1 // uses bitcode sequences and annotated regular 

     2 // expressions

     3 //

     4 // for basic regular expressions and RECD

     5 //

     6 // the main difference is that it includes

     7 // also character sets

     8 //

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

    10 // tokens (this is tail-recursive) 

    11 //

    12 // can match 200 copies of the fib-program (size 35000)

    13 // in about 20 secs

    14 

    15 

    16 import scala.language.implicitConversions    

    17 import scala.language.reflectiveCalls

    18 import scala.annotation.tailrec   

    19 

    20 // standard regular expressions

    21 abstract class Rexp 

    22 case object ZERO extends Rexp

    23 case object ONE extends Rexp

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

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

    26 case class STAR(r: Rexp) extends Rexp 

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

    28 case class CHARSET(f: Char => Boolean) extends Rexp

    29 

    30 //abbreviation for single characters

    31 def CHAR(c: Char) = CHARSET(_ == c)

    32 

    33 // bit-codes with additional information about 

    34 // matched characters

    35 abstract class Bit

    36 case object Z extends Bit

    37 case object S extends Bit

    38 case class C(c: Char) extends Bit

    39 

    40 type Bits = List[Bit]

    41 

    42 // annotated regular expressions

    43 abstract class ARexp 

    44 case object AZERO extends ARexp

    45 case class AONE(bs: Bits) extends ARexp

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

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

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

    49 case class ACHARSET(bs: Bits, f: Char => Boolean) extends ARexp

    50 

    51 // an abbreviation for binary alternatives

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

    53 

    54 abstract class Val

    55 case object Empty extends Val

    56 case class Chr(c: Char) extends Val

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

    58 case class Left(v: Val) extends Val

    59 case class Right(v: Val) extends Val

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

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

    62    

    63 // some convenience for typing in regular expressions

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

    65   case Nil => ONE

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

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

    68 }

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

    70 

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

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

    73   def % = STAR(r)

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

    75 }

    76 

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

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

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

    80   def % = STAR(s)

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

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

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

    84 }

    85 

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

    87   case ZERO => 1

    88   case ONE => 1

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

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

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

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

    93   case CHARSET(_) => 1

    94 }

    95 

    96 

    97 

    98 // Bitcoded + Annotation

    99 //=======================

   100 

   101 //erase function: extracts the Rexp from ARexp

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

   103   case AZERO => ZERO

   104   case AONE(_) => ONE

   105   case AALTS(bs, Nil) => ZERO

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

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

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

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

   110   case ACHARSET(bs, f) => CHARSET(f)

   111 }

   112 

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

   114   case AZERO => AZERO

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

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

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

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

   119   case ACHARSET(cs, f) => ACHARSET(bs ++ cs, f)

   120 }

   121 

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

   123   case ZERO => AZERO

   124   case ONE => AONE(Nil)

   125   case ALT(r1, r2) => 

   126     AALT(Nil, fuse(List(Z), internalise(r1)), fuse(List(S), internalise(r2)))

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

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

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

   130   case CHARSET(f) => ACHARSET(Nil, f)

   131 }

   132 

   133 // example

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

   135 

   136 

   137 // decoding of a value from a bitsequence

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

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

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

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

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

   143     (Left(v), bs1)

   144   }

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

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

   147     (Right(v), bs1)

   148   }

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

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

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

   152     (Sequ(v1, v2), bs2)

   153   }

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

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

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

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

   158   }

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

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

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

   162     (Recd(s, v), bs1)

   163   case (CHARSET(_), C(c)::bs) => (Chr(c), bs)

   164 }

   165 

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

   167   case (v, Nil) => v

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

   169 }

   170 

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

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

   173 // vdecode)

   174 @tailrec

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

   176   case (Nil, _) => acc

   177   case (_, Nil) => acc

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

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

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

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

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

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

   184   case (RECD(s, r1)::rest, bs) => sdecode_aux(r1::rest, bs, s"$s:"::acc)

   185   case (CHARSET(_)::rest, C(c)::bs) => 

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

   187 }

   188 

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

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

   191 

   192 

   193 

   194 // nullable function: tests whether the a (annotated) 

