--- a/progs/token2.scala Fri Nov 06 08:52:16 2015 +0000
+++ b/progs/token2.scala Sat Nov 07 21:37:03 2015 +0000
@@ -1,25 +1,35 @@
-import scala.language.implicitConversions
+import scala.language.implicitConversions
import scala.language.reflectiveCalls
-import scala.util._
-import scala.annotation.tailrec
+import scala.annotation.tailrec
-sealed abstract class Rexp
-
+abstract class Rexp
case object NULL extends Rexp
case object EMPTY extends Rexp
case class CHAR(c: Char) 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 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
+case class CRANGE(cs: String) extends Rexp
+case class PLUS(r: Rexp) extends Rexp
-def charlist2rexp(s : List[Char]) : Rexp = s match {
+abstract class Val
+case object Empty extends Val
+case class Chr(c: Char) extends Val
+case class Seq(v1: Val, v2: 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(x: String, v: Val) extends Val
+
+// some convenience for typing in regular expressions
+def charlist2rexp(s : List[Char]): Rexp = s match {
case Nil => EMPTY
case c::Nil => CHAR(c)
case c::s => SEQ(CHAR(c), charlist2rexp(s))
}
implicit def string2rexp(s : String) : Rexp = charlist2rexp(s.toList)
-
implicit def RexpOps(r: Rexp) = new {
def | (s: Rexp) = ALT(r, s)
def % = STAR(r)
@@ -32,58 +42,11 @@
def % = STAR(s)
def ~ (r: Rexp) = SEQ(s, r)
def ~ (r: String) = SEQ(s, r)
+ def $ (r: Rexp) = RECD(s, r)
}
-def Range(s : List[Char]) : Rexp = s match {
- case Nil => NULL
- case c::Nil => CHAR(c)
- case c::s => ALT(CHAR(c), Range(s))
-}
-def RANGE(s: String) = Range(s.toList)
-
-def PLUS(r: Rexp) = SEQ(r, STAR(r))
-
-val SYM = RANGE("ABCDEFGHIJKLMNOPQRSTUVXYZabcdefghijklmnopqrstuvwxyz_")
-val DIGIT = RANGE("0123456789")
-val ID = SYM ~ (SYM | DIGIT).%
-val NUM = PLUS(DIGIT)
-val KEYWORD : Rexp = "skip" | "while" | "do" | "if" | "then" | "else" | "read" | "write"
-val SEMI: Rexp = ";"
-val OP: Rexp = ":=" | "=" | "-" | "+" | "*" | "!=" | "<" | ">"
-val WHITESPACE = PLUS(RANGE(" \n"))
-val RPAREN: Rexp = ")"
-val LPAREN: Rexp = "("
-val BEGIN: Rexp = "{"
-val END: Rexp = "}"
-
-abstract class Token
-case object T_WHITESPACE extends Token
-case object T_SEMI extends Token
-case object T_LPAREN extends Token
-case object T_RPAREN extends Token
-case object T_BEGIN extends Token
-case object T_END extends Token
-case class T_ID(s: String) extends Token
-case class T_OP(s: String) extends Token
-case class T_NUM(s: String) extends Token
-case class T_KWD(s: String) extends Token
-case class T_ERR(s: String) extends Token // special error token
-
-type TokenFun = String => Token
-type LexRules = List[(Rexp, TokenFun)]
-val lexing_rules: LexRules =
- List((KEYWORD, (s) => T_KWD(s)),
- (ID, (s) => T_ID(s)),
- (OP, (s) => T_OP(s)),
- (NUM, (s) => T_NUM(s)),
- (SEMI, (s) => T_SEMI),
- (LPAREN, (s) => T_LPAREN),
- (RPAREN, (s) => T_RPAREN),
- (BEGIN, (s) => T_BEGIN),
- (END, (s) => T_END),
- (WHITESPACE, (s) => T_WHITESPACE))
-
-
+// nullable function: tests whether the regular
+// expression can recognise the empty string
def nullable (r: Rexp) : Boolean = r match {
case NULL => false
case EMPTY => true
@@ -91,77 +54,239 @@
case ALT(r1, r2) => nullable(r1) || nullable(r2)
case SEQ(r1, r2) => nullable(r1) && nullable(r2)
case STAR(_) => true
+ case RECD(_, r) => nullable(r)
+ case CRANGE(_) => false
+ case PLUS(r) => nullable(r)
}
-def zeroable (r: Rexp) : Boolean = r match {
- case NULL => true
- case EMPTY => false
- case CHAR(_) => false
- case ALT(r1, r2) => zeroable(r1) && zeroable(r2)
- case SEQ(r1, r2) => zeroable(r1) || zeroable(r2)
- case STAR(_) => false
-}
-
+// derivative of a regular expression w.r.t. a character
def der (c: Char, r: Rexp) : Rexp = r match {
case NULL => NULL
- case EMPTY => NULL
+ case EMPTY => NULL
case CHAR(d) => if (c == d) EMPTY else NULL
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)
+ case CRANGE(cs) => if (cs.contains(c)) EMPTY else NULL
+ case PLUS(r) => SEQ(der(c, r), STAR(r))
+}
+
+// derivative w.r.t. a string (iterates der)
+def ders (s: List[Char], r: Rexp) : Rexp = s match {
+ case Nil => r
+ case c::s => ders(s, der(c, r))
+}
+
+// extracts a string from value
+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 Seq(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 Seq(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 EMPTY => Empty
