type rexp = NULL | EMPTY | CHAR of char | ALT of rexp * rexp | SEQ of rexp * rexp | STAR of rexp | RECD of string * rexp;;type value = Void | Chr of char | Sequ of value * value | Left of value | Right of value | Stars of value list | Rec of string * value;;(* some helper functions for strings *) let explode s = [for c in s -> c];;let string_repeat s n = String.replicate n s;;(* some helper functions for rexps *)let rec seq s = match s with | [] -> EMPTY | [c] -> CHAR(c) | c::cs -> SEQ(CHAR(c), seq cs);;let chr c = CHAR(c)let str s = seq(explode s);;let plus r = SEQ(r, STAR(r));;let (++) r1 r2 = ALT(r1, r2);;let (--) r1 r2 = SEQ(r1, r2);;let ($) x r = RECD(x, r);;let alts rs = match rs with | [] -> NULL | [r] -> r | r::rs -> List.fold (++) r rs;;(* size of a regular expressions - for testing purposes *)let rec size r = match r with | NULL -> 1 | EMPTY -> 1 | CHAR(_) -> 1 | ALT(r1, r2) -> 1 + (size r1) + (size r2) | SEQ(r1, r2) -> 1 + (size r1) + (size r2) | STAR(r) -> 1 + (size r) | RECD(_, r) -> 1 + (size r);;(* nullable function: tests whether the regular expression can recognise the empty string *)let rec nullable r = match r with | NULL -> false | EMPTY -> true | CHAR(_) -> false | ALT(r1, r2) -> nullable(r1) || nullable(r2) | SEQ(r1, r2) -> nullable(r1) && nullable(r2) | STAR(_) -> true | RECD(_, r) -> nullable(r);;(* derivative of a regular expression r w.r.t. a character c *)let rec der c r = match r with | NULL -> NULL | EMPTY -> NULL | CHAR(d) -> if c = d then EMPTY else NULL | ALT(r1, r2) -> ALT(der c r1, der c r2) | SEQ(r1, r2) -> if nullable r1 then ALT(SEQ(der c r1, r2), der c r2) else SEQ(der c r1, r2) | STAR(r) -> SEQ(der c r, STAR(r)) | RECD(_, r) -> der c r;;(* derivative w.r.t. a list of chars (iterates der) *)let rec ders s r = match s with | [] -> r | c::s -> ders s (der c r);;(* extracts a string from value *)let rec flatten v = match v with | Void -> "" | Chr(c) -> System.Convert.ToString(c) | Left(v) -> flatten v | Right(v) -> flatten v | Sequ(v1, v2) -> flatten v1 ^ flatten v2 | Stars(vs) -> String.concat "" (List.map flatten vs) | Rec(_, v) -> flatten v;;(* extracts an environment from a value *)let rec env v = match v with | Void -> [] | Chr(c) -> [] | Left(v) -> env v | Right(v) -> env v | Sequ(v1, v2) -> env v1 @ env v2 | Stars(vs) -> List.fold (@) [] (List.map env vs) | Rec(x, v) -> (x, flatten v) :: env v;;let string_of_pair (x, s) = "(" ^ x ^ "," ^ s ^ ")";;let string_of_env xs = String.concat "," (List.map string_of_pair xs);;(* the value for a nullable rexp *)let rec mkeps r = match r with | EMPTY -> Void | ALT(r1, r2) -> if nullable r1 then Left(mkeps r1) else Right(mkeps r2) | SEQ(r1, r2) -> Sequ(mkeps r1, mkeps r2) | STAR(r) -> Stars([]) | RECD(x, r) -> Rec(x, mkeps r);;(* injection of a char into a value *)let rec inj r c v = match r, v with | STAR(r), Sequ(v1, Stars(vs)) -> Stars(inj r c v1 :: vs) | SEQ(r1, r2), Sequ(v1, v2) -> Sequ(inj r1 c v1, v2) | SEQ(r1, r2), Left(Sequ(v1, v2)) -> Sequ(inj r1 c v1, v2) | SEQ(r1, r2), Right(v2) -> Sequ(mkeps r1, inj r2 c v2) | ALT(r1, r2), Left(v1) -> Left(inj r1 c v1) | ALT(r1, r2), Right(v2) -> Right(inj r2 c v2) | CHAR(d), Void -> Chr(d) | RECD(x, r1), _ -> Rec(x, inj r1 c