I have this pattern for a compiler, and in the expr parser I want to do this
let (|InfixOperator|_|) tokens = ...
let (|UnaryValue|_|) tokens = ...
let infixExpr tokens =
match tokens with
| Value v::Operator op::tokens -> ...
| Value v::tokens -> ...
The problem is that either the tokens for the match expr have a signature of Token list list or the pattern Value (needs to receive a list since it parses unary operations) does not give back the remaining tokens, the same happens for the Operator pattern
The ugly way around it would be something like this
let (|InfixOperator|_|) tokens = ...
let (|UnaryValue|_|) tokens = ...
let infixExpr tokens =
match tokens with
| Value (v, Operator (op, tokens)) -> ...
| Value (v, tokens) -> ...
Does anyone know any cleaner way of doing this with pattern matching?
I wrote a parser for a subset of BASIC that uses pattern matching (source code is available on GitHub). This is not heavily using active patterns, but it works in two steps - first it tokenizes the input (turning list<char> into list<Token>) and then it parses the list of tokens (turning list<Token> into list<Statement>).
This way, you mostly avoid the issue with nesting, because most interesting things become just a single token. For example, for operators, the tokenization looks like:
let rec tokenize toks = function
(* First character is number, collect all remaining number characters *)
| c::cs when isNumber c -> number toks [c] cs
(* First character is operator, collect all remanining operator characters *)
| c::cs when isOp c -> operator toks [c] cs
(* more cases omitted *)
| [] -> List.rev toks
and number toks acc = function
| c::cs when isNumber c -> number toks (c::acc) cs
| input -> tokenize (Number(float (str acc))::toks) input
and operator toks acc = function
| c::cs when isOp c -> operator toks (c::acc) cs
| input -> tokenize (Operator(List.rev acc)::toks) input
If you now have a list of tokens, then parsing a binary expression becomes:
let rec parseBinary left = function
| (Operator o)::toks ->
let right, toks = parseExpr toks
Binary(o, left, right), toks
| toks -> left, toks
and parseExpr = function
| (Number n)::toks -> parseBinary (Const(NumberValue n)) toks
(* more cases omitted *)
There are some places where I used active patterns, for example to parse a BASIC range expression which can be N1-N2 or -N1 or N1-
let (|Range|_|) = function
| (Number lo)::(Operator ['-'])::(Number hi)::[] ->
Some(Some (int lo), Some (int hi))
| (Operator ['-'])::(Number hi)::[] -> Some(None, Some(int hi))
| (Number lo)::(Operator ['-'])::[] -> Some(Some(int lo), None)
| [] -> Some(None, None)
| _ -> None
So, I guess the answer is that if you want to write a simple parser using pattern matching (without getting into more sophisticated and more powerful monadic parser libraries), then it is quite doable, but it is good idea to separate tokenization from parsing.
Related
I am working on making a Lisp interpreter in OCaml. I naturally started with the front-end. So far I have an S-expression parsing algorithm that works most of the time. For both simple S-expressions like (a b) and ((a b) (c d)) my function, ast_as_str, shows that the output list structure is incorrect. I've documented that below. After trying countless variations on parse nothing seems to work. Does someone adept in writing parsers in OCaml have a suggestion as to how I can fix my code?
type s_expression = Nil | Atom of string | Pair of s_expression * s_expression
let rec parse tokens =
match tokens with
| [] -> Nil
| token :: rest ->
match token with
| "(" -> parse rest
| ")" -> Pair(Nil, parse rest)
| atom -> Pair(Atom atom, parse rest)
let rec ast_as_str ast =
match ast with
| Nil -> "nil"
| Atom a -> Printf.sprintf "%s" a
| Pair(a, b) -> Printf.sprintf "(%s %s)" (ast_as_str a) (ast_as_str b);;
let check_output test = print_endline (ast_as_str (parse test));;
(*
Input:
(a b)
Output:
(a (b (nil nil)))
Almost correct...
*)
check_output ["("; "a"; "b"; ")"];;
(*
Input:
((w x) (y z))
Output:
(w (x (nil (y (z (nil (nil nil)))))))
Incorrect.
