1 

     2 // regular expressions including NOT

     3 abstract class Rexp

     4 

     5 case object NULL extends Rexp

     6 case object EMPTY extends Rexp

     7 case class CHAR(c: Char) extends Rexp

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

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

    10 case class STAR(r: Rexp) extends Rexp

    11 case class NOT(r: Rexp) extends Rexp

    12 

    13 

    14 // some convenience for typing in regular expressions

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

    16   case Nil => EMPTY

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

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

    19 }

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

    21 

    22 

    23 // nullable function: tests whether the regular 

    24 // expression can recognise the empty string

    25 def nullable (r: Rexp) : Boolean = r match {

    26   case NULL => false

    27   case EMPTY => true

    28   case CHAR(_) => false

    29   case ALT(r1, r2) => nullable(r1) || nullable(r2)

    30   case SEQ(r1, r2) => nullable(r1) && nullable(r2)

    31   case STAR(_) => true

    32   case NOT(r) => !(nullable(r))

    33 }

    34 

    35 // tests whether a regular expression 

    36 // cannot recognise more

    37 def no_more (r: Rexp) : Boolean = r match {

    38   case NULL => true

    39   case EMPTY => false

    40   case CHAR(_) => false

    41   case ALT(r1, r2) => no_more(r1) && no_more(r2)

    42   case SEQ(r1, r2) => if (nullable(r1)) (no_more(r1) && no_more(r2)) else no_more(r1)

    43   case STAR(_) => false

    44   case NOT(r) => !(no_more(r))

    45 }

    46 

    47 

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

    49 def der (c: Char, r: Rexp) : Rexp = r match {

    50   case NULL => NULL

    51   case EMPTY => NULL  case CHAR(d) => if (c == d) EMPTY else NULL

    52   case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))

    53   case SEQ(r1, r2) => 

    54     if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))

    55     else SEQ(der(c, r1), r2)

    56   case STAR(r) => SEQ(der(c, r), STAR(r))

    57   case NOT(r) => NOT(der (c, r))

    58 }

    59 

    60 // regular expression for specifying 

    61 // ranges of characters

    62 def RANGE(s : List[Char]) : Rexp = s match {

    63   case Nil => NULL

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

    65   case c::s => ALT(CHAR(c), RANGE(s))

    66 }

    67 

    68 // one or more

    69 def PLUS(r: Rexp) = SEQ(r, STAR(r))

    70 

    71 // some regular expressions

    72 val DIGIT = RANGE("0123456789".toList)

    73 val NONZERODIGIT = RANGE("123456789".toList)

    74 

    75 val NUMBER = ALT(SEQ(NONZERODIGIT, STAR(DIGIT)), "0")

    76 val LPAREN = CHAR('(')

    77 val RPAREN = CHAR(')')

    78 val WHITESPACE = PLUS(RANGE(" \n".toList))

    79 val OPS = RANGE("+-*".toList)

    80 

    81 // for classifying the strings that have been recognised

    82 abstract class Token

    83 

    84 case object T_WHITESPACE extends Token

    85 case object T_NUM extends Token

    86 case class T_OP(s: String) extends Token

    87 case object T_LPAREN extends Token

    88 case object T_RPAREN extends Token

    89 case class T_NT(s: String) extends Token

    90 

    91 type Rule = (Rexp, List[Char] => Token)

    92 

    93 def error (s: String) = throw new IllegalArgumentException ("Cannot tokenize: " + s)

    94 

    95 def munch(r: Rexp, action: List[Char] => Token, s: List[Char], t: List[Char]) : Option[(List[Char], Token)] = 

    96   s match {

    97     case Nil if (nullable(r)) => Some(Nil, action(t))

    98     case Nil => None

    99     case c::s if (no_more(der (c, r)) && nullable(r)) => Some(c::s, action(t))

   100     case c::s if (no_more(der (c, r))) => None

   101     case c::s => munch(der (c, r), action, s, t ::: List(c))

   102   }

   103 

   104 def one_token (rs: List[Rule], s: List[Char]) : (List[Char], Token) = {

   105  val somes = rs.map { (r) => munch(r._1, r._2, s, Nil) } .flatten

   106  if (somes == Nil) error(s.mkString) else (somes sortBy (_._1.length) head)

   107 }

   108 

   109 def tokenize (rs: List[Rule], s: List[Char]) : List[Token] = s match {

   110   case Nil => Nil

   111   case _ => one_token(rs, s) match {

   112     case (rest, token) => token :: tokenize(rs, rest) 

   113   }

   114 }

   115 

   116 def tokenizer(rs: List[Rule], s: String) : List[Token] = 

   117   tokenize(rs, s.toList).filterNot(_ match {

   118     case T_WHITESPACE => true

   119     case _ => false

   120   })

   121 

   122 

   123 

   124 // lexing rules for arithmetic expressions

   125 val lexing_rules: List[Rule]= 

   126   List((NUMBER, (s) => T_NUM),

   127        (WHITESPACE, (s) => T_WHITESPACE),

   128        (LPAREN, (s) => T_LPAREN),

   129        (RPAREN, (s) => T_RPAREN),

   130        (OPS, (s) => T_OP(s.mkString)))

   131 

   132 

   133 // examples

   134 println(tokenizer(lexing_rules, "2 + 3 * 4 + 1"))

   135 println(tokenizer(lexing_rules, "(2 + 3) * (4 + 1)"))

   136 

   137 

   138 type Grammar = List[(String, List[Token])]

   139 

   140 // grammar for arithmetic expressions

   141 val grammar = 

   142   List ("E" -> List(T_NUM),

   143         "E" -> List(T_NT("E"), T_OP("+"), T_NT("E")),

   144         "E" -> List(T_NT("E"), T_OP("-"), T_NT("E")),

   145         "E" -> List(T_NT("E"), T_OP("*"), T_NT("E")),    

   146         "E" -> List(T_LPAREN, T_NT("E"), T_RPAREN))

   147 

   148 

   149 def chop[A](ts1: List[A], prefix: List[A], ts2: List[A]) : Option[(List[A], List[A])] = 

   150   ts1 match {

   151     case Nil => None

   152     case t::ts => 

   153       if (ts1.startsWith(prefix)) Some(ts2.reverse, ts1.drop(prefix.length))

   154       else chop(ts, prefix, t::ts2)

   155   }

   156 

   157 // examples

   158 chop(List(1,2,3,4,5,6,7,8,9), List(4,5), Nil)  

   159 chop(List(1,2,3,4,5,6,7,8,9), List(3,5), Nil)  

   160 

   161 def replace[A](ts: List[A], out: List[A], in: List [A]) = 

   162   chop(ts, out, Nil) match {

   163     case None => None

   164     case Some((before, after)) => Some(before ::: in ::: after)

   165   }  

   166 

   167 def parse1(g: Grammar, ts: List[Token]) : Boolean = {

   168   //println(ts)

   169   if (ts == List(T_NT("E"))) true

   170   else {

   171     val tss = for ((lhs, rhs) <- g) yield replace(ts, rhs, List(T_NT(lhs)))

   172     tss.flatten.exists(parse1(g, _))

   173   }

   174 }

   175  

   176 

   177 println() ; parse1(grammar, tokenizer(lexing_rules, "2 + 3 * 4 + 1"))

   178 println() ; parse1(grammar, tokenizer(lexing_rules, "(2 + 3) * (4 + 1)"))

   179 println() ; parse1(grammar, tokenizer(lexing_rules, "(2 + 3) * 4 (4 + 1)"))

   180 

   181 

   182