I'm using a Discriminating Union to define an algebraic expression, and then implementing a simplification function that performs basic Algebraic Simplification with a recursive Match-Case algorithm. I ran into a roadblock involving nested addition/subtraction/multiplication.
The issue is that the match-case for adding/subtracting/etc two or more nested Expression objects wont simplify all the way down to a single constant when it should. IE:
simplify Add( Add( Const(1), Const(2) ), Add( Const(1), Const(2) ) )
Returns an Expression object containing Add(Const(3), Const(3)) when it should return one containing Const(6)
The following code will make what's happening very clear, for brevity I only included the cases for Addition, since the structure (and the problem) is identical for Subtraction and Multiplication:
// Expression type
type Expression =
| X
| Y
| Const of float
| Neg of Expression
| Add of Expression * Expression
| Sub of Expression * Expression
| Mul of Expression * Expression
let rec simplify expr =
match expr with
| X -> expr
| Y -> expr
| Const(n) -> Const(n)
| Add(X, Const(0.0)) -> X
| Add(Const(0.0), X) -> X
| Add(Const(0.0), Y) -> Y
| Add(Y, Const(0.0)) -> Y
| Add(Const(n1), Const(n2)) -> Const(n1+n2) // PROBLEM_1
| Add(expr1, expr2) -> Add(simplify(expr1), simplify(expr2)) // PROBLEM_2
The issue is that with the way it's currently structured, an input that matches // PROBLEM_2 won't fully recursively simplify to a // PROBLEM_1 case, even if expr1 and expr2 contain only Const values. As mentioned above, it eventually will return an Expression containing -> Add(Const(n2), Const(n2)) instead of actually adding those final two values together as in -> Const(n1+n2)
What can I change about the way this is structured so that Add(expr1, expr2) will return a single Const in the event that that all sub expressions contain and are reducible to Const values, ie: that contain no variables or irreducible expressions?
I think the last line changed to
| Add(expr1, expr2) -> simplify(Add(simplify(expr1), simplify(expr2)))
// ^^^^^^^^^ ^
is the only change needed?
The use of type typedef<typename> as a literal target within a match expression or when supported by an active expression as shown here shows syntax error on the typeof<> portion of the expression. Am I doing something wrong?
(Note that the match is against an instance of Type and not an actual value of that type.)
let (|IsType|_|) (et:Type) t = if et.Equals(t) then Some() else None
let someFunc someType =
match someType with
| IsType (typeof<string>) _ -> "Yes"
| _ -> "No"
Interestingly, within an if..then expression it does work.
let someFunc someType =
if typeof<string> = someType then "Yes"
else "No"
)
This circumstance would appear to be an oversight in the language.
This is a restriction of the F# language syntax - the things that can appear in a pattern (even when it is an active pattern) have to be syntactically patterns. So you can write IsSomething 123 because IsSomething and 123 are valid patterns, but you cannot write IsType typeof<Foo> because typeof<Foo> cannot be parsed as a pattern.
One workaround is to pass the argument as a quotation - I would probably not do this because it looks uglier and it is slower than doing a normal if, but it works:
open Microsoft.FSharp.Quotations
let (|IsType|_|) (et:Expr<'T>) (t:System.Type) =
if t.Equals(typeof<'T>) then Some() else None
let someFunc someType =
match someType with
| IsType <# Unchecked.defaultof<string> #> -> "Yes"
| _ -> "No"
It uses Unchecked.defaultof just to get a quotation of type string - but you could provide any other quotation of the type, so IsType <# "nada" #> would work too.
EDIT: In the F# language specification, the allowed syntax is defined in section 7:
pat-param :=
| const
| long-ident
| [ pat-param ; ... ; pat-param ]
| ( pat-param, ..., pat-param )
| long-ident pat-param
| pat-param : type
| <# expr #>
| <## expr ##>
| null
The recommendation is to use the when class to perform a dynamic test as demonstrated here:
match typeValue with
| t when t = typeof<string> -> ...
| t when t = typeof<int> -> ...
...
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 function, which can returns different types, and I use discriminated union for this. What I need, is to have conversion from one type in discriminated union to another type.
Also some of the types can be convertable to all other types (String), but some of the types can be converted only to String (MyCustomType)
For this I've added member method ConvertTo to the ResultType:
type MyTypes =
| Boolean = 1
| Integer = 2
| Decimal = 3
| Double = 4
| String = 5
| MyCustomType = 6
type ResultType =
| Boolean of bool
| Integer of int
| Decimal of decimal
| Double of double
| String of string
| MyCustomType of MyCustomType
with
member this.ConvertTo(newType: MyTypes) =
match this with
| ResultType.Boolean(value) ->
match newType with
| MyTypes.Boolean ->
this
| MyTypes.Integer ->
ResultType.Integer(if value then 1 else 0)
...
| ResultType.MyCustomType(value) ->
match newType with
| MyTypes.MyCustomType ->
this
| MyTypes.String ->
ResultType.String(value.ToString())
| _ ->
failwithf "Conversion from MyCustomType to %s is not supported" (newType.ToString())
I don't like such construction, because if I add more types, this requires me to do many changes: MyTypes, ResultType and also in several places in the ConvertTo member function.
Can anybody suggest better solution for such types conversion?
Thanks in advance
With a slightly different design, it is possible to exploit System.Convert.ChangeType and the fact that the constructors of discriminated unions are actually functions:
// statically typed wrapper for System.Convert.ChangeType
let conv a : 'T = System.Convert.ChangeType(a, typeof<'T>) :?> 'T
type MyCustomType() = class end
type ResultType =
| Boolean of bool
| Integer of int
| Decimal of decimal
| Double of double
| String of string
| MyCustomType of MyCustomType
with
member this.ConvertTo (newType:'T->ResultType) =
match this with
| Boolean b -> newType( conv b )
| Integer i -> newType( conv i )
| Decimal d -> newType( conv d )
| Double d -> newType( conv d )
| String s -> newType( conv s )
| MyCustomType m ->
if typeof<'T> <> typeof<string> then
raise (new System.InvalidCastException("MyCustomType can only be converted to String"))
else
String (m.ToString())
let i = Integer 42
let b = i.ConvertTo Boolean
printfn "%A" b
let d = i.ConvertTo Decimal
printfn "%A" d
let d2 = i.ConvertTo Double
printfn "%A" d2
let s = i.ConvertTo String
printfn "%A" s
//let mi = i.ConvertTo MyCustomType // throws InvalidCastException
let m = MyCustomType (new MyCustomType())
let sm = m.ConvertTo String
printfn "%A" sm
//let im = m.ConvertTo Integer // throws InvalidCastException
EDIT: Once you add more custom types, this will not help much.
Maybe you should make your custom types implement IConvertible. Then you can remove the special case code from ConvertTo and completely rely on System.Convert.ChangeType.
You would still have to extend every custom type's ToObject implementation whenever you add a new custom type. Whether that really is better than a central ConvertTofunction is debatable.
Why are you wanting to do type conversion to begin with? Discriminated Unions are a good way of hiding type information until you need it and abstract complexity away. Generally you have a match statement in a function that consumes this type and then you only cast if you need to.
If you're trying to make some type of parser or language engine then you have no choice but to define all the cast or at least their error states. If you wouldn't mind elaborating on why / what you would use this for, maybe I could suggest another approach.
An aside: F# and .NET in general doesn't support overloading of return types.
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.