*)
check_output ["("; "("; "w"; "x"; ")"; "("; "y"; "z"; ")"; ")"]
I'm going to assume this isn't homework. If it is, I will change the answer to some less specific hints.
A recursive descent parser works by recognizing the beginning token of a construct, then parsing the contents of the construct, then (very often) recognizing the ending token of the construct. S-expressions have just one construct, the parenthesized list. Your parser isn't doing the recognizing of the end of the construct.
If you assume your parser works correctly, then encountering a right parenthesis ) is a syntax error. There shouldn't be any unmatched right parentheses, and matching right parentheses are parsed as part of the parenthesized list construct (as I described above).
If you swear that this is just a personal project I'd be willing to write up a parser. But you should try writing up something as described above.
Note that when you see an atom, you aren't seeing a pair. It's not correct to return Pair (Atom xyz, rest) when seeing an atom.
Update
The way to make things work in a functional setting is to have the parsing functions return not only the construct that they saw, but also the remaining tokens that haven't been parsed yet.
The following code works for your examples and is probably pretty close to correct:
let rec parse tokens =
match tokens with
| [] -> failwith "Syntax error: end of input"
| "(" :: rest ->
(match parselist rest with
| (sexpr, ")" :: rest') -> (sexpr, rest')
| _ -> failwith "Syntax error: unmatched ("
)
| ")" :: _ -> failwith "Syntax error: unmatched )"
| atom :: rest -> (Atom atom, rest)
and parselist tokens =
match tokens with
| [] | ")" :: _ -> (Nil, tokens)
| _ ->
let (sexpr1, rest) = parse tokens in
let (sexpr2, rest') = parselist rest in
(Pair (sexpr1, sexpr2), rest')
You can define check_output like this:
let check_output test =
let (sexpr, toks) = parse test in
if toks <> [] then
Printf.printf "(extra tokens in input)\n";
print_endline (ast_as_str sexpr)
Here's what I see for your two test cases:
# check_output ["("; "a"; "b"; ")"];;
(a (b nil))
- : unit = ()
# check_output ["("; "("; "w"; "x"; ")"; "("; "y"; "z"; ")"; ")"];;
((w (x nil)) ((y (z nil)) nil))
- : unit = ()
I think these are the right results.
sexp is like this: type sexp = Atom of string | List of sexp list, e.g., "((a b) ((c d) e) f)".
I have written a parser to parse a sexp string to the type:
let of_string s =
let len = String.length s in
let empty_buf () = Buffer.create 16 in
let rec parse_atom buf i =
if i >= len then failwith "cannot parse"
else
match s.[i] with
| '(' -> failwith "cannot parse"
| ')' -> Atom (Buffer.contents buf), i-1
| ' ' -> Atom (Buffer.contents buf), i
| c when i = len-1 -> (Buffer.add_char buf c; Atom (Buffer.contents buf), i)
| c -> (Buffer.add_char buf c; parse_atom buf (i+1))
and parse_list acc i =
if i >= len || (i = len-1 && s.[i] <> ')') then failwith "cannot parse"
else
match s.[i] with
| ')' -> List (List.rev acc), i
| '(' ->
let list, j = parse_list [] (i+1) in
parse_list (list::acc) (j+1)
| c ->
let atom, j = parse_atom (empty_buf()) i in
parse_list (atom::acc) (j+1)
in
if s.[0] <> '(' then
let atom, j = parse_atom (empty_buf()) 0 in
if j = len-1 then atom
else failwith "cannot parse"
else
let list, j = parse_list [] 1 in
if j = len-1 then list
else failwith "cannot parse"
But I think it is too verbose and ugly.
Can someone help me with an elegant way to write such a parser?
Actually, I always have problems in writing code of parser and what I could do only is write such a ugly one.
Any tricks for this kind of parsing? How to effectively deal with symbols, such as (, ), that implies recursive parsing?
You can use a lexer+parser discipline to separate the details of lexical syntax (skipping spaces, mostly) from the actual grammar structure. That may seem overkill for such a simple grammar, but it's actually better as soon as the data you parse has the slightest chance of being wrong: you really want error location (and not to implement them yourself).
