How can I edit the following code so that I don't have so many evaluators (which right now reflect various result types)? I want a single evaluator so that I don't know in advance the type of the result. I want one evaluator for the whole language. Do I need to add a value type for Exp to do this? How would this look like? What kind of value type and how would I edit the current eval functions in order to reflect this new polymorphic type?
{-# LANGUAGE TypeSynonymInstances, FlexibleInstances #-}
data Exp = V Var
| B Bool
| MyInt Int
| And Exp Exp
| Or Exp Exp
| Not Exp
| Mult Exp Exp
| UnaryNeg Exp
| LEQ Exp Exp
| LESST Exp Exp
| Add Exp Exp
| POLYEQ Exp Exp
data Var = VZ |VS Var
eval:: Exp -> Int
eval (MyInt e1) = e1
eval (UnaryNeg e1) = - (eval e1)
eval (Mult e1 e2) = eval e1 * eval e2
eval (Add e1 e2) = eval e1 + eval e2
eval0:: Exp -> Bool
eval0 (B e1) = e1
eval0 (Not e1) = not (eval0 e1)
eval0 (And e1 e2) = (eval0 e1) && (eval0 e2)
eval0 (Or e1 e2) = (eval0 e1) || (eval0 e2)
eval0 (LEQ e1 e2) = eval e1 <= eval e2
eval0 (LESST e1 e2) = eval e1 < eval e2
eval1:: Exp -> Bool
eval1 (POLYEQ e1 e2) = eval0 e1 == eval0 e2
The standard solution is to have the expression type annotated with what type of result it represents. This will then be a Generalised Algebraic Data Type:
{-# LANGUAGE GADTs #-}
data Exp a where
V :: Var -> Exp a
B :: Bool -> Exp Bool
MyInt :: Int -> Exp Int
And :: Exp Bool -> Exp Bool -> Exp Bool
POLYEQ :: Exp a -> Exp a -> Exp Bool
...
Then you need only one evaluation function, whose result will be whatever the type represented by the expression:
eval :: Exp a -> a
eval (MyInt e1) = e1
eval (B e1) = e1
...
Related
I have added | Lit Int and | Add Exp Exp to the data type as seen below, along with the evaluation. However I get an error "Couldn't match expected type ‘Var’ with actual type ‘Int’".
data Exp = V Var
| B Bool
| L Exp
| A Exp Exp
| Lit Int
| Add Exp Exp
data Var = VZ |VS Var
eval:: Exp -> Var
eval (Lit n) = n
eval (Add e1 e2) = eval e1 + eval e2
How can I add Int and Add to the data type, along with the evaluation, but maintain the following code as is. Is this possible?
data Exp = V Var
| B Bool
| L Exp
| A Exp Exp
data Var = VZ |VS Var
I have added an instance to resolve this, as seen below, but now I have the error "Pattern bindings (except simple variables) not allowed in instance declaration: Add e1 e2 = (Lit e1) + (Lit e2)":
{-# LANGUAGE TypeSynonymInstances, FlexibleInstances #-}
data Exp = V Var
| B Bool
| L Exp
| A Exp Exp
| Lit Int
| Add Exp Exp
data Var = VZ |VS Var
eval:: Exp -> Var
eval (Lit n) = n
eval (Add e1 e2) = eval e1 + eval e2
instance Num Var where
Lit e = e
instance Num Var where
Add e1 e2 = (Lit e1) + (Lit e2)
You seem to have gotten mixed up along the way. I believe Var represents a free variable in De Bruijn notation. You're missing an encoding of values. This could look something vaguely like
data Value
= IntVal !Int
| BoolVal !Bool
| Closure { environment :: [Value]
, expr :: Exp }
Now you'll want eval :: Exp -> Value to evaluate expressions. There's lots of work left.
The If data constructor (If BoolType Expr Expr) should evaluate the Boolean expression (first argument) and return the value of the second argument if it is true or return the third argument. I can construct an evaluation of the expression Expr but I don't understand how to incorporate the nested expression BoolType in order to evaluate an expression. A bit too twisty, I can't get my head around it.