v);;(* some "rectification" functions for simplification *)let f_id v = v;;let f_right f = fun v -> Right(f v);;let f_left f = fun v -> Left(f v);;let f_alt f1 f2 = fun v -> match v with Right(v) -> Right(f2 v) | Left(v) -> Left(f1 v);;let f_seq f1 f2 = fun v -> match v with Sequ(v1, v2) -> Sequ(f1 v1, f2 v2);;let f_seq_Void1 f1 f2 = fun v -> Sequ(f1 Void, f2 v);;let f_seq_Void2 f1 f2 = fun v -> Sequ(f1 v, f2 Void);;let f_rec f = fun v -> match v with Rec(x, v) -> Rec(x, f v);;(* simplification of regular expressions returning also an rectification function; no simplification under STARs *)let rec simp r = match r with ALT(r1, r2) -> let (r1s, f1s) = simp r1 in let (r2s, f2s) = simp r2 in (match r1s, r2s with NULL, _ -> (r2s, f_right f2s) | _, NULL -> (r1s, f_left f1s) | _, _ -> if r1s = r2s then (r1s, f_left f1s) else (ALT (r1s, r2s), f_alt f1s f2s)) | SEQ(r1, r2) -> let (r1s, f1s) = simp r1 in let (r2s, f2s) = simp r2 in (match r1s, r2s with NULL, _ -> (NULL, f_right f2s) | _, NULL -> (NULL, f_left f1s) | EMPTY, _ -> (r2s, f_seq_Void1 f1s f2s) | _, EMPTY -> (r1s, f_seq_Void2 f1s f2s) | _, _ -> (SEQ(r1s, r2s), f_seq f1s f2s)) | RECD(x, r1) -> let (r1s, f1s) = simp r1 in (RECD(x, r1s), f_rec f1s) | r -> (r, f_id);;(* matcher function *)let matcher r s = nullable(ders (explode s) r);;(* lexing function (produces a value) *)exception LexError;;let rec lex r s = match s with [] -> if (nullable r) then mkeps r else raise LexError | c::cs -> inj r c (lex (der c r) cs);;let lexing r s = lex r (explode s);;(* lexing with simplification *)let rec lex_simp r s = match s with [] -> if (nullable r) then mkeps r else raise LexError | c::cs -> let (r_simp, f_simp) = simp (der c r) in inj r c (f_simp (lex_simp r_simp cs));;let lexing_simp r s = lex_simp r (explode s);;(* Lexing rules for a small WHILE language *)let sym = alts (List.map chr (explode "abcdefghijklmnopqrstuvwxyz"));;let digit = alts (List.map chr (explode "0123456789"));;let idents = sym -- STAR(sym ++ digit);;let nums = plus(digit);;let keywords = alts (List.map str ["skip"; "while"; "do"; "if"; "then"; "else"; "read"; "write"; "true"; "false"]);;let semicolon = str ";"let ops = alts (List.map str [":="; "=="; "-"; "+"; "*"; "!="; "<"; ">"; "<="; ">="; "%"; "/"]);;let whitespace = plus(str " " ++ str "\n" ++ str "\t");;let rparen = str ")";;let lparen = str "(";;let begin_paren = str "{";;let end_paren = str "}";;let while_regs = STAR(("k" $ keywords) ++ ("i" $ idents) ++ ("o" $ ops) ++ ("n" $ nums) ++ ("s" $ semicolon) ++ ("p" $ (lparen ++ rparen)) ++ ("b" $ (begin_paren ++ end_paren)) ++ ("w" $ whitespace));;(* Some Tests ============ *)let time f x = let t = System.DateTime.Now in let f_x = f x in (printfn "%O" (System.DateTime.Now - t); f_x);;let prog0 = "read n";;string_of_env (env (lexing while_regs prog0));;let prog1 = "read n; write (n)";;string_of_env (env (lexing_simp while_regs prog1));;let prog2 = "i := 2;max := 100;while i < max do { isprime := 1; j := 2; while (j * j) <= i + 1 do { if i % j == 0 then isprime := 0 else skip; j := j + 1 }; if isprime == 1 then write i else skip; i := i + 1}";;for i = 1 to 100 do printf "%i: " i ; time (lexing_simp while_regs) (string_repeat prog2 i);done;;