A technique that is easy and gives short parsers is to use stream parsers (using a Camlp4 extension for them described in the Developping Applications with Objective Caml book); you may even get a lexer for free by using the Genlex module.
If you want to do really do it manually, as in your example above, here is my recommendation to have a nice parser structure. Have mutually recursive parsers, one for each category of your syntax, with the following interface:
parsers take as input the index at which to start parsing
they return a pair of the parsed value and the first index not part of the value
nothing more
Your code does not respect this structure. For example, you parser for atoms will fail if it sees a (. That is not his role and responsibility: it should simply consider that this character is not part of the atom, and return the atom-parsed-so-far, indicating that this position is not in the atom anymore.
Here is a code example in this style for you grammar. I have split the parsers with accumulators in triples (start_foo, parse_foo and finish_foo) to factorize multiple start or return points, but that is only an implementation detail.
I have used a new feature of 4.02 just for fun, match with exception, instead of explicitly testing for the end of the string. It is of course trivial to revert to something less fancy.
Finally, the current parser does not fail if the valid expression ends before the end of the input, it only returns the end of the input on the side. That's helpful for testing but you would do it differently in "production", whatever that means.
let of_string str =
let rec parse i =
match str.[i] with
| exception _ -> failwith "unfinished input"
| ')' -> failwith "extraneous ')'"
| ' ' -> parse (i+1)
| '(' -> start_list (i+1)
| _ -> start_atom i
and start_list i = parse_list [] i
and parse_list acc i =
match str.[i] with
| exception _ -> failwith "unfinished list"
| ')' -> finish_list acc (i+1)
| ' ' -> parse_list acc (i+1)
| _ ->
let elem, j = parse i in
parse_list (elem :: acc) j
and finish_list acc i =
List (List.rev acc), i
and start_atom i = parse_atom (Buffer.create 3) i
and parse_atom acc i =
match str.[i] with
| exception _ -> finish_atom acc i
| ')' | ' ' -> finish_atom acc i
| _ -> parse_atom (Buffer.add_char acc str.[i]; acc) (i + 1)
and finish_atom acc i =
Atom (Buffer.contents acc), i
in
let result, rest = parse 0 in
result, String.sub str rest (String.length str - rest)
Note that it is an error to reach the end of input when parsing a valid expression (you must have read at least one atom or list) or when parsing a list (you must have encountered the closing parenthesis), yet it is valid at the end of an atom.
This parser does not return location information. All real-world parsers should do so, and this is enough of a reason to use a lexer/parser approach (or your preferred monadic parser library) instead of doing it by hand. Returning location information here is not terribly difficult, though, just duplicate the i parameter into the index of the currently parsed character, on one hand, and the first index used for the current AST node, on the other; whenever you produce a result, the location is the pair (first index, last valid index).
Myello! So I am looking for a concise, efficient an idiomatic way in F# to parse a file or a string. I have a strong preference to treat the input as a sequence of char (char seq). The idea is that every function is responsible to parse a piece of the input, return the converted text tupled with the unused input and be called by a higher level function that chains the unused input to the following functions and use the results to build a compound type. Every parsing function should therefore have a signature similar to this one: char seq -> char seq * 'a . If, for example, the function's responsibility is simply to extract the first word, then, one approach would be the following:
let parseFirstWord (text: char seq) =
let rec forTailRecursion t acc =
let c = Seq.head t
if c = '\n' then
(t, acc)
else
forTailRecursion (Seq.skip 1 t) (c::acc)
let rest, reversedWord = forTailRecursion text []
(rest, List.reverse reversedWord)
Now, of course the main problem with this approach is that it extracts the word in reverse order and so you have to reverse it. Its main advantages however are that is uses strictly functional features and proper tail recursion. One could avoid the reversing of the extracted value while losing tail recursion:
let rec parseFirstWord (text: char seq) =
let c = Seq.head t
if c = '\n' then
(t, [])
else
let rest, tail = parseFirstWord (Seq.skip 1 t)
(rest, (c::tail))
Or use a fast mutable data structure underneath instead of using purely functional features, such as:
let parseFirstWord (text: char seq) =
let rec forTailRecursion t queue =
let c = Seq.head t
if c = '\n' then
(t, queue)
else
forTailRecursion (Seq.skip 1 t) (queue.Enqueu(c))
forTailRecursion text (new Queue<char>())
I have no idea how to use OO concepts in F# mind you so corrections to the above code are welcome.