Here's two data types:
data Expr = Val Int
| Add Expr Expr
| Sub Expr Expr
| Mul Expr Expr
| Div Expr Expr
| If BoolType Expr Expr
data BoolType = Lit Bool
| Or BoolType BoolType
| EqualTo Expr Expr
| LessThan Expr Expr
I wrote an expression that evaluates the type:
eval :: BoolType -> Bool
eval (Lit n) = n
eval (Or e1 e2) = eval e1 || eval e2
eval (EqualTo e1 e2) = num e1 == num e2
eval (LessThan e1 e2) = num e1 < num e2
num :: Expr -> Int
num (Val n) = n
num (Add e1 e2) = num e1 + num e2
num (Sub e1 e2) = num e1 - num e2
num (Mul e1 e2) = num e1 * num e2
num (Div e1 e2) = num e1 `div` num e2
It should evaluate out to any normal equation but how do I even incorporate an If boolean data type into the total equation?
The evaluator functions currently have incomplete pattern matches:
*Q55835635> :l 55835635.hs
[1 of 1] Compiling Q55835635 ( 55835635.hs, interpreted )
55835635.hs:22:1: warning: [-Wincomplete-patterns]
Pattern match(es) are non-exhaustive
In an equation for `num': Patterns not matched: (If _ _ _)
|
22 | num (Val n) = n
| ^^^^^^^^^^^^^^^^^^^^^^^^^...
Ok, one module loaded.
Just add the missing pattern to num:
num :: Expr -> Int
num (Val n) = n
num (Add e1 e2) = num e1 + num e2
num (Sub e1 e2) = num e1 - num e2
num (Mul e1 e2) = num e1 * num e2
num (Div e1 e2) = num e1 `div` num e2
num (If b e1 e2) = if (eval b) then num e1 else num e2
You can now evaluate expressions, including those with If expressions:
*Q55835635> num $ Add (Val 40) (If (Lit True) (Val 2) (Val 0))
42
The rest of the functions aren't necessary.
You can proceed with your approach, adding a suitable equation:
value (If cond e1 e2) = ifHelper (eval cond) (value e1) (value e2)
Then, you need to define your helper:
ifHelper :: Bool -> Maybe Int -> Maybe Int -> Maybe Int
ifHelper True m1 m2 = ...
ifHelper False m1 m2 = ...
By the way, it is usually recommended to define the type of any function before starting to write the function itself. This helps both the programmer (who can reason about what type are the arguments) and the compiler (which can produce better error messages if something goes wrong).
Turning on warnings with the -Wall flag is also a good idea, since warnings can spot several potential errors.
There's an exercise on this page of the Haskell Wikibook which gets you to work through a scenario using Maybe and Either (presumably to show that it is quite painful for the use case).
The exercise is:
data Expr = I Int
| B Bool -- boolean constants
| Add Expr Expr
| Mul Expr Expr
| Eq Expr Expr -- equality test
eval :: Expr -> Maybe (Either Int Bool)
-- Your implementation here.
The first lines of the solution are - I think - straightforward:
data Expr = I Int -- integer constants
| B Bool -- boolean constants
| Add Expr Expr -- add two expressions
| Mul Expr Expr -- multiply two expressions
| Eq Expr Expr -- equality test
deriving (Show)
eval :: Expr -> Maybe (Either Int Bool)
eval (I n) = Just $ Left n
eval (B b) = Just $ Right b
eval (Add e1 e2) = ...
eval (Mul e1 e2) = ...
eval (Eq e1 e2) = ...
But I'm not exactly sure how to define the rest. As an example I guess for add I need to unpack the fromLeft, fromJust of each expression, but I'm not sure how to do this properly (with pattern matching?)
Thanks in advance!
Yes, with pattern matching and perhaps even the Maybe monad.