+ case ALT(r1, r2) =>
+ if (nullable(r1)) Left(mkeps(r1)) else Right(mkeps(r2))
+ case SEQ(r1, r2) => Seq(mkeps(r1), mkeps(r2))
+ case STAR(r) => Stars(Nil)
+ case RECD(x, r) => Rec(x, mkeps(r))
+ case PLUS(r) => Stars(List(mkeps(r)))
}
+def inj(r: Rexp, c: Char, v: Val) : Val = (r, v) match {
+ case (STAR(r), Seq(v1, Stars(vs))) => Stars(inj(r, c, v1)::vs)
+ case (SEQ(r1, r2), Seq(v1, v2)) => Seq(inj(r1, c, v1), v2)
+ case (SEQ(r1, r2), Left(Seq(v1, v2))) => Seq(inj(r1, c, v1), v2)
+ case (SEQ(r1, r2), Right(v2)) => Seq(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(_), Empty) => Chr(c)
+ case (CRANGE(_), Empty) => Chr(c)
+ case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))
+ case (PLUS(r), Seq(v1, Stars(vs))) => Stars(inj(r, c, v1)::vs)
+}
-// calculates derivatives until all of them are zeroable
-@tailrec
-def munch(s: List[Char],
- pos: Int,
- rs: LexRules,
- last: Option[(Int, TokenFun)]): Option[(Int, TokenFun)] = rs match {
- case Nil => last
- case rs if (s.length <= pos) => last
- case rs => {
- val ders = rs.map({case (r, tf) => (der(s(pos), r), tf)})
- val rs_nzero = ders.filterNot({case (r, _) => zeroable(r)})
- val rs_nulls = ders.filter({case (r, _) => nullable(r)})
- val new_last = if (rs_nulls != Nil) Some((pos, rs_nulls.head._2)) else last
- munch(s, 1 + pos, rs_nzero, new_last)
+// main lexing function (produces a value)
+def lex(r: Rexp, s: List[Char]) : Val = s match {
+ 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(("ab" | "ab") ~ ("b" | EMPTY), "ab")
+
+// 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 Seq(v1, v2) => Seq(f1(v1), f2(v2))
+}
+def F_SEQ_Empty1(f1: Val => Val, f2: Val => Val) =
+ (v:Val) => Seq(f1(Empty), f2(v))
+def F_SEQ_Empty2(f1: Val => Val, f2: Val => Val) =
+ (v:Val) => Seq(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")
+
+// 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 (NULL, _) => (r2s, F_RIGHT(f2s))
+ case (_, NULL) => (r1s, F_LEFT(f1s))
+ case _ => if (r1s == r2s) (r1s, F_LEFT(f1s))
+ else (ALT (r1s, r2s), F_ALT(f1s, f2s))
+ }
+ }
+ case SEQ(r1, r2) => {
+ val (r1s, f1s) = simp(r1)
+ val (r2s, f2s) = simp(r2)
+ (r1s, r2s) match {
+ case (NULL, _) => (NULL, F_ERROR)
+ case (_, NULL) => (NULL, F_ERROR)
+ case (EMPTY, _) => (r2s, F_SEQ_Empty1(f1s, f2s))
+ case (_, EMPTY) => (r1s, F_SEQ_Empty2(f1s, f2s))
+ 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 c::cs => {
+ val (r_simp, f_simp) = simp(der(c, r))
+ inj(r, c, f_simp(lex_simp(r_simp, cs)))
}
}
-// iterates the munching function and returns a Token list
-def tokenize(s: String, rs: LexRules) : List[Token] = munch(s.toList, 0, rs, None) match {
- case None if (s == "") => Nil
- case None => List(T_ERR("Lexing error: $s"))
- case Some((n, tf)) => {
- val (head, tail) = s.splitAt(n + 1)
- tf(head)::tokenize(tail, rs)
- }
+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
+
+val SYM = CRANGE("abcdefghijklmnopqrstuvwxyz")
+val DIGIT = CRANGE("0123456789")
+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.% ~ "\""
+
+
+val WHILE_REGS = (("k" $ KEYWORD) |
+ ("i" $ ID) |
+ ("o" $ OP) |
+ ("n" $ NUM) |
+ ("s" $ SEMI) |
+ ("str" $ STRING) |
+ ("p" $ (LPAREN | RPAREN)) |
+ ("b" $ (BEGIN | END)) |
+ ("w" $ WHITESPACE)).%
+
+// filters out all white spaces
+def tokenise(r: Rexp, s: String) =
+ env(lexing_simp(r, s)).filterNot { (s) => s._1 == "w"}.mkString("\n")
+
+
+// Testing
+//============
+
+def time[T](code: => T) = {
+ val start = System.nanoTime()
+ val result = code
+ val end = System.nanoTime()
+ println((end - start)/1.0e9)
+ result
}
-val test_prog = """
-start := XXX;
-x := start;
-y := start;
-z := start;
-while 0 < x do {
- while 0 < y do {
- while 0 < z do {
- z := z - 1
- };
- z := start;
- y := y - 1
- };
- y := start;
- x := x - 1
+val r1 = ("a" | "ab") ~ ("bcd" | "c")
+println(lexing(r1, "abcd"))
+
+val r2 = ("" | "a") ~ ("ab" | "b")
+println(lexing(r2, "ab"))
+
+
+// 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)))
+
+
+// Big 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 x;
-write y;
-write z
+write "result";
+write minus2
"""
-println(tokenize(test_prog, lexing_rules).mkString("\n"))
+println("Tokens")
+println(tokenise(WHILE_REGS, prog2))
+for (i <- 1 to 120 by 10) {
+ print(i.toString + ": ")
+ time(lexing_simp(WHILE_REGS, prog2 * i))
+}
+
+