Being new to this language, I would like to be guided in terms of the usual compromises that an F# developer makes. Among the suggested approaches and your own, which should I consider more idiomatic and why? Also, in that particular case, how would you encapsulate the return value: char seq * char seq, char seq * char list or evenchar seq * Queue<char>? Or would you even consider char seq * String following a proper conversion?
I would definitely have a look at FSLex. FSYacc, FParsec. However if you just want to tokenize a seq<char> you can use a sequence expression to generate tokens in the right order. Reusing your idea of a recursive inner function, and combinining with a sequence expression, we can stay tail recursive like shown below, and avoid non-idiomatic tools like mutable data structures.
I changed the separator char for easy debugging and the signature of the function. This version produces a seq<string> (your tokens) as result, which is probably easier to consume than a tuple with the current token and the rest of the text. If you just want the first token, you can just take the head. Note that the sequence is generated 'on demand', i.e. the input is parsed only as tokens are consumed through the sequence. Should you need the remainder of the input text next to each token, you can yield a pair in loop instead, but I'm guessing the downstream consumer most likely wouldn't (furthermore, if the input text is itself a lazy sequence, possibly linked to a stream, we don't want to expose it as it should be iterated through only in one place).
let parse (text : char seq) =
let rec loop t acc =
seq {
if Seq.isEmpty t then yield acc
else
let c, rest = Seq.head t, Seq.skip 1 t
if c = ' ' then
yield acc
yield! loop rest ""
else yield! loop rest (acc + string c)
}
loop text ""
parse "The FOX is mine"
val it : seq<string> = seq ["The"; "FOX"; "is"; "mine"]
This is not the only 'idiomatic' way of doing this in F#. Every time we need to process a sequence, we can look at the functions made available in the Seq module. The most general of these is fold which iterates through a sequence once, accumulating a state at each element by running a given function. In the example below accumulate is such a function, that progressively builds the resulting sequence of tokens. Since Seq.fold doesn't run the accumulator function on an empty sequence, we need the last two lines to extract the last token from the function's internal accumulator.
This second implementation keeps the nice characteriestics of the first, i.e. tail recursion (inside the fold implementation, if I'm not mistaken) and processing of the input sequence on demand. It also happens to be shorter, albeit a bit less readable probably.
let parse2 (text : char seq) =
let accumulate (res, acc) c =
if c = ' ' then (Seq.append res (Seq.singleton acc), "")
else (res, acc + string c)
let (acc, last) = text |> Seq.fold accumulate (Seq.empty, "")
Seq.append acc (Seq.singleton last)
parse2 "The FOX is mine"
val it : seq<string> = seq ["The"; "FOX"; "is"; "mine"]
One way of lexing/parsing in a way truly unique to F# is by using active patterns. The following simplified example shows the general idea. It can process a calculation string of arbitrary length without producing a stack overflow.
let rec (|CharOf|_|) set = function
| c :: rest when Set.contains c set -> Some(c, rest)
| ' ' :: CharOf set (c, rest) -> Some(c, rest)
| _ -> None
let rec (|CharsOf|) set = function
| CharOf set (c, CharsOf set (cs, rest)) -> c::cs, rest
| rest -> [], rest
let (|StringOf|_|) set = function
| CharsOf set (_::_ as cs, rest) -> Some(System.String(Array.ofList cs), rest)
| _ -> None
type Token =
| Int of int
| Add | Sub | Mul | Div | Mod
| Unknown
let lex: string -> _ =
let digits = set ['0'..'9']
let ops = Set.ofSeq "+-*/%"
let rec lex chars =
seq { match chars with
| StringOf digits (s, rest) -> yield Int(int s); yield! lex rest
| CharOf ops (c, rest) ->
let op =
match c with
| '+' -> Add | '-' -> Sub | '*' -> Mul | '/' -> Div | '%' -> Mod
| _ -> failwith "invalid operator char"
yield op; yield! lex rest
| [] -> ()
| _ -> yield Unknown }
List.ofSeq >> lex
lex "1234 + 514 / 500"
// seq [Int 1234; Add; Int 514; Div; Int 500]
I have been searching around the interwebs for a couple of days, trying to get an answer to my questions and i'm finally admitting defeat.