You could implement the eval (Add e1 e2) branch using just pattern matching:
eval (Add e1 e2) = case eval e1 of
Just (Left i1) -> case eval e2 of
Just (Left i2) -> Just (Left (i1 + i2))
_ -> Nothing
_ -> Nothing
Pattern matching on a pair is one good way of reducing the amount of nested case statements:
eval (Add e1 e2) = case (eval e1, eval e2) of
(Just (Left i1), Just (Left i2)) -> Just (Left (i1 + i2))
_ -> Nothing
Or, you could use the Maybe monad as an abstraction over those case statements. It will automatically return Nothing if any of the pattern matching fails in the do block bindings (due to how the Maybe monad implements fail).
eval (Add e1 e2) = do
Left i1 <- eval e1
Left i2 <- eval e2
return (Left (i1 + i2))
eval (Add e1 e2) = ...
You will want to evaluate e1 and e2 then pattern match on those results. There's various ways of doing that. For instance you can use let bindings.
let ev1 = eval e1
ev2 = eval e2
in
And then pattern match using the case construction. Or with no let bindings if you prefer, you can just do
case (eval e1, eval e2) of
and pattern match on that pair.
So let me define a few things:
type Name = String
data Exp = Var Name
| App Exp Exp
| Lam Name Exp
deriving (Eq,Show,Read)
I want to define alpha-equivalence, which is
alpha_eq :: Exp -> Exp -> Bool
-- The terms x and y are not alpha-equivalent, because they are not bound in a lambda abstraction
alpha_eq (Var x) (Var y) = False
alpha_eq (Lam x e1) (Lam y e2) = False
alpha_eq (App e1 e2) (App e3 e4) = False
For example Lam "x" (Var "x") and Lam "y" (Var "y") are both equivalent. However I'm both new and horrible at Haskell. Could someone give a clue of how to implement alpha_eq? One thing I thought about was to use Map Name Int so in this case I would have:
['x' -> 0] ['y' -> 0]
so in this case Map['x'] == Map['y']. But again I'm horrible at Haskell. Could you someone give me a clue how to implement it?
Yes, using a Map a correct idea (though think on what the key and value types should be; with Map Name Int you need two extra arguments instead of one). You need to add it as the argument of a helper function, I won't give the full implementation since you asked for a clue only:
alpha_eq e1 e2 = alpha_eq' e1 e2 env0 where
env0 = ???
alpha_eq' (Var x) (Var y) env = ???
alpha_eq' (Lambda x e1) (Lambda y e2) env = ???
alpha_eq' (App e1 e2) (App e3 e4) env = ???
-- you don't want to throw an error in all other cases
alpha_eq' _ _ env = ???
You could also make separate function subst :: Name -> Exp -> Exp -> Exp. Then, alpha_eq Lam-case becomes
alpha_eq :: Exp -> Exp -> Bool
...
alpha_eq (Lam x xb) (Lam y yb) = xb `alpha_eq` subst y (Var x) yb
...
Excersise: figure out other alpha_eq cases and implementation of subst.
I am trying to write a program in Haskell to compute the denotational semantics of an imperative language program with integer variables, 1-dimensional (integer) arrays and functions. The function I am starting with is of the type:
progsem :: Prog -> State -> State
where State is the following:
type State (Name -> Int, Name -> Int -> Int)
The first part is the value of integer variables, while the second part is the value of an array variable at a particular index.