   195 // regular expression can recognise the empty string

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

   197   case AZERO => false

   198   case AONE(_) => true

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

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

   201   case ASTAR(_, _) => true

   202   case ACHARSET(_, _) => false

   203 }

   204 

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

   206   case AONE(bs) => bs

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

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

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

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

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

   212   case _ => throw new Exception("trying to bmkeps from non-nullable regex")

   213 }

   214 

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

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

   217   case AZERO => AZERO

   218   case AONE(_) => AZERO

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

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

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

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

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

   224   case ACHARSET(bs, f) => if(f(c)) AONE(bs :+ C(c)) else AZERO

   225 }

   226 

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

   228 @tailrec

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

   230   case Nil => r

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

   232 }

   233 

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

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

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

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

   238 }

   239 

   240 // calls blex and decodes the value

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

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

   243 

   244 

   245 // example by Tudor

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

   247 

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

   249 

   250 

   251 //=======================

   252 // simplification 

   253 //

   254 

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

   256   case Nil => Nil

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

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

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

   260 }

   261 

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

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

   264       case (AZERO, _) => AZERO

   265       case (_, AZERO) => AZERO

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

   267       // needed in order to keep the size down

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

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

   270   }

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

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

   273       case Nil => AZERO

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

   275       case rs => AALTS(bs1, rs)

   276   }

   277   case r => r

   278 }

   279 

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

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

   282               else throw new Exception("Not matched")

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

   284 }

   285 

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

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

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

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

   290 

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

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

   293 

   294 

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

   296 

   297 

   298 // extracts a string from value

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

   300   case Empty => ""

   301   case Chr(c) => c.toString

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

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

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

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

   306 }

   307 

   308 // extracts an environment from a value

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

   310   case Empty => Nil

   311   case Chr(c) => Nil

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

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

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

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

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

   317 }

   318 

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

   320 

   321 // Some Tests

   322 //============

   323 

   324 

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

   326   val start = System.nanoTime()

   327   for (j <- 1 to i) code

   328   val end = System.nanoTime()

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

   330 }

   331 

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

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

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

   335 

   336 /*

   337 println("evil1")

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

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

   340 }

   341 */

   342 

   343 /*

   344 println("evil2")

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

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

   347 }

   348 */

   349 

   350 /*

   351 println("evil3")

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

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

   354 }

   355 */

   356 

   357 // WHILE LANGUAGE

   358 //================

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

   360 def RANGE(s: String) = CHARSET(s.toSet)

   361 

   362 val SYM = RANGE("ABCDEFGHIJKLMNOPQRSTUVXYZabcdefghijklmnopqrstuvwxyz_")

   363 val DIGIT = RANGE("0123456789")

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

   365 val NUM = PLUS(DIGIT)

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

   367 val SEMI: Rexp = ";"

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

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

   370 val RPAREN: Rexp = "{"

   371 val LPAREN: Rexp = "}"

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

   373 

   374 

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

   376                   ("i" $ ID) | 

   377                   ("o" $ OP) | 

   378                   ("n" $ NUM) | 

   379                   ("s" $ SEMI) | 

   380                   ("str" $ STRING) |

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

   382                   ("w" $ WHITESPACE)).%

   383 

   384 

   385 // Some Simple While Tests

   386 //========================

   387 

   388 val prog0 = """read n"""

   389 println(s"test: $prog0")

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

   391 println(blexing2_simp(WHILE_REGS, prog0))

   392 

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

   394 println(s"test: $prog1")

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

   396 println(blexing2_simp(WHILE_REGS, prog1))

   397 

   398 val prog2 = """

   399 write "Fib";

   400 read n;

   401 minus1 := 0;

   402 minus2 := 1;

   403 while n > 0 do {

   404   temp := minus2;

   405   minus2 := minus1 + minus2;

   406   minus1 := temp;

   407   n := n - 1

   408 };

   409 write "Result";

   410 write minus2

   411 """

   412 

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

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

   415 

   416 val n = 200

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

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