I have been given a grammar:
Dig ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
Int ::= Dig | Dig Int
Var ::= a | b | ... z | A | B | C | ... | Z
Expr ::= Int | - Expr | + Expr Expr | * Expr Expr | Var | let Var = Expr in Expr
And i have been told to parse, evaluate and print expressions using this grammar
where the operators * + - has their normal meaning
The specific task is to write a function parse :: String -> AST
that takes a string as input and returns an abstract syntax tree when the input is in the correct format (which i can asume it is).
I am told that i might need a suitable data type and that data type might need to derive from some other classes.
Following an example output
data AST = Leaf Int | Sum AST AST | Min AST | ...
Further more, i should consider writing a function
tokens::String -> [String]
to split the input string into a list of tokens
Parsing should be accomplished with
ast::[String] -> (AST,[String])
where the input is a list of tokens and it outputs an AST, and to parse sub-expressions i should simply use the ast function recursively.
I should also make a printExpr method to print the result so that
printE: AST -> String
printE(parse "* 5 5") yields either "5*5" or "(5*5)"
and also a function to evaluate the expression
evali :: AST -> Int
I would just like to be pointed in the right direction of where i might start. I have little knowledge of Haskell and FP in general and trying to solve this task i made some string handling function out of Java which made me realize that i'm way off track.
So a little pointer in the right direction, and maybe an explantion to 'how' the AST should look like
Third day in a row and still no running code, i really appreciate any attempt to help me find a solution!
Thanks in advance!
Edit
I might have been unclear:
I'm wondering how i should go about from having read and tokenized an input string to making an AST.
Parsing tokens into an Abstract Syntax Tree
OK, let's take your grammar
Dig ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
Int ::= Dig | Dig Int
Var ::= a | b | ... z | A | B | C | ... | Z
Expr ::= Int | - Expr | + Expr Expr | * Expr Expr | Var | let Var = Expr in Expr
This is a nice easy grammar, because you can tell from the first token what sort of epression it will be.
(If there was something more complicated, like + coming between numbers, or - being used for subtraction as
well as negation, you'd need the list-of-successes trick, explained in
Functional Parsers.)
Let's have some sample raw input:
rawinput = "- 6 + 45 let x = - 5 in * x x"
Which I understand from the grammar represents "(- 6 (+ 45 (let x=-5 in (* x x))))",
and I'll assume you tokenised it as
tokenised_input' = ["-","6","+","4","5","let","x","=","-","5","in","*","x","x"]
which fits the grammar, but you might well have got
tokenised_input = ["-","6","+","45","let","x","=","-","5","in","*","x","x"]
which fits your sample AST better. I think it's good practice to name your AST after bits of your grammar,
so I'm going to go ahead and replace
data AST = Leaf Int | Sum AST AST | Min AST | ...
with
data Expr = E_Int Int | E_Neg Expr | E_Sum Expr Expr | E_Prod Expr Expr | E_Var Char
| E_Let {letvar::Char,letequal:: Expr,letin::Expr}
deriving Show
I've named the bits of an E_Let to make it clearer what they represent.
Writing a parsing function
You could use isDigit by adding import Data.Char (isDigit) to help out:
expr :: [String] -> (Expr,[String])
expr [] = error "unexpected end of input"
expr (s:ss) | all isDigit s = (E_Int (read s),ss)
| s == "-" = let (e,ss') = expr ss in (E_Neg e,ss')
| s == "+" = (E_Sum e e',ss'') where
(e,ss') = expr ss
(e',ss'') = expr ss'
-- more cases
Yikes! Too many let clauses obscuring the meaning,
and we'll be writing the same code for E_Prod and very much worse for E_Let.
Let's get this sorted out!