The program will have the following qualities:
progsem will return the resulting state after the program executes
functions have two parameter lists, one for integer variables, and one for array variables.
functions are call by value result
Here is the abstract syntax for the imperative language:
-- names (variables) are just strings.
type Name = String
-- a program is a series (list) of function definitions, followed by a
-- series of statements.
type Prog = ([FunDefn],[Stmt])
-- a statement is either...
data Stmt =
Assign Name Exp -- ...assignment (<name> := <exp>;)
| If BExp [Stmt] [Stmt] -- ...if-then-else (if <bexp> { <stmt>* } else { <stmt>* })
| While BExp [Stmt] -- ...or a while-loop (while <bexp> { <stmt>*> })
| Let Name Exp [Stmt] -- ...let bindings (let <name>=<exp> in { <stmt> *})
| LetArray Name Exp Exp [Stmt] -- ...let-array binding (letarray <name> [ <exp> ] := <exp> in { <stmt>* })
| Case Exp [(Int,[Stmt])] -- ...case statements
| For Name Exp Exp [Stmt] -- ...for statements
| Return Exp -- ...return statement
| ArrayAssign Name Exp Exp -- ...or array assignment (<name> [ <exp> ] := <exp>;)
deriving Show
-- an integer expression is either...
data Exp =
Add Exp Exp -- ...addition (<exp> + <exp>)
| Sub Exp Exp -- ...subtract (<exp> - <exp>)
| Mul Exp Exp -- ...multiplication (<exp> * <exp>)
| Neg Exp -- ...negation (-<exp>)
| Var Name -- ...a variable (<name>)
| LitInt Int -- ...or an integer literal (e.g. 3, 0, 42, 1999)
| FunCall Name [Exp] [Name] -- ...or a function call (<name> (<exp>,...,<exp> [; <name>,...,<name>]))
| VarArray Name Exp -- ...or an array lookup (<name> [ <exp> ])
deriving Show
-- a boolean expression is either...
data BExp =
IsEq Exp Exp -- ...test for equality (<exp> == <exp>)
| IsNEq Exp Exp -- ...test for inequality (<exp> != <exp>)
| IsGT Exp Exp -- ...test for greater-than (<exp> > <exp>)
| IsLT Exp Exp -- ...test for less-than (<exp> < <exp>)
| IsGTE Exp Exp -- ...test for greater-or-equal (<exp> >= <exp>)
| IsLTE Exp Exp -- ...test for less-or-equal (<exp> <= <exp>)
| And BExp BExp -- ...boolean and (<bexp> && <bexp>)
| Or BExp BExp -- ...boolean or (<bexp> || <bexp>)
| Not BExp -- ...boolean negation (!<bexp>)
| LitBool Bool -- ... or a boolean literal (true or false)
deriving Show
type FunDefn = (Name,[Name],[Name],[Stmt])
Now, I do not have a specific question, but I was wondering if someone could point me in the right direction on how to go about writing the semantics.
In the past I have done something similar for an imperative programming language without arrays and functions. It looked something like this:
expsem :: Exp -> State -> Int
expsem (Add e1 e2) s = (expsem e1 s) + (expsem e2 s)
expsem (Sub e1 e2) s = (expsem e1 s) - (expsem e2 s)
expsem (Mul e1 e2) s = (expsem e1 s) * (expsem e2 s)
expsem (Neg e) s = -(expsem e s)
expsem (Var x) s = s x
expsem (LitInt m) _ = m
boolsem :: BExp -> State -> Bool
boolsem (IsEq e1 e2) s = expsem e1 s == expsem e2 s
boolsem (IsNEq e1 e2) s= not(expsem e1 s == expsem e2 s)
boolsem (IsGT e1 e2) s= expsem e1 s > expsem e2 s
boolsem (IsGTE e1 e2) s= expsem e1 s >= expsem e2 s
boolsem (IsLT e1 e2) s= expsem e1 s < expsem e2 s
boolsem (IsLTE e1 e2) s= expsem e1 s <= expsem e2 s
boolsem (And b1 b2) s= boolsem b1 s && boolsem b2 s
boolsem (Or b1 b2) s= boolsem b1 s || boolsem b2 s
boolsem (Not b) s= not (boolsem b s)
boolsem (LitBool x) _= x
stmtsem :: Stmt -> State -> State
stmtsem (Assign x e) s = (\z -> if (z==x) then expsem e s else s z)
stmtsem (If b pt pf) s = if (boolsem b s) then (progsem pt s) else (progsem pf s)
stmtsem (While b p) s = if (boolsem b s) then stmtsem (While b p) (progsem p s) else s
stmtsem (Let x e p) s = s1 where
initvalx = s x
letvalx = expsem e s
snew = progsem p (tweak s x letvalx)
s1 z = if (z == x) then initvalx else snew z
tweak :: State->Name->Int->State
tweak s vr n = \y -> if vr == y then n else s y
progsem :: Prog -> State -> State
progsem smlist s0 = (foldl (\s -> \sm -> (stmtsem sm s)) (s0) ) smlist
s :: State
s "x" = 10
Where states were of the type
type State= Name -> Int
Like I said, I do not need an in depth answer, but any help/hints/pointers would be much appreciated.
