(_) => AZERO

   209   case ACHAR(bs, d) => if (c == d) AONE(bs) else AZERO

   210   case AALTS(bs, rs) => AALTS(bs, rs.map(bder(c, _)))

   211   case ASEQ(bs, r1, r2) => 

   212     if (bnullable(r1)) AALT(bs, ASEQ(Nil, bder(c, r1), r2), fuse(bmkeps(r1), bder(c, r2)))

   213     else ASEQ(bs, bder(c, r1), r2)

   214   case ASTAR(bs, r) => ASEQ(bs, fuse(List(Z), bder(c, r)), ASTAR(Nil, r))

   215 }

   216 

   217 // derivative w.r.t. a string (iterates bder)

   218 @tailrec

   219 def bders (s: List[Char], r: ARexp) : ARexp = s match {

   220   case Nil => r

   221   case c::s => bders(s, bder(c, r))

   222 }

   223 

   224 // main unsimplified lexing function (produces a bitsequence)

   225 def blex(r: ARexp, s: List[Char]) : Bits = s match {

   226   case Nil => if (bnullable(r)) bmkeps(r) else throw new Exception("Not matched")

   227   case c::cs => blex(bder(c, r), cs)

   228 }

   229 

   230 // calls blex and decodes the value

   231 def blexing(r: Rexp, s: String) : Val = 

   232   vdecode(r, blex(internalise(r), s.toList))

   233 

   234 

   235 // example by Tudor

   236 //val reg = (STAR("a") ~ ("b" | "c")).%

   237 

   238 //println(blexing(reg, "aab"))

   239 

   240 

   241 //=======================

   242 // simplification 

   243 //

   244 

   245 def flts(rs: List[ARexp]) : List[ARexp] = rs match {

   246   case Nil => Nil

   247   case AZERO :: rs => flts(rs)

   248   case AALTS(bs, rs1) :: rs => rs1.map(fuse(bs, _)) ++ flts(rs)

   249   case r1 :: rs => r1 :: flts(rs)

   250 }

   251 

   252 def bsimp(r: ARexp): ARexp = r match {

   253   case ASEQ(bs1, r1, r2) => (bsimp(r1), bsimp(r2)) match {

   254       case (AZERO, _) => AZERO

   255       case (_, AZERO) => AZERO

   256       case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s)

   257       // needed in order to keep the size down

   258       case (AALTS(bs, rs), r2) => AALTS(bs1 ++ bs, rs.map(ASEQ(Nil, _, r2)))

   259       case (r1s, r2s) => ASEQ(bs1, r1s, r2s)

   260   }

   261   // distinctBy deletes copies of the same  "erased" regex

   262   case AALTS(bs1, rs) => (flts(rs.map(bsimp))).distinctBy(erase) match {

   263       case Nil => AZERO

   264       case r::Nil => fuse(bs1, r)

   265       case rs => AALTS(bs1, rs)

   266   }

   267   case r => r

   268 }

   269 

   270 def blex_simp(r: ARexp, s: List[Char]) : Bits = s match {

   271   case Nil => if (bnullable(r)) bmkeps(r) 

   272               else throw new Exception("Not matched")

   273   case c::cs => blex_simp(bsimp(bder(c, r)), cs)

   274 }

   275 

   276 // blexing_simp decodes a value from the bitsequence (potentially slow)

   277 // blexing2_simp decodes a string-list from the bitsequence

   278 def blexing_simp(r: Rexp, s: String) : Val = 

   279   vdecode(r, blex_simp(internalise(r), s.toList))

   280 

   281 def blexing2_simp(r: Rexp, s: String) : List[String] = 

   282   sdecode(r, blex_simp(internalise(r), s.toList))

   283 

   284 

   285 //println(blexing_simp(reg, "aab"))

   286 

   287 

   288 // extracts a string from value

   289 def flatten(v: Val) : String = v match {

   290   case Empty => ""

   291   case Chr(c) => c.toString

   292   case Left(v) => flatten(v)

   293   case Right(v) => flatten(v)

   294   case Sequ(v1, v2) => flatten(v1) + flatten(v2)

   295   case Stars(vs) => vs.map(flatten).mkString

   296 }

   297 

   298 // extracts an environment from a value

   299 def env(v: Val) : List[(String, String)] = v match {

   300   case Empty => Nil

   301   case Chr(c) => Nil

   302   case Left(v) => env(v)