The standard way of dealing with this is to write some combinators;
instead of tiresomely threading the input [String]s through our definition, define ways to
mess with the output of parsers (map) and combine
multiple parsers into one (lift).
Clean up the code 1: map
First we should define pmap, our own equivalent of the map function so we can do pmap E_Neg (expr1 ss)
instead of let (e,ss') = expr1 ss in (E_Neg e,ss')
pmap :: (a -> b) -> ([String] -> (a,[String])) -> ([String] -> (b,[String]))
nonono, I can't even read that! We need a type synonym:
type Parser a = [String] -> (a,[String])
pmap :: (a -> b) -> Parser a -> Parser b
pmap f p = \ss -> let (a,ss') = p ss
in (f a,ss')
But really this would be better if I did
data Parser a = Par [String] -> (a,[String])
so I could do
instance Functor Parser where
fmap f (Par p) = Par (pmap f p)
I'll leave that for you to figure out if you fancy.
Clean up the code 2: combining two parsers
We also need to deal with the situation when we have two parsers to run,
and we want to combine their results using a function. This is called lifting the function to parsers.
liftP2 :: (a -> b -> c) -> Parser a -> Parser b -> Parser c
liftP2 f p1 p2 = \ss0 -> let
(a,ss1) = p1 ss0
(b,ss2) = p2 ss1
in (f a b,ss2)
or maybe even three parsers:
liftP3 :: (a -> b -> c -> d) -> Parser a -> Parser b -> Parser c -> Parser d
I'll let you think how to do that.
In the let statement you'll need liftP5 to parse the sections of a let statement,
lifting a function that ignores the "=" and "in". You could make
equals_ :: Parser ()
equals_ [] = error "equals_: expected = but got end of input"
equals_ ("=":ss) = ((),ss)
equals_ (s:ss) = error $ "equals_: expected = but got "++s
and a couple more to help out with this.
Actually, pmap could also be called liftP1, but map is the traditional name for that sort of thing.
Rewritten with the nice combinators
Now we're ready to clean up expr:
expr :: [String] -> (Expr,[String])
expr [] = error "unexpected end of input"
expr (s:ss) | all isDigit s = (E_Int (read s),ss)
| s == "-" = pmap E_Neg expr ss
| s == "+" = liftP2 E_Sum expr expr ss
-- more cases
That'd all work fine. Really, it's OK. But liftP5 is going to be a bit long, and feels messy.
Taking the cleanup further - the ultra-nice Applicative way
Applicative Functors is the way to go.
Remember I suggested refactoring as
data Parser a = Par [String] -> (a,[String])
so you could make it an instance of Functor? Perhaps you don't want to,
because all you've gained is a new name fmap for the perfectly working pmap and
you have to deal with all those Par constructors cluttering up your code.
Perhaps this will make you reconsider, though; we can import Control.Applicative,
then using the data declaration, we can
define <*>, which sort-of means then and use <$> instead of pmap, with *> meaning
<*>-but-forget-the-result-of-the-left-hand-side so you would write
expr (s:ss) | s == "let" = E_Let <$> var *> equals_ <*> expr <*> in_ *> expr
Which looks a lot like your grammar definition, so it's easy to write code that works first time.
This is how I like to write Parsers. In fact, it's how I like to write an awful lot of things.
You'd only have to define fmap, <*> and pure, all simple ones, and no long repetative liftP3, liftP4 etc.
Read up about Applicative Functors. They're great.
Applicative in Learn You a Haskell for Great Good!
Applicative in Haskell wikibook
Hints for making Parser applicative: pure doesn't change the list.
<*> is like liftP2, but the function doesn't come from outside, it comes as the output from p1.
To make a start with Haskell itself, I'd recommend Learn You a Haskell for Great Good!, a very well-written and entertaining guide. Real World Haskell is another oft-recommended starting point.
Edit: A more fundamental introduction to parsing is Functional Parsers. I wanted How to replace a failure by a list of successes by PHilip Wadler. Sadly, it doesn't seem to be available online.
To make a start with parsing in Haskell, I think you should first read monadic parsing in Haskell, then maybe this wikibook example, but also then the parsec guide.