I'll deviate a bit from your given code and indicate how you might start to write a monadic interpreter which encodes the evaluation semantics for an toy imperative language, much like the one you specified. You'll need a frontend AST like you have:
import Control.Monad.State
import qualified Data.Map as Map
data Expr = Var Var
| App Expr Expr
| Fun Var Expr
| Lit Ground
| If Expr Expr Expr
-- Fill in the rest
deriving (Show, Eq, Ord)
data Ground = LInt Integer
| LBool Bool
deriving (Show, Eq, Ord)
We will evaluate via a function eval into concrete Value types.
data Value = VInt Integer
| VBool Bool
| VUnit
| VAddress Int
| VClosure String Expr TermEnv
type TermEnv = Map.Map String Value
type Memory = [Value]
type Interpreter t = State Memory t
eval :: TermEnv -> Expr -> State Memory Value
eval _ (Lit (LInt k)) = return $ VInt k
eval _ (Lit (LBool k)) = return $ VBool k
eval env (Fun x body) = return (VClosure x body env)
eval env (App fun arg) = do
VClosure x body clo <- eval env fun
res <- eval env fun
args <- eval env arg
let nenv = Map.insert x args clo
eval nenv body
eval env (Var x) = case (Map.lookup x env) of
Just v -> return v
Nothing -> error "prevent this statically!"
eval env (If cond tr fl) = do
VBool br <- eval env cond
if br == True
then eval env tr
else eval env fl
-- Finish with the rest of your syntax.
programs will return the resulting state after the program executes
To run the interpreter we need to feed it two values: the binding environment of variables and the values on the heap. This will return a tuple of the resulting value and the memory state which you can then feed back to the function itself if building a REPL-like loop.
runInterpreter :: TermEnv -> Memory -> Expr -> (Value, [Value])
runInterpreter env mem x = runState (eval env x) mem
initMem = []
initTermEnv = Map.empty
Since you're writing an imperative language likely you'll want to add state and references, so you can create new AST nodes working allocating and mutating Refs. Left for you to do is to add the logic for allocating an Array as a sequence of Refs and using pointer arithmetic when indexing and assigning into it.
data Expr = -- Same as above
| Ref Expr
| Access Expr
| Assign Expr Expr
eval :: TermEnv -> Expr -> State Memory Value
...
eval env (Ref e) = do
ev <- eval env e
alloc ev
eval env (Access a) = do
VAddress i <- eval env a
readAddr i
eval env (Assign a e) = do
VAddress i <- eval env a
ev <- eval env e
updateAddr ev i
With this we'll need some helper functions for dealing with the values on the heap which are just thin wrappers around the State monad functions.
access :: Int -> Memory -> Value
access i mem = mem !! i
update :: Int -> Value -> Memory -> Memory
update addr val mem = a ++ [val] ++ b
where
(a, _:b) = splitAt addr mem
alloc :: Value -> Interpreter Value
alloc val = do
mem <- get
put $ mem ++ [val]
return $ VAddress (length mem)
readAddr :: Int -> Interpreter Value
readAddr i = do
mem <- get
return $ access i mem
updateAddr :: Value -> Int -> Interpreter Value
updateAddr val i = do
mem <- get
put $ update i val mem
return VUnit
Hope that helps get you started.