   303   case Right(v) => env(v)

   304   case Sequ(v1, v2) => env(v1) ::: env(v2)

   305   case Stars(vs) => vs.flatMap(env)

   306   case Recd(x, v) => (x, flatten(v))::env(v)

   307 }

   308 

   309 def bsize(a: ARexp) = size(erase(a))

   310 

   311 // Some Tests

   312 //============

   313 

   314 

   315 def time_needed[T](i: Int, code: => T) = {

   316   val start = System.nanoTime()

   317   for (j <- 1 to i) code

   318   val end = System.nanoTime()

   319   (end - start)/(i * 1.0e9)

   320 }

   321 

   322 /*

   323 val evil1 = STAR(STAR("a")) ~ "b"

   324 val evil2 = STAR(STAR(STAR("a"))) ~ "b"

   325 val evil3 = STAR("aa" | "a")

   326 

   327 for(i <- 0 to 10000 by 1000) {

   328     println(time_needed(1, blexing2_simp(evil1, "a"*i ++ "b")))

   329 }

   330 

   331 for(i <- 0 to 10000 by 1000) {

   332     println(time_needed(1, blexing2_simp(evil2, "a"*i ++ "b")))

   333 }

   334 

   335 for(i <- 0 to 10000 by 1000) {

   336     println(time_needed(1, blexing2_simp(evil3, "a"*i)))

   337 }

   338 */

   339 

   340 // WHILE LANGUAGE

   341 //================

   342 def PLUS(r: Rexp) = r ~ r.%

   343 

   344 def Range(s : List[Char]) : Rexp = s match {

   345   case Nil => ZERO

   346   case c::Nil => CHAR(c)

   347   case c::s => ALT(CHAR(c), Range(s))

   348 }

   349 def RANGE(s: String) = Range(s.toList)

   350 

   351 val SYM = RANGE("ABCDEFGHIJKLMNOPQRSTUVXYZabcdefghijklmnopqrstuvwxyz_")

   352 val DIGIT = RANGE("0123456789")

   353 val ID = SYM ~ (SYM | DIGIT).% 

   354 val NUM = PLUS(DIGIT)

   355 val KEYWORD : Rexp = "skip" | "while" | "do" | "if" | "then" | "else" | "read" | "write" 

   356 val SEMI: Rexp = ";"

   357 val OP: Rexp = ":=" | "=" | "-" | "+" | "*" | "!=" | "<" | ">"

   358 val WHITESPACE = PLUS(" " | "\n" | "\t")

   359 val RPAREN: Rexp = "{"

   360 val LPAREN: Rexp = "}"

   361 val STRING: Rexp = "\"" ~ SYM.% ~ "\""

   362 

   363 

   364 val WHILE_REGS = (("k" $ KEYWORD) | 

   365                   ("i" $ ID) | 

   366                   ("o" $ OP) | 

   367                   ("n" $ NUM) | 

   368                   ("s" $ SEMI) | 

   369                   ("str" $ STRING) |

   370                   ("p" $ (LPAREN | RPAREN)) | 

   371                   ("w" $ WHITESPACE)).%

   372 

   373 

   374 // Some Simple While Tests

   375 //========================

   376 

   377 val prog0 = """read n"""

   378 println(s"test: $prog0")

   379 println(env(blexing_simp(WHILE_REGS, prog0)))

   380 println(blexing2_simp(WHILE_REGS, prog0))

   381 

   382 val prog1 = """read  n; write n"""  

   383 println(s"test: $prog1")

   384 println(env(blexing_simp(WHILE_REGS, prog1)))

   385 println(blexing2_simp(WHILE_REGS, prog1))

   386 

   387 val prog2 = """

   388 write "Fib";

   389 read n;

   390 minus1 := 0;

   391 minus2 := 1;

   392 while n > 0 do {

   393   temp := minus2;

   394   minus2 := minus1 + minus2;

   395   minus1 := temp;

   396   n := n - 1

   397 };

   398 write "Result";

   399 write minus2

   400 """

   401 

   402 println("lexing fib program (once)")

   403 println(blexing2_simp(WHILE_REGS, prog2).filter(s => s == "" || !s.startsWith("w")))

   404 

   405 val n = 60

   406 println(s"lexing fib program ($n times, size ${prog2.length * n})")

   407 println(time_needed(1, blexing2_simp(WHILE_REGS, prog2 * n)))