Your grammar
Dig ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
Int ::= Dig | Dig Int
Var ::= a | b | ... z | A | B | C | ... | Z
Expr ::= Int | - Expr | + Expr Expr | * Expr Expr | Var | let Var = Expr in Expr
suggests a few abstract data types:
data Dig = Dig_0 | Dig_1 | Dig_2 | Dig_3 | Dig_4 | Dig_5 | Dig_6 | Dig_7 | Dig_8 | Dig_9
data Integ = I_Dig Dig | I_DigInt Dig Integ
data Var = Var_a | Var_b | ... Var_z | Var_A | Var_B | Var_C | ... | Var_Z
data Expr = Expr_I Integ
| Expr_Neg Expr
| Expr_Plus Expr Expr
| Expr_Times Expr Expr Var
| Expr_Var Var
| Expr_let Var Expr Expr
This is inherently a recursively defined syntax tree, no need to make another one.
Sorry about the clunky Dig_ and Integ_ stuff - they have to start with an uppercase.
(Personally I'd want to be converting the Integs to Ints straight away, so would have done newtype Integ = Integ Int, and would probably have done newtype Var = Var Char but that might not suit you.)
Once you've done with the basic ones - dig and var, and neg_, plus_, in_ etcI'd go with the Applicative interface to build them up, so for example your parser expr for Expr would be something like
expr = Expr_I <$> integ
<|> Expr_Neg <$> neg_ *> expr
<|> Expr_Plus <$> plus_ *> expr <*> expr
<|> Expr_Times <$> times_ *> expr <*> expr
<|> Expr_Var <$> var
<|> Expr_let <$> let_ *> var <*> equals_ *> expr <*> in_ *> expr
So nearly all the time, your Haskell code is clean and closely resembles the grammar you were given.
OK, so it seems you're trying to build lots and lots of stuff, and you're not really sure exactly where it's all going. I would suggest that getting the definition for AST right, and then trying to implement evali would be a good start.
The grammer you've listed is interesting... You seem to want to input * 5 5, but output 5*5, whic is an odd choice. Is that really supposed to be a unary minus, not binary? Simlarly, * Expr Expr Var looks like perhaps you might have meant to type * Expr Expr | Var...
Anyway, making some assumptions about what you meant to say, your AST is going to look something like this:
data AST = Leaf Int | Sum AST AST | Minus AST | Var String | Let String AST AST
Now, let us try to do printE. It takes an AST and gives us a string. By the definition above, the AST has to be one of five possible things. You just need to figure out what to print for each one!
printE :: AST -> String
printE (Leaf x ) = show x
printE (Sum x y) = printE x ++ " + " ++ printE y
printE (Minus x ) = "-" ++ printE x
...
show turns an Int into a String. ++ joins two strings together. I'll let you work out the rest of the function. (The tricky thing is if you want it to print brackets to correctly show the order of subexpressions... Since your grammer doesn't mention brackets, I guess no.)
Now, how about evali? Well, this is going to be a similar deal. If the AST is a Leaf x, then x is an Int, and you just return that. If you have, say, Minus x, then x isn't an integer, it's an AST, so you need to turn it into an integer with evali. The function looks something like
evali :: AST -> Int
evali (Leaf x ) = x
evali (Sum x y) = (evali x) + (evali y)
evali (Minus x ) = 0 - (evali x)
...
That works great so far. But wait! It looks like you're supposed to be able to use Let to define new variables, and refer to them later with Var. Well, in that case, you need to store those variables somewhere. And that's going to make the function more complicated.
My recommendation would be to use Data.Map to store a list of variable names and their corresponding values. You'll need to add the variable map to a type signature. You can do it this way:
evali :: AST -> Int
evali ast = evaluate Data.Map.empty ast
evaluate :: Map String Int -> AST -> Int
evaluate m ast =
case ast of
...same as before...
Let var ast1 ast2 -> evaluate (Data.Map.insert var (evaluate m ast1)) ast2
Var var -> m ! var
So evali now just calls evaluate with an empty variable map. When evaluate sees Let, it adds the variable to the map. And when it sees Var, it looks up the name in the map.
As for parsing a string into an AST in the first place, that is an entire other answer again...
I have a task to write a (toy) parser for a (toy) grammar using OCaml and not sure how to start (and proceed with) this problem.
Here's a sample Awk grammar:
type ('nonterm, 'term) symbol = N of 'nonterm | T of 'term;;
type awksub_nonterminals = Expr | Term | Lvalue | Incrop | Binop | Num;;
let awksub_grammar =
(Expr,
function
| Expr ->
[[N Term; N Binop; N Expr];
[N Term]]
| Term ->
[[N Num];
[N Lvalue];
[N Incrop; N Lvalue];
[N Lvalue; N Incrop];
[T"("; N Expr; T")"]]
| Lvalue ->
[[T"$"; N Expr]]
| Incrop ->
[[T"++"];
[T"--"]]
| Binop ->
[[T"+"];
[T"-"]]
| Num ->
[[T"0"]; [T"1"]; [T"2"]; [T"3"]; [T"4"];
[T"5"]; [T"6"]; [T"7"]; [T"8"]; [T"9"]]);;
And here's some fragments to parse:
let frag1 = ["4"; "+"; "3"];;
let frag2 = ["9"; "+"; "$"; "1"; "+"];;
What I'm looking for is a rulelist that is the result of the parsing a fragment, such as this one for frag1 ["4"; "+"; "3"]:
[(Expr, [N Term; N Binop; N Expr]);
(Term, [N Num]);
(Num, [T "3"]);
(Binop, [T "+"]);
(Expr, [N Term]);
(Term, [N Num]);
(Num, [T "4"])]
The restriction is to not use any OCaml libraries other than List... :/
Here is a rough sketch - straightforwardly descend into the grammar and try each branch in order. Possible optimization : tail recursion for single non-terminal in a branch.
exception Backtrack
let parse l =
let rules = snd awksub_grammar in
let rec descend gram l =
let rec loop = function
| [] -> raise Backtrack
| x::xs -> try attempt x l with Backtrack -> loop xs
in
loop (rules gram)
and attempt branch (path,tokens) =
match branch, tokens with
| T x :: branch' , h::tokens' when h = x ->
attempt branch' ((T x :: path),tokens')
| N n :: branch' , _ ->
let (path',tokens) = descend n ((N n :: path),tokens) in
attempt branch' (path', tokens)
| [], _ -> path,tokens
| _, _ -> raise Backtrack
in
let (path,tail) = descend (fst awksub_grammar) ([],l) in
tail, List.rev path
Ok, so the first think you should do is write a lexical analyser. That's the
function that takes the ‘raw’ input, like ["3"; "-"; "("; "4"; "+"; "2"; ")"],
and splits it into a list of tokens (that is, representations of terminal symbols).
You can define a token to be
type token =
| TokInt of int (* an integer *)
| TokBinOp of binop (* a binary operator *)
| TokOParen (* an opening parenthesis *)
| TokCParen (* a closing parenthesis *)
and binop = Plus | Minus
The type of the lexer function would be string list -> token list and the ouput of
lexer ["3"; "-"; "("; "4"; "+"; "2"; ")"]
would be something like
[ TokInt 3; TokBinOp Minus; TokOParen; TokInt 4;
TBinOp Plus; TokInt 2; TokCParen ]
This will make the job of writing the parser easier, because you won't have to
worry about recognising what is a integer, what is an operator, etc.
This is a first, not too difficult step because the tokens are already separated.
All the lexer has to do is identify them.
When this is done, you can write a more realistic lexical analyser, of type string -> token list, that takes a actual raw input, such as "3-(4+2)" and turns it into a token list.
I'm not sure if you specifically require the derivation tree, or if this is a just a first step in parsing. I'm assuming the latter.
You could start by defining the structure of the resulting abstract syntax tree by defining types. It could be something like this:
type expr =
| Operation of term * binop * term
| Term of term
and term =
| Num of num
| Lvalue of expr
| Incrop of incrop * expression
and incrop = Incr | Decr
and binop = Plus | Minus
and num = int
Then I'd implement a recursive descent parser. Of course it would be much nicer if you could use streams combined with the preprocessor camlp4of...
By the way, there's a small example about arithmetic expressions in the OCaml documentation here.